This document discusses inflation, appreciation, and depreciation. It provides examples of how to calculate future values when given an original price, inflation/appreciation rate, and number of years. A 3-step process is outlined for calculating future values: 1) identify the variables (original price, rate, years), 2) write the compound interest formula, 3) substitute the values into the formula. The document also describes how a graphics calculator can be used to model appreciation over time by creating a table and graph of the changing value.
Community development officer perfomance appraisal 2tonychoper6204
This document contains materials for evaluating the job performance of a community development officer, including:
1) A three-page performance evaluation form addressing an individual's performance in areas like administration, communication, decision-making, customer service, and suggestions for improvement.
2) A list of online resources for performance appraisal materials like sample phrases, forms, and guides.
3) A section with example positive and negative performance review phrases for a community development officer's attitude, creativity, and decision-making.
The document is a chapter about simple and compound interest from a Year 12 math textbook. It includes:
- An introduction explaining the concepts of simple and compound interest and how interest amounts are affected by factors like interest rate, time, and principal invested.
- Examples calculating simple interest earned on investments over different time periods and interest rates.
- An explanation of the simple interest formula and how to calculate total amount from principal and interest.
- A worked example using the simple interest formula to calculate interest and total amount for two investments.
- An explanation of how to calculate simple interest using a graphics calculator.
- Another worked example calculating the semi-annual interest payments and total interest over 5 years
This document provides answers to exercises related to simple and compound interest, appreciation and depreciation. For exercise 1C on graphing simple interest functions, the summary provides graphs of interest versus years for different interest rates. Exercise 2A involves inflation and appreciation word problems, with answers including equations to model changes in value over time. Exercise 2B models depreciation of assets, providing equations and calculating values after a given number of years.
This document provides an overview of chapter 3 from a maths textbook on consumer credit and investments. The chapter covers various topics related to managing money through loans, mortgages, bonds, bank accounts and investing. It includes worked examples on calculating flat rate interest, loan repayments, deposits and total costs for purchases made through financing options. Spreadsheets are also described for calculating loan payments and interest rates given different inputs.
Captain Quinn of the sinking yacht Kestrel needs to determine his position to direct rescuers to his location. The chapter will cover navigation techniques to accurately describe positions on Earth using lines of latitude and longitude. It will also discuss compass use, fixing positions on charts, and how lighthouses and GPS can assist navigators. The ability to rapidly communicate one's position in an emergency could mean the difference between life and death.
1) The document discusses different methods of calculating employee pay, including salaries, wages, commissions, overtime, and allowances. It provides examples of calculating annual salaries from weekly or monthly amounts, and hourly rates from annual salaries.
2) Wages are calculated by multiplying an employee's hourly rate by the number of hours worked. Hourly rates can be determined by dividing a weekly wage amount by the hours worked. Allowances may be added for unfavorable working conditions.
3) The document contains worked examples of calculating salaries, wages, hourly rates, and totals that include allowances for different occupations and scenarios. It also provides practice problems for readers to work through.
This document provides information about probability and the binomial distribution from a math textbook. It includes:
- An introduction discussing using probabilities to analyze outcomes that depend on multiple factors.
- Sections on compound events with independent events, explaining the multiplication rule for calculating probabilities of multiple independent events occurring.
- Examples of using tree diagrams to visually represent and calculate probabilities of compound independent events.
- Discussion of complementary events and finding the probability of an event not occurring.
- Worked examples applying the concepts to probability word problems involving events like coin tosses and weather.
Percentage and its applications /COMMERCIAL MATHEMATICSindianeducation
The document discusses percentage and its applications. It begins by providing examples of percentages used in everyday contexts like sales, voter turnout, exam scores, and interest rates. It then defines percentage as a fraction with a denominator of 100 and discusses converting between percentages, fractions, and decimals. The objectives are to illustrate percentage concepts and solve problems involving profit/loss, discounts, simple/compound interest, rates of growth, and more. Background knowledge expected includes the four basic operations on whole numbers, fractions, and decimals. The document then explains calculating a specified percentage of a given number by converting the percentage to a fraction or decimal and multiplying. It provides several examples like finding percentage of marks scored, expenditures, increases/reductions.
Community development officer perfomance appraisal 2tonychoper6204
This document contains materials for evaluating the job performance of a community development officer, including:
1) A three-page performance evaluation form addressing an individual's performance in areas like administration, communication, decision-making, customer service, and suggestions for improvement.
2) A list of online resources for performance appraisal materials like sample phrases, forms, and guides.
3) A section with example positive and negative performance review phrases for a community development officer's attitude, creativity, and decision-making.
The document is a chapter about simple and compound interest from a Year 12 math textbook. It includes:
- An introduction explaining the concepts of simple and compound interest and how interest amounts are affected by factors like interest rate, time, and principal invested.
- Examples calculating simple interest earned on investments over different time periods and interest rates.
- An explanation of the simple interest formula and how to calculate total amount from principal and interest.
- A worked example using the simple interest formula to calculate interest and total amount for two investments.
- An explanation of how to calculate simple interest using a graphics calculator.
- Another worked example calculating the semi-annual interest payments and total interest over 5 years
This document provides answers to exercises related to simple and compound interest, appreciation and depreciation. For exercise 1C on graphing simple interest functions, the summary provides graphs of interest versus years for different interest rates. Exercise 2A involves inflation and appreciation word problems, with answers including equations to model changes in value over time. Exercise 2B models depreciation of assets, providing equations and calculating values after a given number of years.
This document provides an overview of chapter 3 from a maths textbook on consumer credit and investments. The chapter covers various topics related to managing money through loans, mortgages, bonds, bank accounts and investing. It includes worked examples on calculating flat rate interest, loan repayments, deposits and total costs for purchases made through financing options. Spreadsheets are also described for calculating loan payments and interest rates given different inputs.
Captain Quinn of the sinking yacht Kestrel needs to determine his position to direct rescuers to his location. The chapter will cover navigation techniques to accurately describe positions on Earth using lines of latitude and longitude. It will also discuss compass use, fixing positions on charts, and how lighthouses and GPS can assist navigators. The ability to rapidly communicate one's position in an emergency could mean the difference between life and death.
1) The document discusses different methods of calculating employee pay, including salaries, wages, commissions, overtime, and allowances. It provides examples of calculating annual salaries from weekly or monthly amounts, and hourly rates from annual salaries.
2) Wages are calculated by multiplying an employee's hourly rate by the number of hours worked. Hourly rates can be determined by dividing a weekly wage amount by the hours worked. Allowances may be added for unfavorable working conditions.
3) The document contains worked examples of calculating salaries, wages, hourly rates, and totals that include allowances for different occupations and scenarios. It also provides practice problems for readers to work through.
This document provides information about probability and the binomial distribution from a math textbook. It includes:
- An introduction discussing using probabilities to analyze outcomes that depend on multiple factors.
- Sections on compound events with independent events, explaining the multiplication rule for calculating probabilities of multiple independent events occurring.
- Examples of using tree diagrams to visually represent and calculate probabilities of compound independent events.
- Discussion of complementary events and finding the probability of an event not occurring.
- Worked examples applying the concepts to probability word problems involving events like coin tosses and weather.
Percentage and its applications /COMMERCIAL MATHEMATICSindianeducation
The document discusses percentage and its applications. It begins by providing examples of percentages used in everyday contexts like sales, voter turnout, exam scores, and interest rates. It then defines percentage as a fraction with a denominator of 100 and discusses converting between percentages, fractions, and decimals. The objectives are to illustrate percentage concepts and solve problems involving profit/loss, discounts, simple/compound interest, rates of growth, and more. Background knowledge expected includes the four basic operations on whole numbers, fractions, and decimals. The document then explains calculating a specified percentage of a given number by converting the percentage to a fraction or decimal and multiplying. It provides several examples like finding percentage of marks scored, expenditures, increases/reductions.
This document discusses taxation and tax deductions in Australia. It provides information on:
1. Income tax is deducted from Karla's pay by her employer and sent to the government. Tax deductions are allowed for work-related expenses.
2. Taxable income is calculated as gross income from all sources minus any allowable tax deductions. Deductions include costs incurred earning income, work-related travel, and depreciation of work equipment.
3. Examples are provided to demonstrate how to calculate taxable income by determining gross income from multiple sources, allowable deductions, and subtracting deductions from gross income.
1) Archaeologist Archie uncovered skulls in Egypt and wanted to date them by comparing skull measurements to recorded data from two time periods - 4000 BC and AD 150.
2) The document discusses statistical techniques like mean, median, and mode that can be used to summarize the skull measurement data sets in order to compare them to Archie's findings and potentially date his skulls.
3) It provides examples of calculating the mean using manual calculations and a graphing calculator and discusses interpreting the mean and how it is impacted by outliers.
The document discusses finishing touches for a home, including painting walls and ceilings, wallpapering, and calculating paint and wallpaper needs. It provides examples of calculating the area to be painted on walls and ceilings, determining the quantity of paint or wallpaper rolls required according to coverage rates, and estimating costs. Specific examples include calculating paint needs for ceiling and bedroom wall painting jobs and estimating the number of wallpaper rolls required.
This document is a glossary containing definitions of mathematical and statistical terms. It provides concise explanations of over 150 terms ranging from simple concepts like "adjacent" and "allowance" to more complex ideas like "box-and-whisker plot", "causality", and "frequency histogram". The glossary acts as a reference for understanding key terms across various topics in mathematics and statistics.
This document contains a series of skillsheets covering various mathematical topics related to percentages, ratios, trigonometry, geometry, and other concepts. The skillsheets provide examples and problems for students to practice calculating and applying different formulas. They cover areas such as finding percentages of quantities, increasing/decreasing values by percentages, calculating ratios, using trigonometric functions, measuring areas and volumes, converting between units, and determining averages, medians, and rates of change. The skillsheets are accompanied by answer keys to allow students to check their work.
To find the selling price of softball bats:
- The store buys each bat for $35
- The mark-up on each bat is 40% of the purchase price
- To calculate the mark-up, express 40% as a decimal (0.4) and multiply it by the purchase price of $35
- The mark-up is 0.4 × $35 = $14
- To get the selling price, add the purchase price ($35) to the mark-up ($14)
- Therefore, the selling price of each softball bat is $35 + $14 = $49
This document provides an overview of different methods for collecting and organizing statistical data, including observation, surveys, and experiments. It discusses categorical versus numerical data and discrete versus continuous numerical data. The document also provides examples of collecting data through observation of bottled water prices and designing questionnaires for surveys.
The document provides information about chapter 4 of a math textbook, which covers populations, samples, statistics, and probability. It includes subsections on populations and samples, samples and sampling methods, and random sampling. Key points include:
- A census collects data from the entire population, while a survey collects data from a sample of the population.
- When selecting a sample, it is important for the sample to be representative of the population.
- Random sampling aims to give each member of the population an equal chance of being selected for the sample. Random number tables and calculators can be used to randomly select sample numbers.
This chapter discusses topics related to managing money, including discounts, profit and loss, and budgeting. It provides examples and explanations of calculating discounts as percentages of the original price and finding sale prices. It also demonstrates how to calculate profit and loss as percentages based on cost price. The chapter aims to help readers better understand financial concepts like these so they can make wise use of their money.
This document discusses geometry concepts related to length, area, and volume. It includes examples of calculating perimeters of different shapes and composite figures. Conversions between different units of length are also covered. The document contains instructions for an origami investigation involving folding a square piece of paper into geometric shapes.
This document discusses probability and chance. It introduces key concepts like sample space, tree diagrams, equally likely outcomes, and the fundamental counting principle. It uses the example of a lottery draw to explain concepts like single event probability, writing probabilities as decimals and percentages, and complementary events. The document seeks to help the reader understand probability through investigating a simplified lottery system.
1. The document discusses Pythagoras' theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
2. It provides examples of right-angled triangles and has students measure the sides to see if they follow Pythagoras' theorem.
3. The history section discusses the life and contributions of Pythagoras, the ancient Greek mathematician who is credited with discovering and naming the theorem.
This document provides information about Pythagoras' theorem and how it can be used to solve problems involving right triangles. It begins with a brief history of Pythagoras and the times he lived in. It then explains Pythagoras' theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Examples are provided to demonstrate how to use the theorem to calculate unknown sides of right triangles. The document also notes that rearranging the theorem allows calculating the lengths of the shorter sides given the hypotenuse.
This document contains sample exercises and answers for calculating earnings such as salary, wages, commission, overtime pay, and deductions from pay. It includes multiple choice and short answer questions on topics like calculating taxable income, Medicare levy, GST, and profit/loss calculations involving discounts. The exercises provide practice calculating common payroll and financial scenarios like hourly wages, piecework pay, salary amounts, tax deductions, and profit margins.
This document discusses concepts related to geography and navigation on Earth. It begins by explaining how early explorers navigated using the sun, moon, and stars before Greenwich, England was established as the Prime Meridian. It then explores questions about Greenwich's importance in determining time and position on Earth. The document defines key terms like latitude, longitude, and time zones. It discusses how distances on Earth are measured along arcs rather than straight lines due to the spherical shape. Overall, the document provides background on historical and modern systems for locating positions and measuring distances on the surface of the Earth.
The document provides information about scale drawings and building plans used in the construction of homes. It discusses scale drawings and scale factors, explaining how to convert between plan measurements and actual field measurements. It then describes the different types of plans used in construction, including survey plans, site plans, floor plans, and elevations. Survey plans show the boundaries of a property. Site plans show where a structure will be situated on a lot. Floor plans provide dimensions and details of rooms, doors, windows, and wall thicknesses.
This document provides information about networks and minimal spanning trees. It begins with an introduction to networks, noting they are made up of nodes and arcs. Several worked examples are provided that use networks to model situations involving shortest paths between locations based on distance or time. The document then discusses minimal spanning trees and provides an algorithm to identify the minimum total length of connections needed to link all nodes in a network. A worked example applies this algorithm to determine the minimum cabling length needed to connect buildings on a farm to a transformer.
This document provides an overview of critical path analysis and queuing. It begins with an example of Cameron planning tasks to complete in the morning. A network diagram and activity chart are used to represent Cameron's tasks of downloading email, reading email, and eating breakfast. Forward scanning is then demonstrated to determine the earliest completion time of 6 minutes for these initial tasks. The document extends the example to all of Cameron's morning tasks, identifying a critical path of A-D-E-C-F with an earliest completion time of 20 minutes. It defines float time and latest start time, calculating these for activities B and G in the example network.
1. The document discusses using scatterplots to analyze bivariate data and examine relationships between two variables. It provides an example of data collected on depth of snow and number of skiers at a ski resort over 12 weekends.
2. A scatterplot is created with depth of snow on the x-axis and number of skiers on the y-axis. This shows a general upward trend, indicating higher skier numbers with more snow.
3. The document discusses key aspects of scatterplots, including identifying independent and dependent variables and exploring linear and non-linear relationships between variable pairs. Examples are provided to illustrate these concepts.
This document provides answers to end of chapter questions from chapters 1-3 of a personal finance textbook. The answers cover topics such as calculating rates of return and interest, determining financial ratios like debt ratios and current ratios, preparing personal budgets, and calculating taxable income and tax refund amounts. Formulas and tables from the textbook are referenced in some of the calculations.
This document provides an overview of key concepts from Chapter 1 of a linear functions textbook, including:
- Solving linear equations and using data to create scatterplots and graph lines
- Finding equations of lines from their graphs or intercepts
- Using linear models to represent real-world situations like business costs and revenues
- Identifying the slope, intercepts, domain and range of linear equations and determining if sets of points represent functions
The chapter content is explained through examples like modeling the costs and profits of a golf cart refurbishing business.
ASSESSMENT CASE PAPER ANALYSIS / TUTORIALOUTLET DOT COMjorge0048
A report broken down into the following sections:
Summary results and recommendations—up front, concise, and to the point.
Answers to the 6 questions asked—devote a paragraph to each, with individual headings
This document discusses taxation and tax deductions in Australia. It provides information on:
1. Income tax is deducted from Karla's pay by her employer and sent to the government. Tax deductions are allowed for work-related expenses.
2. Taxable income is calculated as gross income from all sources minus any allowable tax deductions. Deductions include costs incurred earning income, work-related travel, and depreciation of work equipment.
3. Examples are provided to demonstrate how to calculate taxable income by determining gross income from multiple sources, allowable deductions, and subtracting deductions from gross income.
1) Archaeologist Archie uncovered skulls in Egypt and wanted to date them by comparing skull measurements to recorded data from two time periods - 4000 BC and AD 150.
2) The document discusses statistical techniques like mean, median, and mode that can be used to summarize the skull measurement data sets in order to compare them to Archie's findings and potentially date his skulls.
3) It provides examples of calculating the mean using manual calculations and a graphing calculator and discusses interpreting the mean and how it is impacted by outliers.
The document discusses finishing touches for a home, including painting walls and ceilings, wallpapering, and calculating paint and wallpaper needs. It provides examples of calculating the area to be painted on walls and ceilings, determining the quantity of paint or wallpaper rolls required according to coverage rates, and estimating costs. Specific examples include calculating paint needs for ceiling and bedroom wall painting jobs and estimating the number of wallpaper rolls required.
This document is a glossary containing definitions of mathematical and statistical terms. It provides concise explanations of over 150 terms ranging from simple concepts like "adjacent" and "allowance" to more complex ideas like "box-and-whisker plot", "causality", and "frequency histogram". The glossary acts as a reference for understanding key terms across various topics in mathematics and statistics.
This document contains a series of skillsheets covering various mathematical topics related to percentages, ratios, trigonometry, geometry, and other concepts. The skillsheets provide examples and problems for students to practice calculating and applying different formulas. They cover areas such as finding percentages of quantities, increasing/decreasing values by percentages, calculating ratios, using trigonometric functions, measuring areas and volumes, converting between units, and determining averages, medians, and rates of change. The skillsheets are accompanied by answer keys to allow students to check their work.
To find the selling price of softball bats:
- The store buys each bat for $35
- The mark-up on each bat is 40% of the purchase price
- To calculate the mark-up, express 40% as a decimal (0.4) and multiply it by the purchase price of $35
- The mark-up is 0.4 × $35 = $14
- To get the selling price, add the purchase price ($35) to the mark-up ($14)
- Therefore, the selling price of each softball bat is $35 + $14 = $49
This document provides an overview of different methods for collecting and organizing statistical data, including observation, surveys, and experiments. It discusses categorical versus numerical data and discrete versus continuous numerical data. The document also provides examples of collecting data through observation of bottled water prices and designing questionnaires for surveys.
The document provides information about chapter 4 of a math textbook, which covers populations, samples, statistics, and probability. It includes subsections on populations and samples, samples and sampling methods, and random sampling. Key points include:
- A census collects data from the entire population, while a survey collects data from a sample of the population.
- When selecting a sample, it is important for the sample to be representative of the population.
- Random sampling aims to give each member of the population an equal chance of being selected for the sample. Random number tables and calculators can be used to randomly select sample numbers.
This chapter discusses topics related to managing money, including discounts, profit and loss, and budgeting. It provides examples and explanations of calculating discounts as percentages of the original price and finding sale prices. It also demonstrates how to calculate profit and loss as percentages based on cost price. The chapter aims to help readers better understand financial concepts like these so they can make wise use of their money.
This document discusses geometry concepts related to length, area, and volume. It includes examples of calculating perimeters of different shapes and composite figures. Conversions between different units of length are also covered. The document contains instructions for an origami investigation involving folding a square piece of paper into geometric shapes.
This document discusses probability and chance. It introduces key concepts like sample space, tree diagrams, equally likely outcomes, and the fundamental counting principle. It uses the example of a lottery draw to explain concepts like single event probability, writing probabilities as decimals and percentages, and complementary events. The document seeks to help the reader understand probability through investigating a simplified lottery system.
1. The document discusses Pythagoras' theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
2. It provides examples of right-angled triangles and has students measure the sides to see if they follow Pythagoras' theorem.
3. The history section discusses the life and contributions of Pythagoras, the ancient Greek mathematician who is credited with discovering and naming the theorem.
This document provides information about Pythagoras' theorem and how it can be used to solve problems involving right triangles. It begins with a brief history of Pythagoras and the times he lived in. It then explains Pythagoras' theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Examples are provided to demonstrate how to use the theorem to calculate unknown sides of right triangles. The document also notes that rearranging the theorem allows calculating the lengths of the shorter sides given the hypotenuse.
This document contains sample exercises and answers for calculating earnings such as salary, wages, commission, overtime pay, and deductions from pay. It includes multiple choice and short answer questions on topics like calculating taxable income, Medicare levy, GST, and profit/loss calculations involving discounts. The exercises provide practice calculating common payroll and financial scenarios like hourly wages, piecework pay, salary amounts, tax deductions, and profit margins.
This document discusses concepts related to geography and navigation on Earth. It begins by explaining how early explorers navigated using the sun, moon, and stars before Greenwich, England was established as the Prime Meridian. It then explores questions about Greenwich's importance in determining time and position on Earth. The document defines key terms like latitude, longitude, and time zones. It discusses how distances on Earth are measured along arcs rather than straight lines due to the spherical shape. Overall, the document provides background on historical and modern systems for locating positions and measuring distances on the surface of the Earth.
The document provides information about scale drawings and building plans used in the construction of homes. It discusses scale drawings and scale factors, explaining how to convert between plan measurements and actual field measurements. It then describes the different types of plans used in construction, including survey plans, site plans, floor plans, and elevations. Survey plans show the boundaries of a property. Site plans show where a structure will be situated on a lot. Floor plans provide dimensions and details of rooms, doors, windows, and wall thicknesses.
This document provides information about networks and minimal spanning trees. It begins with an introduction to networks, noting they are made up of nodes and arcs. Several worked examples are provided that use networks to model situations involving shortest paths between locations based on distance or time. The document then discusses minimal spanning trees and provides an algorithm to identify the minimum total length of connections needed to link all nodes in a network. A worked example applies this algorithm to determine the minimum cabling length needed to connect buildings on a farm to a transformer.
This document provides an overview of critical path analysis and queuing. It begins with an example of Cameron planning tasks to complete in the morning. A network diagram and activity chart are used to represent Cameron's tasks of downloading email, reading email, and eating breakfast. Forward scanning is then demonstrated to determine the earliest completion time of 6 minutes for these initial tasks. The document extends the example to all of Cameron's morning tasks, identifying a critical path of A-D-E-C-F with an earliest completion time of 20 minutes. It defines float time and latest start time, calculating these for activities B and G in the example network.
1. The document discusses using scatterplots to analyze bivariate data and examine relationships between two variables. It provides an example of data collected on depth of snow and number of skiers at a ski resort over 12 weekends.
2. A scatterplot is created with depth of snow on the x-axis and number of skiers on the y-axis. This shows a general upward trend, indicating higher skier numbers with more snow.
3. The document discusses key aspects of scatterplots, including identifying independent and dependent variables and exploring linear and non-linear relationships between variable pairs. Examples are provided to illustrate these concepts.
This document provides answers to end of chapter questions from chapters 1-3 of a personal finance textbook. The answers cover topics such as calculating rates of return and interest, determining financial ratios like debt ratios and current ratios, preparing personal budgets, and calculating taxable income and tax refund amounts. Formulas and tables from the textbook are referenced in some of the calculations.
This document provides an overview of key concepts from Chapter 1 of a linear functions textbook, including:
- Solving linear equations and using data to create scatterplots and graph lines
- Finding equations of lines from their graphs or intercepts
- Using linear models to represent real-world situations like business costs and revenues
- Identifying the slope, intercepts, domain and range of linear equations and determining if sets of points represent functions
The chapter content is explained through examples like modeling the costs and profits of a golf cart refurbishing business.
ASSESSMENT CASE PAPER ANALYSIS / TUTORIALOUTLET DOT COMjorge0048
A report broken down into the following sections:
Summary results and recommendations—up front, concise, and to the point.
Answers to the 6 questions asked—devote a paragraph to each, with individual headings
This document provides an overview of an analytical methods course for economics and finance. It introduces the course staff and coordinators. It describes how econometrics can be used to answer quantitative questions about economics and business. It also discusses different types of economic data and some basic mathematical and statistical concepts needed for the course, including summation, probability, and random variables. An important note reminds students about class attendance, staff consultation hours, accessing learning materials, and preparing for an upcoming online quiz.
- The document provides guidance and examples for journalists on properly understanding and reporting numbers, statistics, and financial information. It discusses key concepts like percentages, averages, margins, ratios, and taxes.
- Examples are given for calculating percentages, averages, margins, profits, tax amounts, current ratios, and other financial metrics.
- Tips are provided for limiting numbers in stories, interpreting results, and determining when to use mean, median or mode in reporting data.
This document discusses key concepts related to the time value of money, including:
- Calculating the future and present value of a single amount and an annuity using compound interest formulas.
- Examples of using financial calculators and spreadsheets to solve time value of money problems.
- Additional topics covered include annuities due, perpetuities, non-annual periods, and effective annual rates. Students are encouraged to use financial calculators to simplify solving discounted cash flow problems.
This document discusses key concepts related to the time value of money, including:
- Calculating the future and present value of a single amount and an annuity using compound interest formulas.
- Examples of using financial calculators and spreadsheets to solve time value of money problems.
- Additional topics covered include annuities due, perpetuities, non-annual periods, and effective annual rates. Students are encouraged to use financial calculators to simplify solving discounted cash flow problems.
The document discusses key concepts related to the time value of money, including formulas for calculating the future value and present value of single amounts and annuities. It provides examples of using these formulas to solve for unknown values like interest rates, time periods, or cash flow amounts. The document also covers topics like perpetuities, non-annual interest compounding, and effective annual rates.
- Regression models were estimated to examine the determinants of CEO salaries, wages, and exchange rates.
- Explanatory variables like firm performance, inflation rates, and unemployment were found to significantly impact the dependent variables in expected directions based on economic theory.
- While most coefficients were statistically significant, one variable in the wage model was found to be insignificant and could potentially be omitted to improve model specification.
- The interpretations focused on the estimated partial effects of the regressors, both in levels and rates of change, depending on the variable transformations used. Adjusted R-squared values indicated the models explained a substantial portion of the variation in the dependent variables.
The Time Value of Money Future Value and Present Value .docxchristalgrieg
The Time Value of
Money: Future Value
and Present Value
Computations
"If I deposit $10,000 today, how much will I have for a down payment on a house in five years?"
"Will $2,000 saved a year give me enough money when I retire?"
"How much must I save today to have enough for my children's post-secondary education?"
As introduced in Chapter 1 and used to measure financial opportunity costs in other chapters,
the time value of money, more commonly referred to as interest, is the cost of money that is bor-
rowed or lent. Interest can be compared to rent, the cost of using an apartment or other item.
The time value of money is based on the fact that a dollar received today is worth more than a
dollar that will be received one year from today because the dollar received today can be saved
or invested and will be worth more than a dollar a year from today Similarly, a dollar that will
be received one year from today is currently worth less than a dollar today.
The time value of money has two major components: future value and present value. Future
value computations, which are also referred to as compounding, yield the amount to which a
current sum will increase based on a certain interest rate and period of time, Present value,
which is calculated through a process called discounting, is the current value of a future sum
based on a certain interest rate and period of time.
In future value problems, you are given an amount to save or invest and you calculate the
amount that will be available at some future date. With present value problems, you are given the
amount that will be available at some future date and you calculate the current value of that amount
Both future value and present value computations are based on basic interest rate calculations.
FINANCIAL CALCULATORS
Currently, financial calculators, with time value of money functions built in, are widely used to
calculate future value, present values, and annuities, For the following examples, we will use the
Texas Instruments BA II Plus financial calculator, which is recommended by the Canadian
Institute of Financial Planning.
When using the BA II Plus calculator to solve time value of money problems, you will be
working with the TVM keys that include:
CPT — Compute key used to initiate financial calculations once all values are inputted
— Number of periods
— Interest rate per period
— Present value
PMT — Amount of payment, used only for annuities
— Future value
Enter values for PV, PMT, and FV as negative if they represent cash outflows (e.g., investing
a sum of money) or as positive if they represent cash inflows (e.g., receiving the proceeds of an
investment). To convert a positive number to a negative number, enter the number and then
press the +/— key.
37
Part 1 PLANNING YOUR PERSONAL FINANCES
The examples that are shown in this chapter assume that interest is compounded
annually and that there is only one cash flow per p ...
The document provides sample problems and discussion questions for each week of an economics course, including problems related to production functions, demand and elasticity, forecasting, and the global economy. Students are asked to analyze scenarios and data, calculate economic measures, and discuss how managers could apply the economic concepts when making business decisions to maximize profitability.
This document provides an overview of the course materials and assignments for ECO 550 at Strayer University. It includes sample problems from chapters 1-3 and 5-6 that focus on topics like the principal-agent problem, elasticity, demand and supply analysis, and forecasting models. Students are asked to complete problems, discussions, and a quiz on these economics fundamentals each week. The course uses managerial decision-making scenarios to explore how businesses use economic analysis for issues like pricing, costs, demand estimation and forecasting.
This document discusses the time value of money and methods for calculating present value. It begins by distinguishing between simple and compound interest, then identifies the key variables used in present value calculations as the future amount, interest rate, and time period. It provides formulas and examples for calculating the present value of a single amount and an annuity. The document demonstrates how to use present value tables to determine the current worth of future sums of money.
This document provides an overview of how to value bonds and stocks. It discusses:
- Valuing bonds by estimating future cash flows and discounting them at the appropriate rate based on risk.
- How bond prices move inversely to market interest rates.
- Valuing stocks using the dividend discount model, accounting for growth rates and discount rates. Growth can be zero, constant, or differential.
- Estimating growth rates based on retention ratio and return on retained earnings. Estimating discount rates based on dividend yield and growth.
- Additional valuation methods including the net present value of growth opportunities model.
- Common metrics used to analyze stocks like price-earnings ratios and how they are calculated and interpreted.
This document summarizes methods for valuing bonds and stocks. It discusses how to value bonds based on their coupon payments and maturity date by discounting expected cash flows. It also explains models for valuing stocks based on expected future dividends, including the dividend discount model for zero, constant, and differential growth. Key parameters like growth rates and discount rates are discussed. Methods include using the dividend discount model or separating value into a "cash cow" portion and growth opportunities.
Assignment 3 Assignment 3 is due after you complete Lessons 9 to.docxsherni1
Assignment 3
Assignment 3 is due after you complete Lessons 9 to 11. It is worth 20% of your final grade.
Prepare your responses to these assignment problems in a word processing file; put financial data in a spreadsheet file. As you complete the assignment problems for each lesson, add your responses to these files.
Do not submit your answers for grading until you have completed all parts of Assignment 3.
Note: In assignments, show all calculations to 4 decimal places.
Lesson 9: Assignment Problems
9.1 The Constant-Growth-Rate Discounted Dividend Model, as described equation 9.5 on page 247, says that:
P0 = D1 / (k – g)
A. rearrange the terms to solve for:
i. g; and
ii. D1.
As an example, to solve for k, we would do the following:
1. Multiply both sides by (k – g) to get: P0 (k – g) = D1
2. Divide both sides by P0 by to get: (k – g) = D1 / P0
3. Add g to both sides: k = D1 / P0 + g
(8 marks)
9.2 Notation: Let
Pn = Price at time n
Dn = Dividend at time n
Yn = Earnings in period n
r = retention ratio = (Yn– Dn) / Yn = 1 – Dn/ Yn = 1 - dividend payout ratio
En = Equity at the end of year n
k = discount rate
g = dividend growth rate = r x ROE
ROE = Yn / En-1 for all n>0.
We will further assume that k and ROE are constant, and that r and g are constant after the first dividend is paid.
A. Using the Discounted Dividend Model, calculate the price P0 if
D1 = 20, k = .15, g = r x ROE = .8 x .15 = .12, and Y1 = 100 per share
B. What, then, will P5 be if:
D6 = 20, k = .15, and g = r x ROE = .8 x .15 = .12?
C. If P5 = your result from part B, and assuming no dividends are paid until D6, what would be P0? P1? P2?
D. Again, assuming the facts from part B, what is the relationship between P2 and P1 (i.e., P2/P1)? Explain why this is the result.
E. If k = ROE, we can show that the price P0 doesn’t depend on r. To see this, let
g = r x ROE, and ROE = Yn / En-1, and
since r = (Yn – Dn) / Yn , then D1 = (1 – r) x Y1 and
P0
=
D1 / (k – g)
P0
=
[(1 – r) x Y1] / (k – g)
P0
=
[(1 – r) x Y1] / (k – g), but, since k = ROE = Y1 / E0
P0
=
[(1 – r) x Y1] / (ROE – r x ROE)
P0
=
[(1 – r) x Y1] / (Y1 / E0 – r x Y1 / E0)
P0
=
[(1 – r) x Y1] / (1 – r) x Y1 / E0), and cancelling (1 – r)
P0
=
Y1 / (Y1/E0) = Y1 x (E0 / Y1) = E0
So, you see that r is not in the final expression for P0, indicating that r (i.e., retention ration or, equivalently, dividend policy) doesn’t matter if k = ROE.
Check that changing r from .8 to .6 does not change your answer in part A of this question by re-calculating your result using r = .6.
(10 marks)
9.3 You are considering an investment in the shares of Kirk's Information Inc. The company is still in its growth phase, so it won’t pay dividends for the next few years. Kirk’s accountant has determined that their first year's earnings per share (EPS) is expected to be $20. The company expects a return on equity (ROE) of 25% in each of the next 5 years but in the six ...
The document discusses unemployment levels in the United States and job creation efforts by President Obama. It then covers topics related to macroeconomics including real and nominal GDP, consumer price indexes, and using CPI to adjust for inflation. Examples are provided to illustrate the importance of considering inflation when evaluating costs over time. The document concludes by mentioning issues related to accurately measuring prices and inflation.
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Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
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Chapter 4
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Chapter 5
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Chapter 6
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1. Maths A Yr 12 - Ch. 02 Page 53 Friday, September 13, 2002 9:01 AM
2
Appreciation
and
depreciation
syllabus reference
eference
Strand
Financial mathematics
Core topic
Managing money 2
In this chapter
chapter
2A Inflation and appreciation
2B Modelling depreciation
2C Straight line depreciation
2D Declining balance or
diminishing value method
of depreciation
2E Depreciation tables
2F Future and present value
of an annuity
2. Maths A Yr 12 - Ch. 02 Page 54 Wednesday, September 11, 2002 3:52 PM
54
M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d
Introduction
With the passing years, it seems that the cost
of goods and services continues to rise. This
appreciation in price is known as inflation.
In Australia, these price movements are
measured by the Consumer Price Index
(CPI). When goods lose value, this is called
depreciation. While some assets such as
houses and land increase in value, others,
such as new cars and computers, continue to
decrease. Throughout this chapter we shall
look at methods of calculating appreciation
and depreciation; how these may be represented graphically; and how they can help us
plan for the future.
1 Calculate the following, giving your answer in decimal form.
10
1
3.5
a 1 + -------b 1 + -------c 1 + -------100
100
200
3.75
d 1 − --------100
4.5
e 1 − -------100
2 Calculate each of the following.
a 10% of $40
b 12.5% of $40
d 4.125% of $10
e 5.5% of $20
f
3
1 − ----------1200
c
f
3.95% of $100
3.625% of $120.50
3 Calculate the following, rounding to the nearest cent.
a Increase $40 by 10%.
b Increase $2.50 by 20%.
c Increase $7.45 by 2.5%.
d Decrease $20 by 10%.
e Decrease $145 by 6.25%.
f Decrease $4000 by 1.5%.
4 a
b
c
d
e
f
Express $20 as a percentage of $50.
Express $350 as a percentage of $400.
Express $6.45 as a percentage of $10.
Express 45c as a percentage of $10.
Express $1.80 as a percentage of $3.50.
Express 12.5c as a percentage of $8.
5 Calculate the following, expressing your answer in decimal form correct to 2 decimal
places.
3 2
a 1 + --------
100
3 2
b 1 – --------
100
4.2 5
d 1 – --------
600
e
10
1 + 3.875
-----------
1200
c
4.2 5
1 + --------
600
f
10
1 – 3.875
-----------
1200
3. Maths A Yr 12 - Ch. 02 Page 55 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
es
in
ion v
t i gat
n inv
io
es
55
Consumer Price Index
The Consumer Price Index (CPI) measures price movements in Australia. Let us
investigate this further to gain an understanding of how this index is calculated.
A ‘basket’ of goods and services, representing a high proportion of household
expenditure is selected. The prices of these goods are recorded each quarter. The
basket on which the CPI is based is divided into eight groups, which are further
divided into subgroups. The groups are: food, clothing, tobacco/alcohol, housing,
health/personal care, household equipment, transportation and recreation/
education. Weights are attached to each of these groups to reflect the importance of
each in relation to the total household expenditure. The following table shows the
weights of the eight groups.
The weights indicate that a
CPI group
Weight
‘typical’ Australian household
(% of total)
spends 19% of its income on food
purchases, 7% on clothing and so
Food
19
on. The CPI is regarded as an
Clothing
7
indication of the cost of living as it
records changes in the level of
Tobacco/alcohol
8.2
retail prices from one period to
another.
Housing
14.1
Let us consider a simplified
Health/personal care
5.6
example showing how this CPI is
calculated and how we are able to
Household equipment
18.3
compare prices between one period
and another.
Transportation
17
Take three items with prices as
Recreation/education
10.8
follows:
• A pair of jeans costing $75
• A hamburger costing $3.90
• A CD costing $25.
Let us say that during the next period of time, the jeans sell for $76, the hamburger
for $4.20 and the CD for $29. This can be summarised in a table.
Period 1
Item
Weight (W)
Jeans
7
Price (P)
$75
Hamburger
19
$ 3.90
CD
10.8
$25
Total
W×P
525
74.1
270
869.1
Period 2
Price (P)
$76
$ 4.20
$29
W×P
532
79.8
313.2
925
The appropriate weight of each item is multiplied by the price of the item to
determine what proportion the household would spend on each item in each period.
The total weighted expenditure is then calculated.
t i gat
4. Maths A Yr 12 - Ch. 02 Page 56 Wednesday, September 11, 2002 3:52 PM
M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d
The first period is regarded as the base and is allocated an index number of 100.
The second period is compared with the first period and is expressed as a
percentage of it. So, period 2 is 925 ÷ 869.1 × 100%; that is, 106.4% compared
with period 1 as 100%. This means that the average price of these items has risen
by 6.4% (the inflation factor) from period 1 to period 2 and the CPI for period 2 is
106.4.
Task 1
1 Take 8 items with which you are quite familiar, one item from each of the CPI
groups (similar to the ones chosen in the previous example — jeans,
hamburger, CD). Slot them into their correct categories and note the weighting
for each.
2 Draw up a table similar to the previous one, entering the item names and
corresponding weightings.
3 Let the period 2 price be the typical prices of the items at this period of time.
Enter these values into the table.
4 Let the period 1 price be the prices of these items one year ago (use your
knowledge of these items to determine an educated realistic estimate). Enter
these values into the table in the period 1 price column.
5 Complete the table as demonstrated above.
6 Determine the CPI for period 2 compared with period 1 as the base period.
7 What is your inflation factor?
Task 2
1 Conduct a search of the World Wide Web using the
words ‘inflation CPI Australia’.
2 Research the history and use of the CPI in Australia.
What items are in the subgroups? You should
discover that separate CPI indices are calculated for
each Australian State. These are then combined to
produce an average CPI for the whole of Australia,
like that shown in the graph below and the table
at right.
CPI — Inflation — Australia
7%
6%
5%
4%
3%
2%
1%
0%
–1%
–2%
GST
Introduced
July 2000
Sep-93
Dec-93
Mar-94
Jun-94
Sep-94
Dec-94
Mar-95
Jun-95
Sep-95
Dec-95
Mar-96
Jun-96
Sep-96
Dec-96
Mar-97
Jun-97
Sep-97
Dec-97
Mar-98
Jun-98
Sep-98
Dec-98
Mar-99
Jun-99
Sep-99
Dec-99
Mar-00
Jun-00
Sep-00
Dec-00
Mar-01
Jun-01
56
3 Write a report of your findings. Remember to
reference the source of your material.
Year
CPI
1988–89
7.3%
1989–90
8.0%
1990–91
5.3%
1991–92
1.9%
1992–93
1.0%
1993–94
1.8%
1994–95
3.2%
1995–96
4.2%
1996–97
1.3%
1997–98
0.0%
1998–99
1.3%
5. Maths A Yr 12 - Ch. 02 Page 57 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
57
Inflation and appreciation
One of the measures of how an economy is performing is the rate of inflation. Inflation
is the rise in prices within an economy and is generally measured as a percentage. In
Australia this percentage is called the Consumer Price Index (CPI). By looking at the
inflation rate we can estimate what the cost of various goods and services will be at
some time in the future.
To estimate the future price of an item one year ahead, we increase the price of an
item by the rate of inflation. The financial functions of graphics calculators can assist
these calculations.
WORKED Example
1
The cost of a new car is $35 000. If the inflation rate is 5% p.a., estimate the price of the
car after one year.
THINK
WRITE
Increase $35 000 by 5%.
Future price = 105% of $35 000
Future price = 105 ÷ 100 × $35 000
Future price = $36 750
When calculating the future cost of an item several years ahead, the method of calculation is the same as for compound interest. This is because we are adding a percentage
of the cost to the cost each year.
R
Remember the compound interest formula is A = P 1 + -----------------
100 × n
n×T
and so in these
examples P is the original price, R is the inflation rate expressed as a % p.a., n is the
number of rests per year and T is the time in years. Generally the inflation is expressed
as a yearly rate, so the value of n is 1.
WORKED Example
2
The cost of a television set is $800. If the average inflation rate is 4% p.a., estimate the cost
of the television after 5 years.
THINK
WRITE
1
Write the values of P, R, T and n.
2
Write down the compound interest
formula.
P = $800, R = 4, T = 5, n = 1
n×T
R
A = P 1 + -----------------
100 × n
Substitute the values of P, R, T and n.
Calculate.
4
A = 800 1 + -----------------
100 × 1
A = $800 × (1.04)5
A = $973.32
3
4
1×5
6. Maths A Yr 12 - Ch. 02 Page 58 Wednesday, September 11, 2002 3:52 PM
58
M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d
A similar calculation can be made to anticipate the future value of collectable items, such
as stamp collections and memorabilia from special occasions. This type of item increases
in value over time if it becomes rare, and rises at a much greater rate than inflation. The
amount by which an item grows in value over time is known as appreciation.
WORKED Example
3
Jenny purchases a rare stamp for $250. It is
anticipated that the value of the stamp will rise
by 20% per year. Calculate the value of the stamp
after 10 years, correct to the nearest $10.
THINK
WRITE
1
Write the values of
P, R, T and n.
2
Write down the
compound interest
formula.
4
in
ion v
t i gat
n inv
io
es
Substitute the values
of P, R, T and n.
Calculate and round
off to the nearest $10.
es
3
P = $250
R = 20
T = 10
n=1
n×T
R
A = P 1 + -----------------
100 × n
20
A = 250 1 + -----------------
100 × 1
1 × 10
A = $250 × (1.2)10
A = $1550
Modelling appreciation with the aid
of a graphics calculator
You have purchased a rare coin that the coin dealer told you should appreciate by
15% each year. You paid $850 for the coin and hope that its value will treble within
the next 10 years. The coin dealer is not sure whether this is the case, so you offer
to produce a graph for him displaying the value of the coin over the next 10 years.
Using a graphics calculator greatly simplifies the calculations and will produce a
graph that can be used to determine the value of the coin at any period of time. The
screens displayed and instructions supplied are those of the TI-83 and Casio CFX9850GB PLUS.
1 Enter the appropriate formula into Y1, with the variable being time (X).
TI: Press Y= then enter the
formula.
Casio: Enter TABLE from MENU
then enter the formula.
t i gat
7. Maths A Yr 12 - Ch. 02 Page 59 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
59
2 Set up the table to start at a minimum of 0 years, increasing in increments of 1
year.
TI: Press 2nd [TBLSET] then enter
the data.
Casio: Press F5 (RANG) then
enter the data.
3 Look at the table produced.
TI: Press 2nd [TABLE].
Press F6 (TABL).
4 Scrolling down the table, notice that the coin has trebled in value by Year 8
($850 × 3 = $2550).
TI: Press the
M
key to navigate.
Casio: Press the
navigate.
M
key to
5 Set the window to cope with the range of x- and y-values.
TI: Press WINDOW then enter the
data.
Casio: Press SHIFT [V-Window]
then enter the data.
6 Graph the function.
TI: Press GRAPH .
Casio: Press MENU , then select the
GRAPH section. Press F6
(DRAW).
8. Maths A Yr 12 - Ch. 02 Page 60 Wednesday, September 11, 2002 3:52 PM
60
M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d
7 Use the TRACE option to locate the point on the curve where the value of the
coin trebles.
TI: Press TRACE then move the
pointer along the curve.
Casio: Press SHIFT (TRACE) then
move the pointer along the curve.
8 Investigate further to determine the time when the coin will be 4 times its
original value.
9 Prepare a report to the coin dealer explaining how the value of the coin
changes over time. Provide a table and a graph to support your figures.
10 The coin dealer asks you if he could use your graph for values other than $850
which was the initial value of the coin. Explain how the graph could be used
to determine the time when any sum of money doubles, trebles, and so on, as
long as the appreciation rate remains at 15% p.a.
remember
remember
1. Inflation is the measure of the rate at which prices increase.
2. The inflation rate is given as a percentage and is called the Consumer Price
Index.
3. To estimate the cost of an item after one year, we increase the price by the
percentage inflation rate.
4. To estimate the cost of an item after several years, we use the compound
interest formula, using the inflation rate as the value of R.
5. Rare items such as collectibles and memorabilia increase in value as time goes
on at a rate that is usually greater than inflation.
2A
2.1
WORKED
Example
1
Inflation and appreciation
1 The cost of a yacht is $20 000. If the inflation rate is 4% p.a., estimate the cost of the
yacht after one year.
2 For each of the following, estimate the cost of the item after one year, with the given
inflation rate.
a A CD player costing $600 with an inflation rate of 3% p.a.
b A toaster costing $45 with inflation at 7% p.a.
c A loaf of bread costing $1.80 with inflation at 6% p.a.
d An airline ticket costing $560 with inflation at 3.5% p.a.
e A washing machine costing $925 with inflation at 0.8% p.a.
9. Maths A Yr 12 - Ch. 02 Page 61 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
61
3 An electric guitar is priced at $850 at the beginning of 2001.
a If the inflation rate is 3.3% p.a., estimate the cost of the guitar at the beginning of
2002.
b The government predicts inflation to fall to 2.7% p.a. in 2002. Estimate the cost of
the guitar at the beginning of 2003.
4 When the Wilson family go shopping, the weekly basket of groceries costs $112.50.
The inflation rate is predicted to be 4.8% p.a. for the next year. How much should the
Wilson’s budget per week be for groceries for the next year?
WORKED
Example
2
5 The cost of a skateboard is $550. If the average inflation rate
is predicted to be 3% p.a. estimate the cost of a new
skateboard in 4 years’ time.
6 The cost of a litre of milk is $1.70. If the inflation rate is an
average 4% p.a., estimate the cost of a litre of milk after 10
years.
7 A daily newspaper costs $1.00. With an average inflation
rate of 3.4% p.a., estimate the cost of a newspaper after 5
years (to the nearest 5c).
8 If a basket of groceries costs $98.50 in 2001, what would
the estimated cost of the groceries be in 2008 if the average
inflation rate for that period is 3.2% p.a.?
9 multiple choice
A bottle of soft drink costs $2.50. If the inflation rate is predicted to average 2% p.a. for the next five years, the cost of
the soft drink in five years will be:
A $2.60
B $2.70
C $2.75
D $2.76
10 Veronica bought a shirt signed by the Australian cricket team after it won the 1999
World Cup for $200. If the value of the shirt increases by 20% per annum for the next
3
5 years, calculate the value of the shirt (to the nearest $10).
WORKED
Example
11 Ken purchased a rare bottle of wine for $350. If the value of the wine is predicted to
increase at 10% per annum, estimate the value of the wine in 20 years (to the nearest
$10).
12 The 1968 Australian 2c piece is very rare. If a coin collector purchased one in 1999
for $400 and the value of the coin increases by 15% p.a., calculate its value in 2012
(to the nearest $10).
Modelling depreciation
An asset is an item that has value to its owner. Many assets such as cars and computers
lose value over time. This is called depreciation.
Consider the case of a new motor vehicle. The value of the car depreciates the
moment that you drive the car away from the showroom. This is because the motor
vehicle is no longer new and if it were sold it would have to be sold as a used car. The
car then continues to lose value steadily each year.
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Depreciation of motor vehicles
Choose a make of car and find out the price for a new vehicle of this make and
model. Then go through the classified advertisements in the newspaper and complete
the table below or look in the RACQ publication The Road Ahead.
Age of car (years)
Price
New (0)
1
2
3
4
5
6
7
8
9
10
Draw a graph that shows the price of this car as it ages.
There are two types of depreciation, the straight line method and the declining
balance or diminishing value method. Depreciation is a significant factor in the operation of a business. The Australian Taxation Office (ATO) allows depreciation on items
necessary for the operation of the business as a legitimate tax deduction.
Straight line depreciation
The straight line method is where the asset depreciates by a constant amount each year.
When this type of depreciation is graphed, a straight line occurs and the asset will
reduce to a value of 0.
In such a case, a linear function can be derived that will allow us to calculate the
value of the item at any time. The function can be found using the gradient–intercept
method. The initial value of the item (Vo) will be the vertical intercept and the gradient
will be the negative of the amount that the item depreciates, D, each year. The equation
of this linear function will be:
V = Vo − Dt
where V is the present value of the item and t is the age of the asset, in years.
Note that gradients for depreciation will always be negative.
t i gat
11. Maths A Yr 12 - Ch. 02 Page 63 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
63
WORKED Example 4
The table below shows the declining value
of a computer. Graph the value against time
and write an equation for this function.
Age (years)
Value ($)
New (0)
4000
1
3500
2
3000
3
2500
4
2000
5
1500
WRITE
1
Draw a set of axes with age on the
horizontal axis and value on the
vertical.
2
Plot each point given by the table.
3
Join all points to graph the function.
Value ($)
THINK
4500
4000
3500
3000
2500
2000
1500
1000
500
0
0123456789
Age (years)
4
5
Write the initial value as Vo and use the
gradient to state D.
Vo = 4000, D = 500
Write the equation using V = Vo − Dt.
V = Vo − Dt
V = 4000 − 500t
In the previous worked example, how long does it take for the computer to
depreciate to a value of $0? The computer is said to be ‘written off’ when it
reaches this value.
Declining balance or diminishing value depreciation
The other method of depreciation used is the declining balance or diminishing value
method of depreciation. Here, the value of the item depreciates each year by a
percentage of its current value. Under such depreciation, the value of the item never
actually becomes zero.
This type of depreciation is an example of exponential decay. A graph depicting the
value over time is non-linear, showing a downward-falling curve which never actually
reaches a zero value.
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WORKED Example 5
The table at right shows the value of a car that is
purchased new for $40 000.
Plot the points on a set of axes and graph
the depreciation of the car. Use the graph to
estimate the value of the car after 10 years.
Age of car
(years)
Value ($)
25 600
20 480
16 384
5
13 107
WRITE
Draw a set of axes with age on the
horizontal axis and value on the
vertical.
Plot the points from the table.
Join the points with a smooth curve.
Value ($)
3
32 000
4
2
1
3
1
40 000
2
THINK
New (0)
40 000
35 000
30 000
25 000
20 000
15 000
10 000
5 000
0
0 1 2 3 4 5 6 7 8 9 10
Age (years)
4
Estimate the value after 10 years from
the graph you have drawn.
From the graph, the approximate value of the
car after 10 years is $4000.
A graphics calculator can be used to plot the values in the tables in worked examples 4
and 5. This can check the shapes of the curves. The two variables (age and value) can
be entered as lists. Because no equations are entered, we can not extrapolate values
from the graph beyond the limits of the values in the lists. Later in this chapter we will
explore a graphics calculator technique which will enable us to extrapolate values
from graphs.
WORKED Example 6
With the aid of a graphics calculator, produce graphs showing the relationship between
the age and value of the following:
a the computer in worked example 4
b the car from worked example 5.
THINK
WRITE/DISPLAY
a
a TI: Press STAT , select
1:edit.
1
Enter the age of the
computer as List1 and
the value as List2.
Casio: Enter MENU, select
STAT.
13. Maths A Yr 12 - Ch. 02 Page 65 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
THINK
2
Set one of the stat
plots ON to graph the
data in L1 and L2.
4
b
WRITE/DISPLAY
Set the window for an
age in the range 0 to
10 and a value of
$4500 down to 0.
3
1
Graph the points. Use the
TRACE function to check
the values of the points on
the line. Notice that the
graph is a straight line and
matches the shape of that
shown in worked
example 4.
Enter the age of the
b
car as List1 and the
value as List2.
2
Set the window for an
age in the range 0 to
10 and a value of
$45 000 down to 0.
3
Set one of the stat
plots ON to graph the
data in L1 and L2.
65
TI: Press WINDOW .
Casio: Press SHIFT [VWINDOW].
TI: Press 2nd [STAT
PLOT].
Casio: Press F1 (GRPH)
and F6 (SET).
TI: Press GRAPH then
TRACE .
Casio: Press F1 (GRPH)
and SHIFT [TRACE].
TI: Press STAT , and
select 1:edit.
Casio: From the main
MENU, select STAT and
enter the information.
TI: Press WINDOW .
Casio: Press SHIFT [VWINDOW].
TI: Press 2nd [STAT
Casio: Press F1 (GRPH)
and F6 (SET).
PLOT].
Continued over page
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THINK
4
WRITE/DISPLAY
Graph the points. Use the
TRACE function
to check the values of the
points on the line. Notice
that the graph
is a curve with the
same shape as shown
in worked example 5.
TI: Press GRAPH then
TRACE .
Casio: Press F1 (GPH1)
and SHIFT [TRACE].
remember
remember
1. Depreciation is the loss in the value of an item over time.
2. Depreciation can be of two types:
(a) straight line depreciation — the item loses a constant amount of value each year
(b) declining balance or diminishing value depreciation — the value of an item
depreciates by a percentage of its value each year.
3. Straight line depreciation can be graphed using a linear function in which the
new value of the item is the vertical intercept and the gradient is the negative of
the annual loss in value.
4. A declining balance or diminishing value depreciation is an example of
exponential decay and is graphed with a smooth curve.
5. A graphics calculator can be used to graph depreciation. Values can be
determined at any point along the straight line or curve.
2B
Modelling depreciation
The following questions may be solved with the aid of
a graphics calculator.
WORKED
Example
4, 6
1 The table below shows the depreciating value of a tractor.
Age (years)
New (0)
Value ($)
100 000
1
90 000
2
80 000
3
70 000
4
60 000
5
50 000
a Draw a graph of the value of the tractor against the age of the tractor.
b Write a function for the value of the tractor.
15. Maths A Yr 12 - Ch. 02 Page 67 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
2 The table at right shows the depreciating value of
a tow truck.
Draw a graph of value against age and, hence,
write the value as a linear function of age.
Age (years)
67
Value ($)
New (0)
50 000
1
42 000
2
34 000
3
26 000
4
18 000
5
10 000
3 The function V = 50 000 − 6000A shows the value, V, of a car when it is A years old.
a Draw a graph of this function.
b Use the graph to calculate the value of the car after 5 years.
c After how many years would the car be ‘written off’ (that is, the value of the car
becomes $0)?
4 A computer is bought new for $6400 and depreciates at the rate of $2000 per year.
a Write a function for the value, V, of the computer against its age, A.
b Draw the graph of this function.
c After how many years does the computer become written off?
5 The table at right shows the declining value of a
new motorscooter.
5, 6
a Plot the points shown by the table and draw
a graph of the value of the motorscooter
against age.
b Use your graph to estimate the value of the
motorscooter after 8 years.
WORKED
Example
Age (years)
Value ($)
20 000
1
15 000
2
11 250
3
8 500
4
6 250
5
6 The table at right shows the declining value of
a semi-trailer.
a Plot the points as given in the table and then
draw a curve of best fit to graph the
depreciation of the semi-trailer.
b Use your graph to estimate the value of the
semi-trailer after 10 years.
c After what number of years will the value of
the semi-trailer fall below $50 000?
New (0)
4 750
Age (years)
Value ($)
New (0)
600 000
1
420 000
2
295 000
3
205 000
4
145 000
5
100 000
7 a A gymnasium values its equipment at $200 000. Each year the value of the equipment depreciates by 20% of the value of the previous year. Calculate the value of
the equipment after:
i 1 year
ii 2 years
iii 3 years
iv 4 years.
b Plot these points on a set of axes and draw a graph of the value of the equipment
against its age.
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8 multiple choice
Which of the tables below shows a straight line depreciation?
A
Age (years)
Value ($)
B
Age (years)
Value ($)
New (0)
New (0)
4000
1
3600
1
3600
2
3240
2
3200
3
2916
3
2800
4
2624
4
2400
5
C
4000
2362
5
2000
Age (years)
Value ($)
D
Age (years)
Value ($)
New (0)
4000
New (0)
4000
1
3600
1
3000
2
3300
2
2500
3
3100
3
1500
4
3000
4
1000
5
2950
5
500
9 A car is bought new for $30 000.
a The straight line method of depreciation sees the car lose $4000 in value each year.
Complete the table below.
Age (years)
New (0)
Value ($)
30 000
1
2
3
4
5
b Draw a graph of this depreciation.
17. Maths A Yr 12 - Ch. 02 Page 69 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
c
69
The declining balance or diminishing value method of depreciation sees the value
of the car fall by 20% of the previous year’s value. Complete the table below.
Age (years)
New (0)
Value ($)
30 000
1
2
3
4
5
d On the same set of axes draw a graph of this depreciation.
e After how many years is the car worth more under declining balance or diminishing
value than under straight line depreciation?
Straight line depreciation
We have already seen that the straight line method of depreciation is where the value of
an item depreciates by a constant amount each year.
The depreciated value of an item is called the salvage value, S. The salvage value of
an asset can be calculated using the formula:
S = Vo − Dn
where Vo is the purchase price of the asset, D is the amount of depreciation per period
and n is the total number of periods.
WORKED Example 7
A laundry buys dry-cleaning
equipment for $30 000. The
equipment depreciates at a
rate of $2500 per year.
Calculate the salvage value
of the equipment after
6 years.
THINK
1
2
3
WRITE
Write the formula.
Substitute the values of Vo, D and n.
Calculate the value of S.
S = Vo − Dn
= $30 000 − $2500 × 6
= $15 000
By solving an equation we are able to calculate when the value of an asset falls below
a particular value.
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WORKED Example 8
A plumber purchases equipment for a total of $60 000. The value of the equipment is
depreciated by $7500 per year. When the value of the equipment falls below $10 000 it
should be replaced. Calculate the number of years after which the equipment should be
replaced.
THINK
WRITE
S = Vo − Dn
1
Write the formula.
2
Substitute for S, Vo and D.
10 000 = 60 000 − 7500n
3
Solve the equation to find the value
of n.
7500n = 50 000
-n = 62
3
4
Give a written answer, taking the value
of n up to the next whole number.
The equipment must be replaced after 7 years.
A graphics calculator can provide the answer to worked example 8 without the need to
solve an equation.
WORKED Example 9
Use a graphics calculator for the problem in worked example 8.
THINK
WRITE/DISPLAY
1
Enter the formula into Y1
with the variable X being
the number of years.
TI: Press Y= to enter data.
Casio: Select the TABLE
function from the MENU.
2
Set up the table for a
minimum of 0 (years)
and increments of 1.
TI: Press 2nd [TBLSET].
Casio: Press F5 (RANG)
to set the range.
3
Display the table
produced by the formula.
TI: Press 2nd [TABLE].
Casio: Press F6 (TABL).
19. Maths A Yr 12 - Ch. 02 Page 71 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
THINK
4
71
WRITE/DISPLAY
Scroll down the table to
locate the first time the
value falls below
$10 000. Note that this
occurs after 7 years. This
is consistent with the
answer using algebra in
worked example 8.
TI: Press
navigate.
M
to
Casio: Press
navigate.
M
to
remember
remember
1. Straight line depreciation occurs when the value of an asset depreciates by a
constant amount each year.
2. The formula to calculate the salvage value, S, of an asset is S = Vo − Dn where
Vo is the purchase price of the asset, D is the amount of depreciation per period
and n is the total number of periods.
3. To calculate a value of Vo, D or n we substitute all known values and solve the
equation that is formed.
2C
WORKED
Example
7
Straight line depreciation
1 A car that is purchased for $45 000 depreciates by $5000 each year. Calculate the
salvage value of the car after 5 years.
2 Calculate the salvage value:
a after 5 years of a computer that is purchased
for $5000 and depreciates by $800 per year
b after 7 years of a motorscooter that is purchased
for $25 000 and depreciates by $2100 per year
c after 6 years of a semi-trailer that is purchased
for $750 000 and depreciates by $80 000 per
year
d after 2 years of a mobile phone that is purchased
for $225 and depreciates by $40 per year
e after 4 years of a farmer’s plough that is
purchased for $80 000 and depreciates by
$12 000 per year.
3 A bus company buys 15 buses for $475 000 each.
a Calculate the total cost of the fleet of buses.
b If each bus depreciates by $25 000 each year, calculate the salvage value of the
fleet of buses after 9 years.
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4 The price of a new car is $25 000. The value of the car depreciates by $300 each
month. Calculate salvage value of the car after 4 years.
WORKED
Example
8, 9
5 An aeroplane is bought by an airline for $60 million. If the aeroplane depreciates by
$4 million each year, calculate when the value of the aeroplane falls below $30 million.
6 Calculate the length of time for each of the following items to depreciate to the value given.
a A computer purchased for $5600 to depreciate to less than $1000 at $900 per year
b An electric guitar purchased for $1200 to depreciate to less than $500 at $150 per year
c An entertainment unit purchased for $6000 to become worthless at $750 per year
d Office equipment purchased for $12 000 to depreciate to less than $2500 at $1500
per year
7 A motor vehicle depreciates from $40 000 to $15 000 in 10 years. Assuming that it is
depreciating in a straight line, calculate the annual amount of depreciation.
8 Calculate the annual amount of depreciation in an asset that depreciates:
a from $20 000 to $4000 in 4 years
b from $175 000 to $50 000 in 10 years
c from $430 000 to $299 500 in 9 years.
9 A computer purchased for $3600 is written off in 4 years. Calculate the annual
amount of depreciation.
10 A car that is 5 years old has an insured value of $12 500. If the car is depreciating at
a rate of $2500 per year, calculate its purchase price.
2.1
11 Calculate the purchase price of each of the following assets given that:
a after 5 years the value is $50 000 and is depreciating at $12 000 per year
b after 15 years the value is $4000 and is depreciating at $1500 per year
c after 25 years the value is $200 and is depreciating at $50 per year.
12 An asset that depreciates at $6500 per year is written off after 12 years. Calculate the
purchase price of that asset.
Declining balance or diminishing value
method of depreciation
The declining balance or diminishing value method of depreciation occurs when the
value of an asset depreciates by a given percentage each period.
Consider the case of a car purchased new for $30 000, which depreciates at the rate
of 20% p.a. Each year the salvage value of the car is 80% of its value at the end of the
previous year.
After 1 year: S = 80% of $30 000
= $24 000
After 2 years: S = 80% of $24 000
= $19 200
After 3 years: S = 80% of $19 200
= $15 360
21. Maths A Yr 12 - Ch. 02 Page 73 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
WORKED Example
73
10
A small truck that was purchased for $45 000 depreciates at a rate of 25% p.a. By
calculating the value at the end of each year, find the salvage value of the truck after 4 years.
THINK
1
2
3
4
5
WRITE
The salvage value at the end of each
year will be 75% of its value at the end
of the previous year.
Find the value after 1 year by
calculating 75% of $45 000.
Find the value after 2 years by
calculating 75% of $33 750.
Find the value after 3 years by
calculating 75% of $25 312.50.
Find the value after 4 years by
calculating 75% of $18 984.38.
S = 75% of $45 000
= $33 750
After 2 years: S = 75% of $33 750
= $25 312.50
After 3 years: S = 75% of $25 312.50
= $18 984.38
After 4 years: S = 75% of $18 984.38
= $14 238.28
After 1 year:
The salvage value under a declining balance can be calculated using the formula:
R T
S = Vo 1 – --------
100
where S is the salvage value, Vo is the purchase price, R is the annual percentage
depreciation and T is the number of years. Note the similarity to the compound interest
formula with a minus sign in place of a plus sign (because the value is decreasing). The
value of n is 1 since depreciation is usually calculated yearly.
WORKED Example 11
The purchase price of a yacht is $15 000. The value of the
yacht depreciates by 10% p.a. Calculate (correct to the
nearest $1) the salvage value of the yacht after 8 years.
THINK
WRITE
1
Write the formula.
2
Substitute values for Vo, R and T.
3
Calculate the salvage value.
R
S = Vo 1 – --------
100
T
10 8
= 15 000 1 – --------
100
= $15 000 × 0.98
= $6457.00
To calculate the amount by which the asset has depreciated, we subtract the salvage
value from the purchase price.
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WORKED Example
12
The purchase price of a computer for a music studio is
$40 000. The computer depreciates by 12% p.a.
Calculate the amount by which the computer
depreciates in 10 years.
THINK
WRITE
R T
S = Vo 1 – --------
100
1
Write the formula.
2
Substitute the value of Vo, R and T.
3
4
Calculate the value of S.
Calculate the amount of depreciation by
subtracting the salvage value from the
purchase price.
10
12
= 40 000 1 – --------
100
= $40 000 × 0.8810
= $11 140.04
Depreciation = $40 000 − $11 140.04
= $28 859.96
remember
remember
1. The declining balance or diminishing value method of depreciation occurs
when the value of an asset depreciates by a fixed percentage each year.
2. The salvage value of an asset can be calculated by subtracting the percentage
depreciation each year.
T
R
3. The salvage value can be calculated using the formula S = Vo 1 – -------- .
100
4. To calculate the amount of depreciation, the salvage value should be subtracted
from the purchase price.
2D
2.2
WORKED
Example
10
Declining balance or
diminishing value method of
depreciation
1 The purchase price of a forklift is $50 000. The value of the forklift depreciates by
20% p.a. By calculating the value of the forklift at the end of each year, find the
salvage value of the forklift after 4 years.
2 A trailer is purchased for $5000. The value of the trailer depreciates by 15% each
year. By calculating the value of the trailer at the end of each year, calculate:
a the salvage value of the trailer after 5 years (to the nearest $10)
b the amount by which the trailer depreciates:
i in the first year
ii in the fifth year.
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Chapter 2 Appreciation and depreciation
75
3 A company purchases a mainframe computer for $3 000 000. The value of the
computer depreciates by 15% p.a. By calculating the value at the end of each year find
the number of years that it takes for the salvage value of the mainframe to fall below
$1 000 000.
WORKED
Example
11
4 Use the declining balance depreciation formula to calculate the salvage value after 7
years of a power generator purchased for $800 000 that depreciates at a rate of
10% p.a. (Give your answer correct to the nearest $1000.)
5 Calculate the salvage value of an asset (correct to the nearest $100) with a purchase
price of:
a $10 000 that depreciates at 10% p.a. for 5 years
b $250 000 that depreciates at 15% p.a. for 8 years
c $5000 that depreciates at 25% p.a. for 5 years
d $2.2 million that depreciates at 30% p.a. for 10 years
e $50 000 that depreciates at 40% p.a. for 5 years.
6 A plumber has tools and equipment valued at $18 000. If the value of the equipment
depreciates by 30% each year, calculate the value of the equipment after 3 years.
WORKED
Example
12
7 A yacht is valued at $950 000. The value of the yacht depreciates by 22% p.a.
Calculate the amount that the yacht will depreciate in value over the first 5 years
(correct to the nearest $1000).
8 A new car is purchased for $35 000. The owner plans to keep the car for 5 years then
trade the car in on another new car. The estimate is that the value of the car will
depreciate by 16% p.a. Calculate:
a the amount the owner can expect as a trade in for the car in 5 years (correct to the
nearest $100)
b the amount by which the car will depreciate in 5 years.
9 multiple choice
A shop owner purchases fittings for her store that cost a total of $120 000. Three years
later, the shop owner is asked to value the fittings for insurance. If the shop owner
allows for depreciation of 15% on the fittings, which of the following calculations
will give the correct estimate of their value?
A 120 000 × 0.853 B 120 000 × 0.153 C 120 000 × 0.55 D 120 000 × 0.45
10 multiple choice
A computer purchased for $3000 will depreciate by 25% p.a. The salvage value of the
computer after 4 years will be closest to:
A $0
B $10
C $950
D $2000
11 An electrician purchases tools of trade for a total of $8000. Each year the electrician
is entitled to a tax deduction for the depreciation of this equipment. If the rate of
depreciation allowed is 33% p.a., calculate:
a the value of the equipment at the end of one year (correct to the nearest $1)
b the tax deduction allowed in the first year
c the value of the equipment at the end of two years (correct to the nearest $1)
d the tax deduction allowed in the second year.
12 An accountant purchased a computer for $6000. The value of the computer depreciates by 33% p.a. When the value of the computer falls below $1000, it is written off
and a new one is purchased. How many years will it take for the computer to be
written off?
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76
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Rates of depreciation
In the previous investigation you chose a make and model of car and researched the
salvage value of this car after each year.
1 Calculate the percentage depreciation for each year.
2 Calculate if this percentage rate is approximately the same each year.
3 Using the average annual depreciation, calculate a table of salvage values for
the first 10 years of the car’s life.
4 Draw a graph showing the depreciating value of the car.
5 Some people claim that it is best to trade your car in on a new model every
three years. With reference to your figures and graph, express your views on
this statement.
1
1 The price of a new CD player is $1250. The CD player will depreciate under straight
line depreciation at a rate of $200 per year. Calculate the value of the CD player after
3 years.
2 An asset that was valued at $39 000 when new depreciates to $22 550 in 7 years.
Calculate the annual amount of depreciation under straight line depreciation.
3 A computer that is purchased new for $9000 depreciates at a rate of $1350 per year.
Calculate the length of time before the computer is written off.
4 A car dealer values a used car at $7000. If the car is 8 years old and the rate of
depreciation is $1750 per year, calculate the value of the car when new.
5 Write the formula for depreciation under the declining balance method.
6 A truck is valued new at $50 000 and depreciates at a rate of 32% p.a. Calculate the
value of the truck after 5 years (correct to the nearest $50).
7 An asset that has a purchase price of $400 000 depreciates at a rate of 45% p.a.
Calculate the asset’s value after 6 years (correct to the nearest $1000).
8 For the asset in question 7, calculate the amount by which it has depreciated in
6 years.
9 Office equipment valued at $250 000 depreciates at a rate of $15% p.a. Calculate the
amount by which it depreciates in the first year.
10 Calculate the length of time it will take for the salvage value of the office equipment
in question 9 to fall below $20 000.
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25. Maths A Yr 12 - Ch. 02 Page 77 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
77
Depreciation tables
The following computer application will prepare a table that will show the depreciated
value of an asset with a purchase price of $1 over various periods of time and various
rates of depreciation.
es
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n inv
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Depreciation table
1 Open a new spreadsheet and type in the headings shown.
2 In cell B3 enter the formula =(1-B$2)^$A3.
3 Highlight the range of cells B3 to K12. Select the Edit function, and then use
(a) the Fill and (b) the Right and Fill and Down functions to copy the formula
throughout the table.
4 The table that you now have should have the values shown in the table.
Time
(years)
Rate of depreciation (per annum)
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
1
0.9500 0.9000 0.8500 0.8000 0.7500 0.7000 0.6500 0.6000 0.5500 0.5000
2
0.9025 0.8100 0.7225 0.6400 0.5625 0.4900 0.4225 0.3600 0.3025 0.2500
3
0.8574 0.7290 0.6141 0.5120 0.4219 0.3430 0.2746 0.2160 0.1664 0.1250
4
0.8145 0.6561 0.5220 0.4096 0.3164 0.2401 0.1785 0.1296 0.0915 0.0625
5
0.7738 0.5905 0.4437 0.3277 0.2373 0.1681 0.1160 0.0778 0.0503 0.0313
6
0.7351 0.5314 0.3771 0.2621 0.1780 0.1176 0.0754 0.0467 0.0277 0.0156
7
0.6983 0.4783 0.3206 0.2097 0.1335 0.0824 0.0490 0.0280 0.0152 0.0078
8
0.6634 0.4305 0.2725 0.1678 0.1001 0.0576 0.0319 0.0168 0.0084 0.0039
9
0.6302 0.3874 0.2316 0.1342 0.0751 0.0404 0.0207 0.0101 0.0046 0.0020
10
0.5987 0.3487 0.1969 0.1074 0.0563 0.0282 0.0135 0.0060 0.0025 0.0010
5 Use the spreadsheet’s graphing facility to draw a depreciation graph for each of
the depreciation rates shown in the table.
6 Print out a copy of the graph for one of these rates.
The table produced shows the depreciated value of $1 and can be used to make
calculations about depreciation.
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26. Maths A Yr 12 - Ch. 02 Page 78 Wednesday, September 11, 2002 3:52 PM
78
M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d
WORKED Example 13
An item is purchased for $500 and depreciates at a rate of 15% p.a. Use the depreciation
table on page 77 to calculate the value of the item after 4 years.
THINK
1
2
WRITE
Look up the table to find the depreciated
value of $1 at 15% p.a. for 4 years.
Multiply the depreciated value of $1 by
$500.
Depreciated value = 0.5220 × $500
= $261
The computer application on page 77 will produce a general table for a declining balance
or diminishing value depreciation. We should be able to use the formula to create a table
and graph showing the salvage value of an asset under both straight line and diminishing
value depreciation.
WORKED Example 14
A car is purchased new for $20 000.
The depreciation can be calculated
under straight-line depreciation at
$2500 per year and under diminishing
value at 20% p.a.
a Complete the table at right.
(Give all values to the nearest $1.)
b Draw a graph of both the straight
line and diminishing value
depreciation and use the graph to
show the point at which the
straight-line value of the car falls
below the diminishing value.
THINK
a 1 Copy the table.
2 Complete the straight line column by
subtracting $2500 from the previous
year’s value.
3 Complete the diminishing value by
multiplying the previous year’s value
by 0.8.
Age of car
(years)
New (0)
1
2
3
4
5
6
7
8
WRITE
a Age of
car
(years)
New (0)
1
2
3
4
5
6
7
8
Straight line
value ($)
20 000
Diminishing
value ($)
20 000
Straight line
value ($)
20 000
17 500
15 000
12 500
10 000
7 500
5 000
2 500
0
Diminishing
value ($)
20 000
16 000
12 800
10 240
8 192
6 554
5 243
4 194
3 355
27. Maths A Yr 12 - Ch. 02 Page 79 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
THINK
WRITE
b
b
79
2
3
4
Plot the points generated by the
table.
Join the points for the straight line
depreciation with a straight line.
Join the points for the diminishing
value depreciation with a smooth
curve.
The graph shows the straight line
going below the curve after 6 years.
WORKED Example
25 000
Value ($)
1
20 000
Straight line
value
Diminishing
value
15 000
10 000
5 000
0
(New)0 1 2 3 4 5 6 7 8
Age (years)
The straight line depreciation becomes less
than the diminishing value depreciation after
6 years.
15
Use a graphics calculator to solve worked example 14.
THINK
WRITE/DISPLAY
Enter a formula for the
straight line
depreciation in Y1 and
a formula for
diminishing value
depreciation in Y2.
TI: Press Y= .
2
Set up the table to
start at 0 and progress
in increments of 1.
TI: Press 2nd and [TBLSET].
Casio: Press F5 (RANG).
3
Display the table and
write down the results
(rounding to the
nearest $).
TI: Press 2nd [TABLE].
Casio: Press F6 (TABL).
1
Casio: From the main
MENU, select TABLE.
Continued over page
28. Maths A Yr 12 - Ch. 02 Page 80 Wednesday, September 11, 2002 3:52 PM
80
M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d
4
5
WRITE/DISPLAY
Set the window to
display the age of the
car (x-value) in the
range 0 to 10 and the
car’s value (y-value)
from $20 000 down to
0.
Graph the functions.
Use the options to
determine the coordinates of the point
of intersection.
Beyond this point the
value using straight
line depreciation
becomes less than the
value using
diminishing value
depreciation. Notice
that this gives a more
accurate result than
the manual method.
TI: Press WINDOW .
Casio: Press SHIFT
[V-WINDOW].
TI: Press GRAPH , then 2nd
[CALC] and select 5:intersect.
At the question ‘First curve’,
press ENTER (to confirm that
the cursor is on one of the
curves), press ENTER again (to
confirm the second curve) and
press ENTER to confirm that
you now want to see the ‘Guess’.
Casio: Press MENU , select
GRAPH, press F6
(DRAW) then SHIFT (GSolv) and F5 (ISCT) to
give the first intersection
point (0, 20 000), Then
press
for the second
intersection point.
M
THINK
Depreciation is an allowable tax deduction for people in many occupations. A tax
deduction for depreciation is allowed when equipment used in earning an income
depreciates in value and will eventually need replacing. Depending on the equipment
and the occupation, either straight line or diminishing value depreciation may be
used.
Under diminishing value depreciation, when the salvage value falls below a certain
point the equipment may be written off. This means that the entire remaining balance
can be claimed as a tax deduction and as such is considered worthless. From this point
on, no further tax deductions can be claimed for this equipment. A visit to the Australian
Taxation Office’s website provides more detailed information on this subject.
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Comparing straight line depreciation
and diminishing value depreciation
using a spreadsheet
Let us look at a solution to worked example 14 using a spreadsheet. The value of
the new car is $20 000. Depreciation using the straight line method is $2500 per
year. Depreciation using the diminishing value method is 20% p.a. Let us draw up
a spreadsheet similar to the one following.
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29. Maths A Yr 12 - Ch. 02 Page 81 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
81
1 Start by entering the main heading in cell A1 and the side headings in cells
A3, A4 and A5.
2 Enter the new value, 20000, in cell B3 and format it as currency with no
decimal places.
3 Enter the value 2500 in cell B4 and format as currency with no decimal
places.
4 Enter the numeric value of 20 in cell B5.
5 Head up the three columns in row 7.
6 In cell position A8, start the years by entering 0.
7 In cell A9, enter the formula =A8+1. Copy this formula down to row 16.
8 In cell B8, enter the formula =$B$3-$B$4*A8 (the formula for straight line
depreciation). Copy this formula down to cell B16.
9 The diminishing value formula to be entered into cell C8 is
=$B$3*(1-$B$5/100)^A8. Copy this formula down to cell C16.
10 Check that all your figures agree with those in the spreadsheet above. Notice
that they are the same as those in the table of worked example 14.
11 Use the graphing facility of the spreadsheet to produce graphs similar to the
ones shown.
12 From the table and the graphs it is evident that a critical point occurs around
the 6-year mark. This is consistent with the graph shown in the worked
example.
13 Try changing values in cells B3, B4 and B5. Notice how the spreadsheet
adjusts.
14 Save your spreadsheet.
30. Maths A Yr 12 - Ch. 02 Page 82 Wednesday, September 11, 2002 3:52 PM
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M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d
WORKED Example 16
A builder has tools of trade that are purchased new for $14 000. He is allowed a
tax deduction of 33% p.a. for depreciation of this equipment. When the salvage
value of the equipment falls below $3000, the electrician is allowed to write the
equipment off on the next year’s return. Complete the depreciation table below.
(Use whole dollars only.)
Years
Salvage value ($)
Tax deduction ($)
1
2
3
4
5
THINK
1
2
3
WRITE
Calculate the salvage value by
multiplying the previous year’s value
by 0.67.
Calculate the tax deduction by
multiplying the previous year’s value
by 0.33.
When the salvage value is less than
$3000, claim the entire amount as a tax
deduction.
Year
Salvage value
($)
Tax deduction
($)
1
9380
4620
2
6285
3095
3
4211
2074
4
2821
1390
5
0
2821
remember
remember
1. Graphs can be drawn to compare the salvage value of an asset under different
rates of depreciation, or to compare diminishing value and straight line
depreciation.
2. The amount by which an asset depreciates can, in many cases, be claimed as a
tax deduction.
2E
Depreciation tables
1 Use the table of depreciated values of $1 to calculate:
a the value of a computer purchased for $5000 after 5 years, given that it depreciates
13
at 20% p.a.
b the value of a car after 8 years with an initial value of $35 000, given that it
depreciates at 15% p.a.
c the value of a boat with an initial value of $100 000 after 10 years, given that it
depreciates at 10% p.a.
WORKED
Example
31. Maths A Yr 12 - Ch. 02 Page 83 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
83
2 A taxi owner purchases a new taxi for $40 000. The taxi depreciates under straightline depreciation at $5000 per year and under diminishing value depreciation at
14, 15
20% p.a.
a Copy and complete the table below. Give all values to the nearest $100.
WORKED
Example
Age of car
(years)
New (0)
Straight line value ($)
Diminishing value ($)
40 000
40 000
2.3
2.4
1
2
3
2.5
4
5
6
7
8
b Draw a graph of the salvage value of the taxi under both methods of depreciation.
c State when the value under straight line depreciation becomes less than under
diminishing value depreciation.
3 A company has office equipment that is valued at $100 000. The value of the equipment can be depreciated at $10 000 each year or by 15% p.a.
a Draw a table that will show the salvage value of the office equipment for the first
ten years. (Give all values correct to the nearest $50.)
b Draw a graph of the depreciating value of the equipment under both methods of
depreciation.
4 A computer purchased new for $4400 can be depreciated at either 20% p.a. or 35% p.a.
Draw a table and a graph that compare the salvage value of the computer at each rate of
depreciation.
5 A teacher purchases a laptop
computer for $6500. A tax
16
deduction for depreciation of the
computer is allowed at the rate of
33% p.a. When the value of the
computer falls below $1000, the
computer can be written off. Copy
and complete the table at right.
(Give all values correct to the
nearest $1.)
WORKED
Example
Year
1
2
3
4
5
6
Salvage value
($)
Tax deduction
($)
32. Maths A Yr 12 - Ch. 02 Page 84 Wednesday, September 11, 2002 3:52 PM
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M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d
6 A plumber purchases a work van
for $45 000. The van can be
depreciated at a rate of 25% p.a.
for tax purposes and the van can
be written off at the end of 8 years.
Copy and complete the depreciation
schedule at right. (Give all answers
correct to the nearest $1.)
Year
Salvage value
($)
Tax deduction
($)
1
2
3
4
5
6
7
8
7 A truck is purchased for $250 000. The truck can be depreciated at the rate of $25 000
each year or over 10 years at 20% p.a.
a Copy and complete the table
Age of
at right. (Give all values
truck
Straight line
Diminishing
correct to the nearest $1.)
(years)
value ($)
value ($)
b Draw a graph of the
depreciating value of the
New (0)
250 000
250 000
truck under both methods of
1
depreciation.
c Complete a depreciation
2
schedule for each method of
3
calculation.
4
5
6
7
8
9
10
8 Leilay is a cinematographer and on 1 March purchases a camera for work purposes.
The cost of the camera is $40 000 and for tax purposes the camera depreciates at the
rate of 25% p.a.
a Calculate the amount that the camera will depreciate in the first year.
b The financial year ends on 30 June. For what fraction of the financial year did
Leilay own the camera?
c Leilay is allowed a tax deduction for depreciation of her camera. Calculate the
amount of tax deduction that Leilay is allowed for the financial year ending on
30 June.
33. Maths A Yr 12 - Ch. 02 Page 85 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
85
9 Calculate the amount of depreciation
on each of the following assets:
a a tractor with an initial value of
$80 000 that depreciates at 15%
p.a. for 3 months
b a bicycle with an initial value of
$600 that depreciates at 25% p.a.
for 6 months
c office furniture with an initial
value of $8000 that depreciates at
30% p.a. for 8 months
d a set of encyclopedias with an
initial value of $2500 that depreciates at 40% p.a. for 9 months.
Future and present value of an annuity
An annuity is a form of investment involving regular periodic contributions to an
account. On such an investment, interest compounds at the end of each period and the
next contribution to the account is then made.
Superannuation is a common example of an annuity. Here, people invest in a fund on
a regular basis, the interest on the investment compounds, while the principal is added
to for each period. The annuity is usually set aside for a person’s entire working life
and is used to fund retirement. It may also be used to fund a long-term goal, such as a
trip in 10 years time.
To understand the growth of an annuity, we first need to revise compound interest.
The compound interest formula is:
n×T
R
A = P 1 + -----------------
100 × n
where A is the amount that an investment will become, P is the principal, R is the
interest rate per annum, n is the number of interest periods per year and T is the
time in years.
WORKED Example 17
Calculate the value of a $5000 investment made at 8% p.a. for 4 years, with interest
compounded annually.
THINK
1
Write the formula.
2
Substitute values for P, R, n and T.
3
Calculate the value of A.
WRITE
R
A = P 1 + -----------------
100 × n
n×T
1×4
8
= 5000 1 + -----------------
100 × 1
= $5000 × (1.08)4
= $6802.44
34. Maths A Yr 12 - Ch. 02 Page 86 Wednesday, September 11, 2002 3:52 PM
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M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d
An annuity takes the form of a sum of compound interest investments. Consider the
case of a person who invests $1000 at 10% p.a. at the end of each year for five years.
To calculate this, we would need to calculate the value of the first $1000 which is
invested for four years, the second $1000 which is invested for three years, the third
$1000 which is invested for two years, the fourth $1000 which is invested for one year
and the last $1000 which is added to the investment.
WORKED Example
18
Calculate the value of an annuity in which $1000 is invested each year at 10% p.a. for 5
years.
THINK
WRITE
1
The value of n is 1 in all cases.
2
Use the compound interest formula to
calculate the amount to which the first
$1000 will grow.
3
Use the compound interest formula to
calculate the amount to which the
second $1000 will grow.
4
Use the compound interest formula to
calculate the amount to which the third
$1000 will grow.
5
Use the compound interest formula to
calculate the amount to which the
fourth $1000 will grow.
6
Find the total of the separate $1000
investments, remembering to add the
final $1000.
n=1
T
R
A = P 1 + --------
100
= $1000 × 1.14
= $1464.10
R T
A = P 1 + --------
100
= $1000 × 1.13
= $1331.00
R T
A = P 1 + --------
100
= $1000 × 1.12
= $1210.00
R T
A = P 1 + --------
100
= $1000 × 1.1
= $1100.00
Total value = $1464.10 + $1331.00 + $1210.00
+ $1100.00 + $1000
= $6105.10
AC
ER T
M
C
D-
IVE
INT
To find out more about the future value of an annuity, click here when using the Maths
Quest Maths A Year 12 CD-ROM.
extension
Future value
extension of an annuity
RO
Annuity calculations
The amount to which an annuity grows is called the future value of an annuity and
can be calculated using the formula:
( 1 + r )n – 1
A = M ---------------------------
r
35. Maths A Yr 12 - Ch. 02 Page 87 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
87
M is the amount of each periodical investment made at the end of the period; r is the
interest rate per period expressed as a decimal and n is the number of deposits.
The present value of an annuity, P, can be calculated using the formula
( 1 + r )n – 1
P = M ---------------------------
n
r(1 + r )
This formula allows us to calculate the single sum needed to be invested to give the
same financial result as an annuity where we are given the size of each contribution.
The financial calculator section of the later model graphics calculators makes short
work of annuity calculations. Let us use the TI-83 and Casio CFX-9850GB PLUS to
demonstrate this facility.
WORKED Example 19
Christina invests $500 into a fund every 6 months at 9% p.a. interest,
compounding six-monthly for 10 years. Calculate the future value of the annuity
after 10 years.
THINK
1
2
3
4
WRITE/DISPLAY
Enter the financial section of the
calculator.
TI: Press 2nd
Enter the following data as shown at
[FINANCE]and select
right.
n = 10 × 2 (Interest is calculated 1:TVMSolver.
twice a year for 10
years.)
I% = 9
(Interest rate is 9% p.a.)
PV = 0
(No deposit is made
initially – only regular
6-monthly payments.)
PMT = 500
(Regular $500
payments are made.)
FV = 0
(This value will be
calculated.)
P/Y = 2
(Regular payments are
made twice a year.)
C/Y = 2
(Interest is calculated
twice a year.)
Solve for FV. Take the cursor to the
FV line and press ALPHA [SOLVE]
on the TI and press F5 (FV) for
the Casio.
Write the answer.
Casio: From MENU,
select TVM, then
F2:Compound Interest.
The future value of the annuity after 10 years is
$15 685.71.
36. Maths A Yr 12 - Ch. 02 Page 88 Wednesday, September 11, 2002 3:52 PM
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M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d
WORKED Example 20
Vicky has the goal of saving $10 000 in the next five years. The best interest rate that she
can obtain is 8% p.a., with interest compounded annually. Calculate the amount of each
annual contribution that Vicky must make.
THINK
2
3
Write the answer.
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Annuity
calculator
n inv
io
es
TI
Enter the following data using the
methods described for the previous
worked example.
n = 5 × 1 (Interest is calculated
once a year for
5 years.)
I% = 8
(Interest rate is 8% p.a.)
PV = 0
(No deposit is made
initially.)
PMT = 0
(This is the unknown
value which will be
calculated.)
FV = 10 000 (The future value is
$10 000.)
P/Y = 1
(Vicky makes one
payment per year.)
C/Y = 1
(Interest is calculated
yearly.)
Solve for PMT. Take the cursor to
the PMT line and press ALPHA
[SOLVE] for the TI. For the Casio,
press F4 (PMT).
es
1
WRITE/DISPLAY
Casio
A payment of $1704.56 is required as the annual
contribution.
Annuity calculator
Start up the Maths Quest Maths A Year 12 CD-ROM which accompanies this book
and download the spreadsheet ‘Annuity Calculator’. The spreadsheet will show you
the growth of an annuity where $1000 is invested at the end of each year for 20
years, at the rate of 8% p.a. interest, compounding annually.
1 The spreadsheet shows that after 20 years the value of this investment is
$45 761.96. Below is the growth of the annuity after each deposit is made. This
will allow you to see the growth for up to 30 deposits. You can use Edit, then
Fill and Down functions on the spreadsheet to see further.
2 Browse the spreadsheet, taking note of the formulas in the various cells.
37. Maths A Yr 12 - Ch. 02 Page 89 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
89
3 Click on the Tab ‘Chart 1’. This is a line graph that shows the growth of the
annuity for up to 30 deposits.
4 Change the size of the deposit to $500 and the compounding periods to 2. This
will show how much benefit can be achieved by reducing the compounding
period.
Annuity values using tables
To compare an annuity with a single sum investment, we need to use the present
value of the annuity. The present value of an annuity is the single sum of money
which, invested on the same terms as the annuity, will produce the same financial
result.
Problems associated with annuities can be simplified by creating a table that will
show either the future value or present value of an annuity of $1 invested per interest
period.
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Future value of $1
Consider $1 is invested into an annuity each interest period. The table we are going
to construct on a spreadsheet shows the future value of that $1.
1 Open a new spreadsheet.
2 Type in the headings shown on the following page.
3 In cell B4 enter the formula =((1+B$3)^$A4-1)/B$3. (This is the future value
formula.) Format the cell, correct to 4 decimal places.
t i gat
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4 Highlight the range of cells B3 to K13. Choose Edit and then the Fill and Down
functions followed by the Fill and Right functions to copy the formula to all
other cells in this range.
5
Save the spreadsheet as ‘Future value of $1’.
This completes the table. The table shows the future value of an annuity of $1
invested for up to 10 interest periods at up to 10% per interest period. You can
extend the spreadsheet further for other interest rates and longer investment
periods.
Future values of $1
Interest rate (per period)
Period
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
1
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
2
2.0100 2.0200 2.0300 2.0400 2.0500 2.0600 2.0700 2.0800 2.0900 2.1000
3
3.0301 3.0604 3.0909 3.1216 3.1525 3.1836 3.2149 3.2464 3.2781 3.3100
4
4.0604 4.1216 4.1836 4.2465 4.3101 4.3746 4.4399 4.5061 4.5731 4.6410
5
5.1010 5.2040 5.3091 5.4163 5.5256 5.6371 5.7507 5.8666 5.9847 6.1051
6
6.1520 6.3081 6.4684 6.6330 6.8019 6.9753 7.1533 7.3359 7.5233 7.7156
7
7.2135 7.4343 7.6625 7.8983 8.1420 8.3938 8.6540 8.9228 9.2004 9.4872
8
8.2857 8.5380 8.8923 9.2142 9.5491 9.8975 10.2598 10.6366 11.0285 11.4359
9
9.3685 9.7546 10.1591 10.5828 11.0266 11.4913 11.9780 12.4876 13.0210 13.5795
10
10.4622 10.9497 11.4639 12.0061 12.5779 13.1808 13.8164 14.4866 15.1929 15.9374
The above table is the set of future values of $1 invested into an annuity. This is the
table you should have obtained.
A table such as this can be used to find the value of an annuity by multiplying
the amount of the annuity by the future value of $1.
39. Maths A Yr 12 - Ch. 02 Page 91 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
WORKED Example
91
21
Use the table to find the future value of an annuity where $1500 is deposited at the end of
each year into an account that pays 7% p.a. interest, compounded annually for 9 years.
THINK
1
2
WRITE
Look up the future value of $1 at
7% p.a. for 9 years.
Multiply this value by 1500.
Future value = $1500 × 11.9780
= $17 967
Just as we have a table for the future value of an annuity, we can create a table for the
present value of an annuity.
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Present value table
The table we are about to make on a spreadsheet shows the present value of an
annuity of $1 invested per interest period.
1 Open a new spreadsheet.
2 Enter the headings shown in the spreadsheet.
3 In cell B4 type the formula =((1+B$3)^$A4-1)/(B$3*(1+B$3)^$A4).
4 Drag from cell B4 to K13 and then use the Edit and then the Fill and Down and
the Fill and Right functions to copy this formula to the remaining cells in your
table.
5 Save your spreadsheet as ‘Present value of $1’.
The table created shows the present value of an annuity of $1 per interest period
for up to 10% per interest period and for up to 10 interest periods.
t i gat
40. Maths A Yr 12 - Ch. 02 Page 92 Wednesday, September 11, 2002 3:52 PM
92
M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d
The result that you should have obtained is shown below.
Present values of $1
Interest rate (per period)
Period
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
1
0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091
2
1.9704 1.9416 1.9135 1.8861 1.8594 1.8334 1.8080 1.7833 1.7591 1.7355
3
2.9410 2.8839 2.8286 2.7751 2.7232 2.6730 2.6243 2.5771 2.5313 2.4869
4
3.9020 3.8077 3.7171 3.6299 3.5460 3.4651 3.3872 3.3121 3.2397 3.1699
5
4.8534 4.7135 4.5797 4.4518 4.3295 4.2124 4.1002 3.9927 3.8897 3.7908
6
5.7955 5.6014 5.4172 5.2421 5.0757 4.9173 4.7665 4.6229 4.4859 4.3553
7
6.7282 6.4720 6.2303 6.0021 5.7864 5.5824 5.3893 5.2064 5.0330 4.8684
8
7.6517 7.3255 7.0197 6.7327 6.4632 6.2098 5.9713 5.7466 5.5348 5.3349
9
8.5660 8.1622 7.7861 7.4353 7.1078 6.8017 6.5152 6.2469 5.9952 5.7590
10
9.4713 8.9826 8.5302 8.1109 7.7217 7.3601 7.0236 6.7101 6.4177 6.1446
This table can be used in the same way as the future value table.
WORKED Example 22
Liam invests $750 per year into an annuity at 6% per annum for 8 years, with interest
compounded annually. Use the table to calculate the present value of Liam’s annuity.
THINK
1
2
WRITE
Use the table to find the present value of a $1
annuity at 6% for 8 interest periods.
Multiply this value by 750.
remember
remember
Present value = $750 × 6.2098
= $4657.35
n×T
R
1. The compound interest formula is A = P 1 + -----------------
, where A is the
100 × n
amount that the investment will grow to, R is the interest rate per year, n is the
number of periods per year and T is the time of the investment in years.
2. An annuity is a form of investment where periodical equal contributions are
made to an account, with interest compounding at the end of each period.
3. The value of an annuity is calculated by adding the value of each amount
contributed as a separate compound interest investment.
4. A table of future values shows the future value of an annuity where $1 is
invested per interest period.
5. A table of present values shows the present value of an annuity where $1 is
invested per interest period.
6. A table of present or future values can be used to compare investments and
determine which will give the greater financial return.
41. Maths A Yr 12 - Ch. 02 Page 93 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
2F
WORKED
Example
17
93
Future and present value of
an annuity
1 Calculate the value after 5 years of an investment of $4000 at 12% p.a., with interest
compounded annually.
2 Calculate the value to which each of the following compound interest investments
will grow.
a $5000 at 6% p.a. for 5 years, with interest calculated annually
b $12 000 at 12% p.a. for 3 years, with interest calculated annually
c $4500 at 8% p.a. for 4 years, with interest compounded six-monthly
d $3000 at 9.6% p.a. for 3 years, with interest compounded six-monthly
e $15 000 at 8.4% p.a. for 2 years, with interest compounded quarterly
f $2950 at 6% p.a. for 3 years, with interest compounded monthly.
WORKED
Example
18
WORKED
Example
19
3 At the end of each year for four years Rodney invests $1000 into an investment fund
that pays 7.5% p.a. interest, compounded annually. By calculating each investment of
$1000 separately, use the compound interest formula to calculate the future value of
Rodney’s investment after four years.
4 At the end of every six months Jason invests $800 into a retirement fund which pays
interest at 6% p.a., with interest compounded six-monthly. Jason does this for
25 years. Calculate the future value of Jason’s annuity after 25 years.
5 Calculate the future value of each of the following annuities on maturity.
a $400 invested at the end of every six months for 12 years at 12% p.a. with interest
compounded six-monthly
b $1000 invested at the end of every quarter for 5 years at 8% p.a. with interest compounded every quarter
c $2500 invested at the end of each quarter at 7.2% p.a. for 4 years with interest
compounded quarterly
d $1000 invested at the end of every month for 5 years at 6% p.a. with interest compounded monthly
6 multiple choice
Tracey invests $500 into a fund at the end of each year for 20 years. The fund pays
12% p.a. interest, compounded annually. The total amount of interest that Tracey
earns on this fund investment is:
A $4323.15
B $4823.23
C $26 026.22
D $36 026.22
WORKED
Example
20
WORKED
Example
21
7 Thomas has the goal of saving $400 000 for his retirement in 25 years. If the best
interest rate that Thomas can obtain is 10% p.a., with interest compounded annually,
calculate the amount of each annual contribution that Thomas will need to make.
8 Calculate the amount of each annual contribution needed to obtain each of the
following amounts.
a $25 000 in 5 years at 5% p.a., with interest compounded annually
b $100 000 in 10 years at 7.5% p.a., with interest compounded annually
c $500 000 in 40 years at 8% p.a., with interest compounded annually
9 Use the table of future values on page 90 to determine the future value of an annuity
of $800 invested per year for 5 years at 9% p.a., with interest compounded annually.
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M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d
10 Use the table of future values on page 90 to determine the future value of each of the
following annuities.
a $400 invested per year for 3 years at 10% p.a., with interest compounded annually
b $2250 invested per year for 8 years at 8% p.a., with interest compounded annually
c $625 invested per year for 10 years at 4% p.a., with interest compounded annually
d $7500 invested per year for 7 years at 6% p.a., with interest compounded annually
11 Samantha invests $500 every 6 months for 5 years into an annuity at 8% p.a., with
interest compounded every 6 months.
a What is the interest rate per interest period?
b How many interest periods are there in Samantha’s annuity?
c Use the table to calculate the future value of Samantha’s annuity.
12 Use the table to calculate the future value of each of the following annuities.
a $400 invested every 6 months for 4 years at 14% p.a., with interest compounded
six-monthly
b $600 invested every 3 months for 2 years at 12% p.a., with interest compounded
quarterly
c $100 invested every month for 5 years at 10% p.a., with interest compounded sixmonthly
13 Use the table of future values to
determine whether an annuity at 5% p.a.
for 6 years or an annuity at 6% p.a. for 5
years will produce the greatest financial
outcome. Explain your answer.
14 multiple choice
Use the table of future values to determine which of the following annuities
will have the greatest financial outcome.
A 6% p.a. for 8 years, with interest
compounded annually
B 8% p.a. for 6 years, with interest
compounded annually
C 7% p.a. for 7 years, with interest
compounded annually
D 10% p.a. for 5 years, with interest
compounded six-monthly
WORKED
Example
22
Work
ET
SHE
2.2
15 Use the table of present values on page 92
to determine the present value of an
annuity of $1250 per year for 8 years
invested at 9% p.a.
16 Use the table of present values to
determine the present value of each of the
following annuities.
a $450 per year for 5 years at 7% p.a., with interest compounded annually
b $2000 per year for 10 years at 10% p.a., with interest compounded annually
c $850 per year for 6 years at 4% p.a., with interest compounded annually
d $3000 per year for 8 years at 9% p.a., with interest compounded annually
43. Maths A Yr 12 - Ch. 02 Page 95 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
95
2
1 Calculate the amount of interest earned on $10 000 invested for 10 years at 10% p.a.,
with interest compounding annually.
2 Calculate the future value of an annuity of $1000 invested every year for 10 years at
10% p.a., with interest compounding annually.
3 Calculate the future value of an annuity where $200 is invested each month for
5 years at 5% p.a., with interest compounding quarterly.
4 Calculate the amount of each annual contribution to an annuity that will have a future
value of $15 000 if the investment is for 8 years at 7.5% p.a., with interest
compounding annually.
5 Calculate the amount of each annual contribution to an annuity that will have a future
value of $500 000 in 25 years when invested at 10% p.a., with interest compounding
annually.
6 Calculate the present value of an annuity that will have a future value of $50 000 in
10 years at 10% p.a., with interest compounding annually.
7 Calculate the present value of an annuity that will have a future value of $1 000 000 in
40 years at 10% p.a., with interest compounding annually.
8 Calculate the present value of an annuity where annual contributions of $1000 are
made at 10% p.a., with interest compounding annually for 20 years.
9 Use the table on page 90 to find the future value of $1 invested at 16% p.a. for
4 years, with interest compounding twice annually.
10 Use the answer to question 9 to calculate the future value of an annuity of $1250
every six months for 4 years, with interest of 16% p.a., compounding twice annually.
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A growing investment
Bindi is investing $20 000 in a fixed term deposit earning 6% p.a. interest. When
Bindi has $30 000 she intends to put a deposit on a house.
1 Write the compound interest function that will model the growth of Bindi’s
investment.
2 Use your graphics calculator to graph this function
3 Find the length of time (correct to the nearest year) that it will take for Bindi’s
investment to grow to $30 000.
4 Suppose that Bindi had been able to invest at 8% p.a. How much quicker would
Bindi’s investment have grown to the $30 000 she needs?
5 Calvin has $15 000 to invest. Find the interest rate at which Calvin must invest
his money, if his investment is to grow to $30 000 in less than 8 years.
t i gat
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M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d
summary
Inflation
• The price of goods and services rises from year to year. To predict the future price
of an item we can use the compound interest formula taking the rate of inflation as R.
• The same method is used to predict the future value of collectibles and of
memorabilia, which tend to rise at a rate greater than inflation.
Modelling depreciation
• Depreciation can be calculated in two ways. The depreciation can be straight line
depreciation or declining balance depreciation.
• Straight line depreciation occurs when the value of an asset decreases by a constant
amount each year. The graph of salvage value is a straight line, the vertical intercept
is the purchase price and the gradient is the negative of the annual depreciation.
• Declining balance depreciation occurs when the salvage value of the item is a
percentage of the previous year’s value. The graph of a declining balance
depreciation will be an exponential decay graph.
Straight line depreciation
• The salvage value of an asset under straight line depreciation can be calculated
using the formula S = Vo − Dn, where S is the salvage value, Vo is the purchase price
of the asset, D is the amount of depreciation per period and n is the number of
periods of depreciation.
• Values of Vo, D or n can be calculated by substitution and solving the equation.
Declining balance method of depreciation
• Under declining balance depreciation the salvage value of an asset can be
R T
calculated using the formula S = Vo 1 – -------- , where R is the percentage
100
depreciation per year.
• To calculate the amount by which an asset depreciates in a year we subtract the
salvage value at the end of the year from the salvage value at the beginning of the year.
Depreciation tables
• Depreciation can be compared using either a table or a graph.
• Tax deductions are allowed for depreciation of assets that are used as part of
earning an income.
• A depreciation schedule is used to calculate tax deductions over a period of years.
Future value of an annuity
• An annuity is where regular equal contributions are made to an investment. The
interest on each contribution compounds as additions are made to the annuity.
• The future value of an annuity is the value that the annuity will have at the end of a
fixed period of time.
Use of tables
• A table can be used to find the present or future value of an annuity.
• The table shows the present or future value of $1 under an annuity.
• The present or future value of $1 must be multiplied by the contribution per period
to calculate its present or future value.
45. Maths A Yr 12 - Ch. 02 Page 97 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
97
CHAPTER
review
1 A walkman is currently priced at $80. If the current inflation rate is 4.3%, estimate the price
of the walkman after one year.
2A
2 It is predicted that the average inflation rate for the next five years will be 3.7%. If a
skateboard currently costs $125, estimate the cost of the skateboard after five years.
2A
3 In 1975, Cherie bought a limited edition photograph autographed by Sir Donald Bradman
for $120. If the photograph appreciates in value by 15% per annum, calculate the value of
the photograph in 2005 (to the nearest $100).
2A
4 The table below shows the depreciating value of a yacht.
2B
Age (years)
Value ($)
New (0)
200 000
1
180 000
2
160 000
3
140 000
4
120 000
5
100 000
a Draw a graph of the value of the yacht against its age.
b Write a function for the value of the yacht.
5 The table below shows the depreciating value of a racing bike.
Age (years)
Value ($)
New (0)
3500
1
3250
2
3000
3
2750
4
2500
5
2250
a Draw a graph of the value of the bike against age.
b Write a function for the straight line depreciation.
c Use your graph to estimate the value of the bike after 9 years.
2B
46. Maths A Yr 12 - Ch. 02 Page 98 Wednesday, September 11, 2002 3:52 PM
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M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d
2B
6 The function V = 15 000 − 900A shows the value, V, of a motorcycle when it is
A years old.
a Draw a graph of this function.
b Use the graph to calculate the value of the motorcycle after 5 years.
c After how many years would the motorcycle be ‘written off’ (the value of the
motorcycle become $0)?
2B
7 The table below shows the declining value of a 4-wheel drive.
Age (years)
Value ($)
New (0)
60 000
1
48 000
2
38 400
3
30 720
4
24 576
5
19 660
a Plot the points as given in the table and then draw a curve of best fit to graph the
depreciation of the 4-wheel drive.
b Use your graph to estimate the value of the 4-wheel drive after 10 years.
c After what number of years will the value of the 4-wheel drive fall below $10 000?
2B
8 A laundry buys dry-cleaning equipment for $8000. Each year the equipment depreciates by
25% of the previous year’s value. Calculate the value of the equipment at the end of the first
five years and use the results to draw a graph of the depreciation.
2C
9 The purchase price of a car is $32 500. The car depreciates by $3250 each year. Use the
formula S = Vo − Dn to calculate the salvage value of the car after 8 years.
2C
10 Calculate the salvage value of an asset:
a after 6 years, that was purchased for $4000
and depreciates by $450 each year
b after 10 years, that was purchased for
$75 000 and depreciates by $6000 each
year
c after 9 years, that was purchased for
$640 000 and depreciates by $45 000 each
year.
2C
11 A movie projector is purchased by a cinema
for $30 000. The projector depreciates by
$2500 each year. Calculate the length of time
it takes for the projector to be written off.
2C
12 A camera that was purchased new for $1500
has a salvage value of $500 four years later.
Calculate the annual amount of depreciation
on the camera.
47. Maths A Yr 12 - Ch. 02 Page 99 Wednesday, September 11, 2002 3:52 PM
Chapter 2 Appreciation and depreciation
99
13 Arthur buys a car for $25 000. The depreciation on the car is $2250 each year. He decides
that he will trade the car in on a new car in the final year before the salvage value falls
below $10 000. When will Arthur trade the car in?
2C
14 The purchase price of a mobile home is $40 000. The value of the mobile home depreciates
by 15% p.a. By calculating the value of the mobile home at the end of each year, find the
salvage value of the mobile home after 4 years. (Give your answer correct to the nearest $1.)
2D
15 Use the declining balance depreciation formula to calculate the salvage value after 7 years of a
crop duster that was purchased for $850 000 and depreciates at 8% p.a. (Give your answer correct
to the nearest $1000.)
2D
16 Calculate the salvage value of an asset (correct to the nearest $10) with a purchase price of:
a $40 000 that depreciates at 10% p.a. for 5 years
b $1500 that depreciates at 4% p.a. for 10 years
c $180 000 that depreciates at 12.5% p.a. for 15 years
d $4.5 million that depreciates at 40% p.a. for 10 years
-e $250 000 that depreciates at 33 1 % p.a. for 4 years.
3
2D
17 A company buys a new bus for $600 000. The company keeps buses for 10 years then trades
them in on a new bus. The estimate is that the value of the bus will depreciate by 12% p.a.
Calculate:
a the amount the owner can expect as a
trade-in for the bus in 10 years
b the amount by which the bus will
depreciate in 10 years.
2D
18 A company has office equipment that
is valued at $100 000. The value of
the equipment can be depreciated at
$10 000 each year or by 15% p.a.
a Draw a table that will show the
salvage value of the office
equipment for the first ten years.
b Draw a graph of the depreciating
value of the equipment under both
methods of depreciation.
2E
48. Maths A Yr 12 - Ch. 02 Page 100 Wednesday, September 11, 2002 3:52 PM
100
2E
M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d
19 A personal computer is purchased for $4500. A tax deduction for depreciation of the
computer is allowed at the rate of 33% p.a. When the value of the computer falls below
$1000, the computer can be written off. Copy and complete the table below.
Year
Salvage value ($)
Tax deduction ($)
1
2
3
4
5
2F
20 Use the table of future values of $1 on page 90 to
calculate the future value of an annuity of $4000
deposited per year at 7% p.a. for 8 years, with
interest compounded annually.
2F
21 Use the table of future values of $1 to calculate the future value of
the following annuities.
a $750 invested per year for 5 years at 8% p.a., with interest compounded annually
b $3500 invested every six months for 4 years at 12% p.a., with interest compounded
six-monthly
c $200 invested every 3 months for 2 years at 16% p.a., with interest compounded
quarterly
d $1250 invested every month for 3 years at 10% p.a., with interest compounded
six-monthly
2F
22 Use the table of present values of $1 on page 92 to calculate the present value of an annuity
of $500 invested per year for 6 years at 9% p.a., with interest compounded
annually.
2F
23 Use the table of present values to calculate the present value of each of the following
annuities.
a $400 invested per year for 5 years at 10% p.a., with interest compounded annually
b $2000 invested every six months for 5 years at 14% p.a., with interest compounded
six-monthly
-c $500 invested every three months for 2 1 years at 16% p.a., with interest compounded
2
quarterly
d $300 invested every month for 4 years at 12% p.a., with interest compounded half-yearly
CHAPTER
test
yourself
2