The document discusses ratios and proportions. It defines a ratio as two related quantities stated side by side, and gives an example of a 3:4 ratio of eggs to flour in a recipe. It explains how to write ratios as fractions and set up proportion equations. Proportions are equal ratios, like 3:4 being proportional to 6:8. The document solves sample proportion word problems, like finding the number of eggs needed given 10 cups of flour using a proportion equation.
The document discusses proportions and ratios. It defines a ratio as two related quantities stated side by side. It provides an example of a recipe ratio of 3 eggs to 4 cups of flour as 3:4. It explains how to set up proportional equations from word problems by ensuring quantities of the same type occupy the same position in fractions. It solves examples, finding x eggs needed given 10 cups of flour is 7.5 eggs. It also solves a map ratio problem of 4 inches on a map equaling 21 miles in real distance.
This document discusses the procedure of cross multiplication. It explains that cross multiplication can be used to rewrite ratios involving fractions as ratios of whole numbers. This is done by writing the fractions as ratios, then multiplying the denominators diagonally to obtain two new numbers. The ratio between these new numbers represents the ratio in whole integers. An example demonstrates taking a ratio of 3/4 to 2/3 and rewriting it as 9:8 using cross multiplication. The document also notes that cross multiplication can be used to compare two fractions, with the fraction corresponding to the larger product being the larger fraction.
The document discusses ratios, proportions, and how to solve proportional equations. It defines a ratio as two related quantities stated side by side, and gives the example of a 3:4 ratio of eggs to flour in a recipe. Proportions are defined as equal ratios. The key steps to solve proportional equations are: 1) write the ratios as fractions set equal to each other, 2) use cross-multiplication to convert the proportions into regular equations, and 3) solve the resulting equation using algebraic techniques. An example problem demonstrates these steps to solve a proportional equation for the variable x.
13 fractions, multiplication and divisin of fractionsalg1testreview
The document defines fractions as numbers of the form p/q where p and q are whole numbers not equal to 0. Fractions represent parts of a whole, for example the fraction 3/6 represents 3 out of 6 equal slices of a pizza. The top number is called the numerator and represents the number of parts, while the bottom number is the denominator and represents the total number of equal parts the whole was divided into. Examples are provided to demonstrate calculating fractional amounts of groups of items. It is noted that fractions with a denominator of 0 are undefined in mathematics.
1 f3 multiplication and division of fractionsmath123a
The document discusses fractions and their multiplication and division. It defines fractions as parts of a whole, with the numerator representing the parts and denominator representing the total parts. Examples show how to multiply and divide fractions by cancelling common factors or dividing the whole number by the denominator. Phrases like "of" are translated to fraction multiplication problems.
The document discusses fractions and their properties. It defines fractions as numbers of the form p/q where p and q are natural numbers. Fractions represent parts of a whole, for example 3/6 represents 3 out of 6 equal slices of a pizza. The numerator is the number on top and represents the parts, while the denominator on bottom represents the total parts of the whole. Equivalent fractions like 1/2, 2/4, and 3/6 represent the same quantity. Dividing the numerator and denominator by a common factor results in an equivalent fraction. The denominator of a fraction cannot be zero, as this results in an undefined fraction.
The document discusses mathematical expressions and how to combine them. It defines an expression as a calculation procedure written with numbers, variables, and operations. Expressions have terms, with the x-term being the variable term and the number term being the constant. To combine expressions, like terms are combined in the same way numbers are, while unlike terms cannot be combined. The simplest expressions are linear expressions of the form ax + b.
The document discusses proportions and ratios. It defines a ratio as two related quantities stated side by side. It provides an example of a recipe ratio of 3 eggs to 4 cups of flour as 3:4. It explains how to set up proportional equations from word problems by ensuring quantities of the same type occupy the same position in fractions. It solves examples, finding x eggs needed given 10 cups of flour is 7.5 eggs. It also solves a map ratio problem of 4 inches on a map equaling 21 miles in real distance.
This document discusses the procedure of cross multiplication. It explains that cross multiplication can be used to rewrite ratios involving fractions as ratios of whole numbers. This is done by writing the fractions as ratios, then multiplying the denominators diagonally to obtain two new numbers. The ratio between these new numbers represents the ratio in whole integers. An example demonstrates taking a ratio of 3/4 to 2/3 and rewriting it as 9:8 using cross multiplication. The document also notes that cross multiplication can be used to compare two fractions, with the fraction corresponding to the larger product being the larger fraction.
The document discusses ratios, proportions, and how to solve proportional equations. It defines a ratio as two related quantities stated side by side, and gives the example of a 3:4 ratio of eggs to flour in a recipe. Proportions are defined as equal ratios. The key steps to solve proportional equations are: 1) write the ratios as fractions set equal to each other, 2) use cross-multiplication to convert the proportions into regular equations, and 3) solve the resulting equation using algebraic techniques. An example problem demonstrates these steps to solve a proportional equation for the variable x.
13 fractions, multiplication and divisin of fractionsalg1testreview
The document defines fractions as numbers of the form p/q where p and q are whole numbers not equal to 0. Fractions represent parts of a whole, for example the fraction 3/6 represents 3 out of 6 equal slices of a pizza. The top number is called the numerator and represents the number of parts, while the bottom number is the denominator and represents the total number of equal parts the whole was divided into. Examples are provided to demonstrate calculating fractional amounts of groups of items. It is noted that fractions with a denominator of 0 are undefined in mathematics.
1 f3 multiplication and division of fractionsmath123a
The document discusses fractions and their multiplication and division. It defines fractions as parts of a whole, with the numerator representing the parts and denominator representing the total parts. Examples show how to multiply and divide fractions by cancelling common factors or dividing the whole number by the denominator. Phrases like "of" are translated to fraction multiplication problems.
The document discusses fractions and their properties. It defines fractions as numbers of the form p/q where p and q are natural numbers. Fractions represent parts of a whole, for example 3/6 represents 3 out of 6 equal slices of a pizza. The numerator is the number on top and represents the parts, while the denominator on bottom represents the total parts of the whole. Equivalent fractions like 1/2, 2/4, and 3/6 represent the same quantity. Dividing the numerator and denominator by a common factor results in an equivalent fraction. The denominator of a fraction cannot be zero, as this results in an undefined fraction.
The document discusses mathematical expressions and how to combine them. It defines an expression as a calculation procedure written with numbers, variables, and operations. Expressions have terms, with the x-term being the variable term and the number term being the constant. To combine expressions, like terms are combined in the same way numbers are, while unlike terms cannot be combined. The simplest expressions are linear expressions of the form ax + b.
The document discusses the mathematical procedure of cross multiplication. It explains that cross multiplication can be used to compare two fractions, with the fraction having the larger product being the greater value. It also describes how cross multiplication allows rewriting fractional ratios in whole numbers. An example shows converting a ratio of 3/4 cups of sugar to 2/3 cups of flour into a whole number ratio of 9:8.
The document discusses cross multiplication, which is a procedure for working with two fractions. It involves multiplying the denominators of the fractions diagonally to obtain the product in the numerator and denominator. This allows fractions to be added, subtracted, or compared by looking at which side has the larger product. Examples are provided to illustrate how to use cross multiplication to rewrite fractional ratios as whole number ratios, add or subtract fractions, and determine if two fractions are equal.
1. Eye color is determined by multiple alleles at a single gene locus. There are multiple versions (alleles) of the eye color gene - blue, green, and brown.
2. Polygenic inheritance is when multiple genes contribute to a single trait. For traits determined by polygenic inheritance, there are many possible combinations of alleles across several gene loci that result in a range of phenotypes.
3. Both multiple allele inheritance and polygenic inheritance can contribute to natural variation in human eye color. Multiple alleles at a single gene locus explain some of the variation, but many genes likely have small effects, making polygenic inheritance important as well.
The document discusses rules for multiplying fractions. It states that to multiply fractions, one should multiply the numerators and multiply the denominators, canceling terms when possible. It then provides examples, such as multiplying 12/25 * 15/8, simplifying to 9/10. It also notes that word problems involving fractions of a quantity can often be solved by translating them into fraction multiplications.
This document provides a lesson on evaluating algebraic expressions and inequalities. It includes examples of writing inequalities to represent word phrases, solving inequalities, graphing inequality solution sets on number lines, writing compound inequalities, and a quiz. The document covers key concepts such as using symbols like <, >, ≤, ≥ to represent inequalities and defining the solution set of an inequality.
The document defines key vocabulary terms related to functions, including function, domain, range, linear function, and vertical line test. It provides an example of writing a linear function to represent the relationship between the number of cups of coffee made and the total number of spoonfuls needed. The function is defined as s=2c+5, and a table is made to graph the relationship between c and s.
Comparing and ordering_fractions_powerpointNeilfieOrit2
This document provides instruction on comparing and ordering fractions. It defines key fraction terms like numerator, denominator, and least common denominator. It includes examples of comparing fractions with like and unlike denominators using <, >, = symbols. It also demonstrates ordering fractions from least to greatest by finding a common denominator. Guided and independent practice problems are provided for students to compare and order fractions.
Ch 7 mathematics class 7 ratio and proportion nandini44
This document provides an introduction to ratios and proportions. It defines key terms like ratio, proportion, and equivalent ratios. It explains how to calculate ratios using the ratio formula a:b and how equivalent ratios are found by multiplying or dividing both terms by the same number. It also discusses continued proportions and using the unitary method to solve word problems involving ratios. Examples are provided to demonstrate finding equivalent ratios, solving proportion problems, and using the unitary method to determine unknown values. In the end, it provides exercises for students to practice applying these ratio and proportion concepts.
The document provides an outline of topics covered in Chapter 6 of The Pharmacy Technician 4E including basic pharmaceutical measurements, calculations, and conversions. Key areas discussed include numbers, fractions, decimals, ratios, proportions, percents, and metric and household conversions. Examples are provided for calculating common denominators, multiplying fractions, setting up and solving proportions, and converting between ratios, percents, and fractions.
The document defines key terms related to functions, including domain, range, and linear functions. It provides an example of writing a function to represent the relationship between the number of cups of coffee made and the total number of spoonfuls needed. The example function is f(c) = 2c + 5, where c is the independent variable representing cups and s is the dependent variable for spoonfuls. A table of values and graph of this linear function are included.
Lesson 1.9 b multiplication and division of rational numbersJohnnyBallecer
The document provides steps for multiplying rational numbers:
1) Write rational numbers in the form a/b ∙ c/d
2) Multiply the numerators and denominators separately
3) Simplify the resulting fraction by reducing if possible
The examples demonstrate changing mixed numbers to improper fractions, multiplying, and simplifying the final answer.
This document provides an overview and objectives for a continuing education learning module on pharmacy calculations for pharmacists and pharmacy technicians. It covers reviewing basic mathematics concepts like numerals, numbers, fractions, decimals, percentages, and units of measure in both the metric and other common systems. It also reviews ratios, proportions, concentration, dilution, and performing intravenous drip rate calculations. A number of example problems and solutions are provided to illustrate these concepts. The document is intended to be an educational resource for reviewing and practicing essential skills for accurately performing pharmacy calculations.
The document provides definitions and examples for various math terms related to numbers, operations, measurement, geometry, and problem solving strategies. It includes 3 or fewer sentence summaries of rational numbers, place value, ordering numbers, estimating, factors and multiples, equivalence, symbols for relationships, fractions, decimals, angles, shapes, and units of measurement. Visual examples and links are provided with several terms.
The document provides examples and definitions related to ratios, proportions, and percents. It defines a ratio as a comparison of two quantities by division, which can be expressed as a fraction. It gives examples of writing ratios as fractions and finding unit rates. Unit rates compare quantities in different units and allow comparing values such as price per ounce when shopping. The document also discusses converting between rates and ratios using dimensional analysis.
This document provides instructions for two experiments:
1) To determine and compare the energy content of two food samples (PA and PB) by measuring the temperature increase of water after burning each food sample. Students are to set up the apparatus, record initial and final temperatures, calculate energy values, and analyze the results.
2) To simulate sex determination in offspring using white, red and black buttons representing ovum/sperm chromosomes. Students mix the buttons in a bag and record the combinations drawn to complete a table showing the possible sex outcomes. They then answer questions about sex determination based on the experiment.
1) The document discusses fractions, alligation, and mixtures. It provides examples of using chain rules or fractions to solve multi-step word problems involving changes in quantities over time or mixing of substances.
2) Alligation problems deal with mixing different quantities or compounds in specific ratios to form a mixture. The rule for alligation uses a formula to determine the ratio of quantities that need to be mixed.
3) Examples demonstrate using alligation to find numbers of items or volumes when quantities change over multiple steps due to reductions in amounts, replacements, or repeated operations.
This document contains notes from a math class that covered several topics:
- Warm-up with Khan Academy on fractions and percents
- Reviewing fraction and percent conversions
- Solving absolute value equations
- The notes provide examples and steps for solving absolute value equations, changing fractions to decimals, and ordering fractions using cross-multiplication. Examples are worked through to demonstrate the process.
Suppose you are interested in selecting a group of six households Experience...pinck336896
FOR MORE CLASSES VISIT
www.tutorialoutlet.com
1. Suppose that in a large metropolitan area, 82% of all households have cable tv. Suppose you are interested in selecting a group of six households from this area. Let X be the number of
households in a group of six households from this area that have cable tv. For what proportion of
groups will at most three of the households have cable tv?
3 multiplication and division of fractions 125sTzenma
The document discusses multiplying and dividing fractions. It provides the rule that to multiply fractions, multiply the numerators and denominators, cancelling terms when possible. It then gives examples of multiplying fractions by reducing or cancelling terms first before multiplying. It notes that fractional expressions like "x of y" translate to fraction multiplications.
This document defines ratios and rates. Ratios compare two quantities using division and can be written as fractions, with a colon, or in words. Rates also compare two quantities but the quantities have different units of measure. A unit rate is when the second term of a rate is 1 unit. Examples are provided for writing ratios and rates as fractions, with colons, and in words. Students are assigned practice problems writing ratios and rates in different formats.
The document discusses techniques for combining fractions with opposite denominators. It explains that we can multiply the numerator and denominator by -1 to change the denominator to its opposite. It provides examples of switching fractions to their opposite denominators and combining fractions with opposite denominators by first switching one denominator so they are the same. It also discusses an alternative approach of pulling out a "-" from the denominator and passing it to the numerator when switching denominators, ensuring the leading term is positive for polynomial denominators.
The document discusses solving rational equations by clearing fractions. It explains that to solve an equation with fractional terms, we first multiply both sides of the equation by the lowest common denominator (LCD) of the fractions. This clears the fractions by distributing the LCD. Then the resulting equation can be solved using normal algebraic techniques. Two examples are provided to demonstrate this process.
The document discusses the mathematical procedure of cross multiplication. It explains that cross multiplication can be used to compare two fractions, with the fraction having the larger product being the greater value. It also describes how cross multiplication allows rewriting fractional ratios in whole numbers. An example shows converting a ratio of 3/4 cups of sugar to 2/3 cups of flour into a whole number ratio of 9:8.
The document discusses cross multiplication, which is a procedure for working with two fractions. It involves multiplying the denominators of the fractions diagonally to obtain the product in the numerator and denominator. This allows fractions to be added, subtracted, or compared by looking at which side has the larger product. Examples are provided to illustrate how to use cross multiplication to rewrite fractional ratios as whole number ratios, add or subtract fractions, and determine if two fractions are equal.
1. Eye color is determined by multiple alleles at a single gene locus. There are multiple versions (alleles) of the eye color gene - blue, green, and brown.
2. Polygenic inheritance is when multiple genes contribute to a single trait. For traits determined by polygenic inheritance, there are many possible combinations of alleles across several gene loci that result in a range of phenotypes.
3. Both multiple allele inheritance and polygenic inheritance can contribute to natural variation in human eye color. Multiple alleles at a single gene locus explain some of the variation, but many genes likely have small effects, making polygenic inheritance important as well.
The document discusses rules for multiplying fractions. It states that to multiply fractions, one should multiply the numerators and multiply the denominators, canceling terms when possible. It then provides examples, such as multiplying 12/25 * 15/8, simplifying to 9/10. It also notes that word problems involving fractions of a quantity can often be solved by translating them into fraction multiplications.
This document provides a lesson on evaluating algebraic expressions and inequalities. It includes examples of writing inequalities to represent word phrases, solving inequalities, graphing inequality solution sets on number lines, writing compound inequalities, and a quiz. The document covers key concepts such as using symbols like <, >, ≤, ≥ to represent inequalities and defining the solution set of an inequality.
The document defines key vocabulary terms related to functions, including function, domain, range, linear function, and vertical line test. It provides an example of writing a linear function to represent the relationship between the number of cups of coffee made and the total number of spoonfuls needed. The function is defined as s=2c+5, and a table is made to graph the relationship between c and s.
Comparing and ordering_fractions_powerpointNeilfieOrit2
This document provides instruction on comparing and ordering fractions. It defines key fraction terms like numerator, denominator, and least common denominator. It includes examples of comparing fractions with like and unlike denominators using <, >, = symbols. It also demonstrates ordering fractions from least to greatest by finding a common denominator. Guided and independent practice problems are provided for students to compare and order fractions.
Ch 7 mathematics class 7 ratio and proportion nandini44
This document provides an introduction to ratios and proportions. It defines key terms like ratio, proportion, and equivalent ratios. It explains how to calculate ratios using the ratio formula a:b and how equivalent ratios are found by multiplying or dividing both terms by the same number. It also discusses continued proportions and using the unitary method to solve word problems involving ratios. Examples are provided to demonstrate finding equivalent ratios, solving proportion problems, and using the unitary method to determine unknown values. In the end, it provides exercises for students to practice applying these ratio and proportion concepts.
The document provides an outline of topics covered in Chapter 6 of The Pharmacy Technician 4E including basic pharmaceutical measurements, calculations, and conversions. Key areas discussed include numbers, fractions, decimals, ratios, proportions, percents, and metric and household conversions. Examples are provided for calculating common denominators, multiplying fractions, setting up and solving proportions, and converting between ratios, percents, and fractions.
The document defines key terms related to functions, including domain, range, and linear functions. It provides an example of writing a function to represent the relationship between the number of cups of coffee made and the total number of spoonfuls needed. The example function is f(c) = 2c + 5, where c is the independent variable representing cups and s is the dependent variable for spoonfuls. A table of values and graph of this linear function are included.
Lesson 1.9 b multiplication and division of rational numbersJohnnyBallecer
The document provides steps for multiplying rational numbers:
1) Write rational numbers in the form a/b ∙ c/d
2) Multiply the numerators and denominators separately
3) Simplify the resulting fraction by reducing if possible
The examples demonstrate changing mixed numbers to improper fractions, multiplying, and simplifying the final answer.
This document provides an overview and objectives for a continuing education learning module on pharmacy calculations for pharmacists and pharmacy technicians. It covers reviewing basic mathematics concepts like numerals, numbers, fractions, decimals, percentages, and units of measure in both the metric and other common systems. It also reviews ratios, proportions, concentration, dilution, and performing intravenous drip rate calculations. A number of example problems and solutions are provided to illustrate these concepts. The document is intended to be an educational resource for reviewing and practicing essential skills for accurately performing pharmacy calculations.
The document provides definitions and examples for various math terms related to numbers, operations, measurement, geometry, and problem solving strategies. It includes 3 or fewer sentence summaries of rational numbers, place value, ordering numbers, estimating, factors and multiples, equivalence, symbols for relationships, fractions, decimals, angles, shapes, and units of measurement. Visual examples and links are provided with several terms.
The document provides examples and definitions related to ratios, proportions, and percents. It defines a ratio as a comparison of two quantities by division, which can be expressed as a fraction. It gives examples of writing ratios as fractions and finding unit rates. Unit rates compare quantities in different units and allow comparing values such as price per ounce when shopping. The document also discusses converting between rates and ratios using dimensional analysis.
This document provides instructions for two experiments:
1) To determine and compare the energy content of two food samples (PA and PB) by measuring the temperature increase of water after burning each food sample. Students are to set up the apparatus, record initial and final temperatures, calculate energy values, and analyze the results.
2) To simulate sex determination in offspring using white, red and black buttons representing ovum/sperm chromosomes. Students mix the buttons in a bag and record the combinations drawn to complete a table showing the possible sex outcomes. They then answer questions about sex determination based on the experiment.
1) The document discusses fractions, alligation, and mixtures. It provides examples of using chain rules or fractions to solve multi-step word problems involving changes in quantities over time or mixing of substances.
2) Alligation problems deal with mixing different quantities or compounds in specific ratios to form a mixture. The rule for alligation uses a formula to determine the ratio of quantities that need to be mixed.
3) Examples demonstrate using alligation to find numbers of items or volumes when quantities change over multiple steps due to reductions in amounts, replacements, or repeated operations.
This document contains notes from a math class that covered several topics:
- Warm-up with Khan Academy on fractions and percents
- Reviewing fraction and percent conversions
- Solving absolute value equations
- The notes provide examples and steps for solving absolute value equations, changing fractions to decimals, and ordering fractions using cross-multiplication. Examples are worked through to demonstrate the process.
Suppose you are interested in selecting a group of six households Experience...pinck336896
FOR MORE CLASSES VISIT
www.tutorialoutlet.com
1. Suppose that in a large metropolitan area, 82% of all households have cable tv. Suppose you are interested in selecting a group of six households from this area. Let X be the number of
households in a group of six households from this area that have cable tv. For what proportion of
groups will at most three of the households have cable tv?
3 multiplication and division of fractions 125sTzenma
The document discusses multiplying and dividing fractions. It provides the rule that to multiply fractions, multiply the numerators and denominators, cancelling terms when possible. It then gives examples of multiplying fractions by reducing or cancelling terms first before multiplying. It notes that fractional expressions like "x of y" translate to fraction multiplications.
This document defines ratios and rates. Ratios compare two quantities using division and can be written as fractions, with a colon, or in words. Rates also compare two quantities but the quantities have different units of measure. A unit rate is when the second term of a rate is 1 unit. Examples are provided for writing ratios and rates as fractions, with colons, and in words. Students are assigned practice problems writing ratios and rates in different formats.
The document discusses techniques for combining fractions with opposite denominators. It explains that we can multiply the numerator and denominator by -1 to change the denominator to its opposite. It provides examples of switching fractions to their opposite denominators and combining fractions with opposite denominators by first switching one denominator so they are the same. It also discusses an alternative approach of pulling out a "-" from the denominator and passing it to the numerator when switching denominators, ensuring the leading term is positive for polynomial denominators.
The document discusses solving rational equations by clearing fractions. It explains that to solve an equation with fractional terms, we first multiply both sides of the equation by the lowest common denominator (LCD) of the fractions. This clears the fractions by distributing the LCD. Then the resulting equation can be solved using normal algebraic techniques. Two examples are provided to demonstrate this process.
The document discusses using rational equations to solve word problems involving costs shared among groups of people. It provides an example where a taxi costs $20 to rent for a group of x people, with the cost shared equally. If one person leaves the group, the remaining people each pay $1 more. Setting up the cost equations and subtracting them allows determining that x must equal 5 people for the equations to hold true. A table is suggested to organize calculations for different inputs when solving similar rational equation word problems.
The document discusses rules and procedures for adding and subtracting rational expressions. It states that fractions can only be directly added or subtracted if they have the same denominator. It provides an example of adding and subtracting fractions with the same denominator and simplifying the results. It also discusses how to convert fractions with different denominators to have a common denominator before adding or subtracting them, using the least common multiple (LCM) of the denominators. It provides an example problem that demonstrates converting fractions to equivalent forms with a specified common denominator.
The document discusses terms, factors, and cancellation in mathematics expressions. It defines a term as one or more quantities that are added or subtracted, and a factor as a quantity that is multiplied to other quantities. Cancellation can be used to simplify fractions by canceling common factors in the numerator and denominator. However, terms cannot be canceled as they represent distinct quantities being added or subtracted. Several examples demonstrate identifying terms and factors and applying cancellation when possible.
The document discusses two methods for simplifying complex fractions. A complex fraction is a fraction with fractions in the numerator or denominator. The first method reduces the complex fraction to an "easy" problem by combining the numerator and denominator terms into single fractions. The second method multiplies the lowest common denominator of all terms to both the numerator and denominator of the complex fraction. An example using each method is provided to demonstrate the simplification process.
The document discusses algebra of radicals. It provides rules for simplifying expressions involving radicals, such as √x·y = √x·√y and √x·√x = x. An example problem is worked through step-by-step, simplifying the expression 3√3 * √2* 2 * √2 * √3 * √2. The concept of conjugates is also introduced, where the conjugate of x + y is x - y.
The document describes the LCD (Least Common Denominator) method for working with fractions. It involves:
1) Finding the simplest expression that can clear all denominators when adding/subtracting fractions.
2) Multiplying each term by the LCD to clear denominators.
3) Solving problems like adding/subtracting fractions, solving fractional equations, and simplifying complex fractions using this method. Examples are provided to illustrate the step-by-step process.
The document discusses rational expressions, which are expressions of the form P/Q where P and Q are polynomials. Polynomials are expressions involving powers of variables with numerical coefficients. Rational expressions include polynomials as a special case where P is viewed as P/1. They may be written in expanded or factored form. The factored form is useful for determining the domain of a rational expression, solving equations involving rational expressions, evaluating expressions for given inputs, and determining the signs of outputs. The domain excludes values of x that make the denominator equal to 0.
The document discusses using the least common multiple (LCD) to convert fractions to equivalent whole numbers. It provides an example of finding the LCD of fractions {2/3, 5/8, 7/12, 3/4} which is 24. Multiplying each fraction by the LCD converts them to the whole number list {16, 15, 14, 18}. The fractions are then listed from largest to smallest as 3/4, 2/3, 5/8, 7/12.
The document discusses methods for finding the least common multiple (LCM) of numbers. It describes the searching method, which involves finding the smallest number that is a multiple of all the given numbers. An example finds the LCM of 18, 24, 16 to be 144. The document also introduces the construction method, which builds the minimum coverage needed to fulfill all requirements.
The document discusses applications of factoring expressions. The main purposes of factoring an expression E into a product E=AB is to utilize properties of multiplication. The most important application of factoring is to solve polynomial equations by setting each factor equal to 0 based on the zero-product property. Examples are provided to demonstrate solving polynomial equations by factoring, setting each factor equal to 0, and extracting the solutions.
The document discusses formulas for multiplying binomial expressions. It states that the conjugate of expressions like (A + B) is (A - B). The difference of squares formula is given as (A + B)(A - B) = A^2 - B^2. Examples of expanding expressions using this formula and the square formulas (A + B)^2 = A^2 + 2AB + B^2 and (A - B)^2 = A^2 - 2AB + B^2 are provided.
The document discusses exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It then provides examples to illustrate four rules for exponents:
1) The multiply-add rule: ANAK = AN+K
2) The divide-subtract rule: AN/AK = AN-K
3) The power-multiply rule: (AN)K = ANK
4) Additional rules including that A0 = 1 and A-K = 1/AK
1 2 2nd-degree equation and word problems-xmath123b
This document discusses solving polynomial equations by factoring. It provides an example of solving the equation x^2 - 2x - 3 = 0 by: 1) factoring the trinomial to get (x - 3)(x + 1) = 0, 2) setting each factor equal to 0 to get x - 3 = 0 and x + 1 = 0, and 3) extracting the solutions x = 3 and x = -1. It then works through another example of solving 2 = 2x^2 - 3x by similar factoring steps to get the solutions x = -1/2, x = 2. Finally, it introduces word problems that can be modeled by quadratic equations of the form AB = C.
1 5 multiplication and division of rational expressionsmath123b
The document discusses methods for finding the least common multiple (LCM) of numbers. It describes the searching method, which involves finding the smallest number that is a multiple of all the given numbers. An example finds the LCM of 18, 24, 16 to be 144. The document also introduces the construction method, which builds the minimum coverage needed to fulfill all requirements.
This document discusses integration and properties of integrals. It contains sections on definite integrals, integral properties in two parts, and two tables of indefinite integrals that contain formulas with constants to represent anti-derivatives. The document provides information on the fundamentals of integration through sections and tables.
3 1 the real line and linear inequalities-xmath123b
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on the real number line, with positive numbers to the right of zero and negative numbers to the left. An inequality relates the positions of two numbers on the real number line, with the number farther to the right said to be greater than the number to its left. The document provides examples of inequalities and how to represent sets of numbers using inequalities, such as all numbers between two values a and b. It also outlines steps for solving inequalities algebraically.
This document discusses indefinite integrals and anti-derivatives. It explains that the indefinite integral of a function is the set of all anti-derivatives of that function. It provides rules for finding the anti-derivative of a function given its derivative, such as adding 1 to the power and dividing by the new power. Examples are given of using these rules to find the function given its derivative.
1 s3 multiplication and division of signed numbersmath123a
The document discusses rules for multiplying signed numbers. It states that to multiply two signed numbers, multiply their absolute values and use rules to determine the sign of the product: two numbers with the same sign yield a positive product, while two numbers with opposite signs yield a negative product. It also discusses that in algebra, operations are often implied rather than written out. The even-odd rule determines the sign of products with multiple factors: an even number of negative factors yields a positive product, while an odd number yields a negative product.
This document discusses the procedure of cross multiplication. It explains that cross multiplication can be used to rewrite ratios involving fractions as ratios of whole numbers. It also describes how cross multiplication can be used to compare two fractions and determine which is larger. Finally, it notes that cross multiplication allows adding or subtracting two fractions by multiplying the numerators and denominators diagonally.
This document discusses the procedure of cross multiplication. It explains that cross multiplication is used to compare two fractions, with the fraction having the larger product corresponding to the larger fraction. It provides examples such as comparing 3/5 and 9/15 by cross multiplying to get 45 and 45, showing they are equal. The document also discusses using cross multiplication to rewrite fractional ratios in whole numbers and explains how the lowest common denominator can be used to clear denominators when adding more than two fractions.
The document discusses the procedure of cross multiplication. It explains that cross multiplication can be used to rewrite ratios involving fractions as ratios of whole numbers. This is done by writing the fractions as ratios, then multiplying the denominators diagonally to obtain two new numbers. The ratio between these new numbers represents the original fractional ratio using whole numbers. An example demonstrates taking a ratio of 3/4 cups sugar to 2/3 cups flour and rewriting it as 9:8 cups sugar to flour using cross multiplication. The document also notes cross multiplication can be used to compare two fractions, with the larger product corresponding to the larger fraction.
India scored 200 runs against Zimbabwe with a knock of 76* runs from Kohli. Zimbabwe was bowled out for 100 runs, so India won by a margin of 100 runs. However, the document goes on to explain how to calculate the exact margin of victory using ratios and proportions. It defines key terms like ratio, proportion, and equivalent ratios. It provides examples of how to find equivalent ratios and use unitary method to solve word problems involving ratios.
The document discusses ratios and how to represent relationships between quantities using ratios expressed in different ways like fractions, colon notation, or "to" language. It provides examples of how to set up tape diagrams and double number lines to represent ratios and use them to solve word problems involving converting between units given a ratio. Ratios can represent relationships between parts and parts, parts and wholes, or wholes and parts.
The document defines and provides examples of ratios. It discusses writing ratios in different forms, equivalent ratios, compound ratios, and solving word problems involving ratios. Examples are provided to show how to write ratios in 3 forms, simplify ratios, determine equivalent ratios, and solve ratio word problems. Readers are given practice problems to work through.
The document provides information about ratios and examples of ratio word problems. It begins with comparing the number of boys and girls in a class to find the ratio. It then defines a ratio as a comparison between two or more quantities. Examples are provided to show ratios can be written in different ways like a:b, a to b, or a/b. Equivalent ratios can be formed by multiplying the original ratio by any number. A compound ratio compares more than two quantities. Sample ratio word problems are worked out with explanations of what the ratios represent.
The document defines and provides examples of ratios. It discusses:
- Ratios compare two quantities and can be written in different forms such as a:b, a to b, or a/b.
- Order matters in ratios - a:b is not the same as b:a.
- Equivalent ratios can be formed by simplifying fractions or multiplying the original ratio.
- A compound ratio compares more than two quantities.
- Word problems demonstrate calculating quantities from a given ratio.
1) The document is a quiz review containing math word problems involving ratios.
2) One question asks students to determine the ratio of fertilizer to water for two fertilizer brands, and which brand would be stronger. The ratios given are 3:2 and 5:3, and the 5:3 brand is identified as stronger.
3) Another question asks how long it would take Martin to read a 100-page book based on a given reading rate. A ratio table and double number line diagram are used, and the answer is determined to be 50 minutes.
1) The document discusses ratios and uses Lucky Charms cereal as an example. It states there are 287 marshmallow pieces and 2,583 oat pieces in one box of Lucky Charms.
2) Ratios are introduced as a comparison of two quantities that can be written as a fraction, using "to", or with a colon. The ratio of marshmallows to oats in one box is given as an example.
3) Students practice writing ratios to compare amounts in different scenarios, such as numbers of lions to birds or people to pizzas. They also work on simplifying ratios by writing them in simplest form.
The document discusses ratios and provides examples using Lucky Charms cereal. It states there are 287 marshmallow pieces and 2,583 oat pieces in one box of Lucky Charms. This ratio of marshmallows to oats can be written in three ways: as a fraction, using the word "to", or using a colon. The document also discusses writing ratios in simplest form and explaining their meanings.
This document discusses converting between units of volume and weight in cooking. It provides two key rules for conversions: 1) the "how many" unit is always the bottom of the fraction and the "are there in" unit is the top, and 2) units must be the same before dividing fractions. Formulas and examples are given for calculating conversion factors and using them to increase, decrease, or adjust recipe yields while maintaining or changing portion sizes. Key steps involve setting up the proper ratio to determine the conversion factor and then multiplying ingredient amounts by the factor.
1) This document contains notes and examples on ratios from a 9th standard algebra class. It defines ratios, shows how to write ratios in different forms, and provides examples of simplifying and equivalent ratios.
2) Examples are given for writing compound ratios that involve more than two quantities. Practice problems are included for writing and interpreting ratios in word problems.
3) The document concludes with a challenge question involving using a ratio to solve for the height of a rectangle given its perimeter and the ratio of its base to height.
This document discusses permutations and combinations. It provides examples of using formulas and the fundamental counting principle to calculate the number of possible outcomes for different scenarios involving selecting items with or without regard for order. Examples include determining the number of possible sandwiches, voicemail passwords, bead necklace designs, and ways to select people or items from a group. The key concepts of permutations, combinations, and factorials are explained.
This document discusses probability and counting principles such as permutations, combinations, and the fundamental counting principle. It provides examples of how to use formulas and tree diagrams to calculate the number of possible outcomes for compound events. Key topics covered include distinguishing between combinations and permutations, using factorials to find the number of arrangements and groupings, and solving probability problems through listing outcomes and applying counting rules.
The document defines and provides examples of ratios. It explains that a ratio compares two or more quantities and can be written in different ways, such as "a to b", "a:b", or "a/b". It also discusses equivalent ratios, compound ratios, and using ratios to solve word problems. An example problem finds that the ratio of blue to black shirts is 2:3, meaning for every 2 blue shirts there are 3 black shirts.
The document discusses mathematical expressions and how to combine them. It defines expressions as calculation procedures written with numbers, variables, and operations. Expressions have terms, with the x-term being the variable term and the number term being the constant. To combine expressions, like terms are combined by adding or subtracting coefficients in the same way numbers are combined. Unlike terms, such as x-terms and number terms, cannot be combined.
A ratio compares two values and can be expressed in different ways such as a fraction (a:b), words (a to b), or a fraction (a/b). Ratios stay the same even if the values are multiplied or divided by the same amount. For example, a recipe ratio of 3 cups flour to 2 cups milk would be 12 cups flour to 8 cups milk if multiplied by 4. Ratios can also compare a part to the whole (part-to-whole ratio) such as 2 boys out of 5 total pups.
4 multiplication and division of rational expressionsmath123b
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which states that the product of two rational expressions is equal to the product of the top expressions divided by the product of the bottom expressions. Examples are provided to demonstrate simplifying rational expression products by factoring and canceling like terms.
2 the least common multiple and clearing the denominatorsmath123b
The document discusses the least common multiple (LCM) and provides examples to illustrate the concept. It describes two methods for finding the LCM - the searching method and the construction method. The searching method involves finding the smallest number that is a multiple of all the given numbers. The construction method builds the minimum that covers all requirements by taking just enough of each specification. An example demonstrates taking the maximum number of years required across different college applications in each subject area.
1. The document provides examples of infinite series that converge to a finite sum. It gives the series 1/2 + 1/4 + 1/8 + 1/16 + ..., which represents taking half of the remaining amount repeatedly, and shows that it converges to 1.
2. It asks the reader to determine the sums of several other infinite series using similar reasoning:
- The series 1/3 + 1/9 + 1/27 + 1/81 + ... is shown to equal 1 by factoring out 1/3 from each term.
- Factoring out 1/4 from the terms shows the series 1/4 + 1/16 + 1/64 + 1/256 + ... equals
5 4 equations that may be reduced to quadratics-xmath123b
The document discusses reducing equations to quadratic equations using substitution. It explains that if a pattern is repeated in an expression, a variable can be substituted to simplify the equation. Examples show substituting expressions like (x/(x-1)) with a variable y, solving the resulting quadratic equation for y, and then substituting back to find values of x. This process of solving two simpler equations through substitution is demonstrated to solve equations that are otherwise difficult to solve directly.
The document discusses graphs of quadratic equations. It explains that quadratic equations form parabolic graphs rather than straight lines. It provides examples of graphing quadratic functions by first finding the vertex using a formula, then making a table of x and y values centered around the vertex to plot points symmetrically. Key properties of parabolas are that they are symmetric around the vertex, which is the highest/lowest point on the center line.
The document discusses three methods for solving second degree equations (ax2 + bx + c = 0):
1) The square-root method, which is used when the x-term is missing. It involves solving for x2 and taking the square root to find x.
2) Factoring, which involves factoring the equation into the form (ax + b)(cx + d) = 0. It is only applicable if b2 - 4ac is a perfect square.
3) The quadratic formula, which can be used to solve any second degree equation.
1) Complex numbers are numbers of the form a + bi, where a and b are real numbers. a is called the real part and bi is called the imaginary part.
2) To add or subtract complex numbers, treat i as a variable and combine like terms.
3) To multiply complex numbers, use FOIL and set i^2 equal to -1 to simplify the result.
Radical equations are equations with an unknown variable under a radical sign. To solve radical equations, each side of the equation is squared repeatedly to remove all radicals. This is done because if two expressions are equal, then their squares are also equal. Once all radicals are removed, the resulting equation can be solved normally for the unknown variable. Examples show how to isolate radical terms, expand squared expressions using formulas, and check solutions. Squaring each side must be done carefully to properly isolate radical terms.
The document introduces exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It provides examples of evaluating exponents like 43. It then introduces rules for exponents, including the multiply-add rule where ANAK = AN+K, and the divide-subtract rule where AN/AK = AN-K. It also covers fractional exponents by defining the 0-power rule where A0 = 1 and the negative power rule where A-K = 1/AK.
The document discusses rules for simplifying expressions involving radicals. It presents the multiplication rule that √x∙y = √x∙√y and the division rule. It then gives examples of simplifying expressions such as √3∙√3 = 3, 3√3∙√3 = 9, and (3√3)2 = 27 using these rules.
The document discusses rules for simplifying radical expressions. It states the square root and multiplication rules, which are that the square root of a perfect square is the number itself, and that the square root of a product is the product of the individual square roots. Examples are provided to demonstrate simplifying radical expressions by extracting square roots from the radicand in steps using these rules.
The document discusses square roots and radicals. It defines the square root operation as finding the number that, when squared, equals the given number. It provides a table of common square numbers and their square roots that should be memorized. It also describes how to estimate the square root of numbers between values in the table by interpolating between the two closest square roots. A scientific calculator is needed to evaluate more complex square roots.
The document discusses solving linear inequalities in two variables (x and y). It explains that the solutions to inequalities in x are segments of the real line, while the solutions to inequalities in both x and y are regions of the plane. It then provides an example of using the graph of y=x to identify the regions defined by y>x and y<x. Finally, it discusses the general process of solving linear inequalities Ax + By > C or Ax + By < C by graphing the line Ax + By = C and using point testing to determine which half-plane satisfies the given inequality.
The document describes the rectangular coordinate system. It defines a coordinate system as assigning positions in a plane or space with addresses. The rectangular coordinate system uses a grid with two perpendicular axes (x and y) intersecting at the origin (0,0). Any point in the plane is located by its coordinates (x,y), where x is the distance right or left of the origin and y is the distance up or down. The four quadrants divided by the axes are labeled based on the signs of the x and y coordinates.
The document discusses absolute value inequalities and provides examples of how to represent them graphically. It explains that an inequality of the form |x| < c can be rewritten as -c < x < c, representing all values within c units of 0. An inequality of the form |x| > c is split into two inequalities x < -c or c < x, representing values more than c units from 0. Examples are given of drawing the solutions to |x| < 7, |x| > 7, and solving |3 - 2x| < 7 algebraically then graphically.
The document discusses solving absolute value inequalities using a geometric method. It introduces absolute value inequalities as statements about distances on the real number line. Example A explains that |x| < 7 represents all numbers within 7 units of 0, or between -7 and 7. Example B translates |x - 2| < 3 to mean the distance between x and 2 must be less than 3, with the solution being -1 < x < 5. The document outlines rules for one-piece and two-piece absolute value inequalities and works through additional examples.
The document discusses the definition and properties of absolute value equations. It defines absolute value as the distance from a number to zero on the real number line. It presents rules for solving absolute value equations, including rewriting an absolute value equation as two separate equations without the absolute value signs, and the property that if the absolute value of an expression is equal to a positive number a, then the expression must equal -a or a. Examples are provided to demonstrate solving absolute value equations using these rules and properties.
The document discusses direct and inverse variations. It defines direct variation as y=kx, where k is a constant, and inverse variation as y=k/x. Examples are given of translating phrases describing variations into equations. For a direct variation problem between variables y and x where y=-4 when x=-6, the specific equation is found to be y=2/3x. For an inverse variation between weight W and distance D from Earth's center, the person's weight 6000 miles above the surface is calculated using the general inverse variation equation W=k/D^2.
This document discusses exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It provides examples like 43 = 64. Rules for exponents are covered, including:
- Multiply-Add Rule: ANAK = AN+K
- Divide-Subtract Rule: AN/AK = AN-K
- Negative exponents represent reciprocals, so A-K = 1/AK
Fractional exponents are introduced, along with the 0-Power Rule that A0 = 1.
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3. Proportions
Two related quantities stated side by side is called a ratio.
For example, if a recipe calls for 3 eggs and 4 cups of flour,
then the ratio of eggs to flour is 3 to 4.
4. 3
4
eggs
cups of flour
Proportions
Two related quantities stated side by side is called a ratio.
For example, if a recipe calls for 3 eggs and 4 cups of flour,
then the ratio of eggs to flour is 3 to 4.
We may write it using fractional notation as:
5. 3
4
eggs
cups of flour
Proportions
This fraction is also the amount of per unit of the given ratio,
in this case, 3/4 egg / per cup of flour.
Two related quantities stated side by side is called a ratio.
For example, if a recipe calls for 3 eggs and 4 cups of flour,
then the ratio of eggs to flour is 3 to 4.
We may write it using fractional notation as:
(egg / per cup of flour)
6. 3
4
eggs
cups of flour
Proportions
This fraction is also the amount of per unit of the given ratio,
in this case, 3/4 egg / per cup of flour.
Two ratios that are equal are said to be in proportion.
Two related quantities stated side by side is called a ratio.
For example, if a recipe calls for 3 eggs and 4 cups of flour,
then the ratio of eggs to flour is 3 to 4.
We may write it using fractional notation as:
(egg / per cup of flour)
7. 3
4
eggs
cups of flour
Proportions
This fraction is also the amount of per unit of the given ratio,
in this case, 3/4 egg / per cup of flour.
Two ratios that are equal are said to be in proportion.
Thus "3 to 4" is proportion to "6 to 8" since
3
4 =
6
8
Two related quantities stated side by side is called a ratio.
For example, if a recipe calls for 3 eggs and 4 cups of flour,
then the ratio of eggs to flour is 3 to 4.
We may write it using fractional notation as:
(egg / per cup of flour)
8. 3
4
eggs
cups of flour
Proportions
This fraction is also the amount of per unit of the given ratio,
in this case, 3/4 egg / per cup of flour.
Two ratios that are equal are said to be in proportion.
Thus "3 to 4" is proportion to "6 to 8" since
3
4 =
6
8
Proportion–equations are the simplest type of fractional
equations.
Two related quantities stated side by side is called a ratio.
For example, if a recipe calls for 3 eggs and 4 cups of flour,
then the ratio of eggs to flour is 3 to 4.
We may write it using fractional notation as:
(egg / per cup of flour)
9. 3
4
eggs
cups of flour
Proportions
This fraction is also the amount of per unit of the given ratio,
in this case, 3/4 egg / per cup of flour.
Two ratios that are equal are said to be in proportion.
Thus "3 to 4" is proportion to "6 to 8" since
3
4 =
6
8
Proportion–equations are the simplest type of fractional
equations. To solve proportional equations, cross-multiply
and change the proportions into regular equations.
Two related quantities stated side by side is called a ratio.
For example, if a recipe calls for 3 eggs and 4 cups of flour,
then the ratio of eggs to flour is 3 to 4.
We may write it using fractional notation as:
(egg / per cup of flour)
12. A
B
C
D ,
=If then AD = BC.
Cross-Multiplication-Rule
Proportions
Example A. Solve for x.
3
x
5
2
=a.
13. A
B
C
D ,
=If then AD = BC.
Cross-Multiplication-Rule
Proportions
Example A. Solve for x.
3
x
5
2
=a. cross multiply
14. A
B
C
D ,
=If then AD = BC.
Cross-Multiplication-Rule
Proportions
Example A. Solve for x.
3
x
5
2
=a. cross multiply
6 = 5x
15. A
B
C
D ,
=If then AD = BC.
Cross-Multiplication-Rule
Proportions
Example A. Solve for x.
3
x
5
2
=a. cross multiply
6 = 5x
= x5
6
16. A
B
C
D ,
=If then AD = BC.
Cross-Multiplication-Rule
Proportions
Example A. Solve for x.
3
x
5
2
=a. cross multiply
6 = 5x
= x
2
3
(x + 2)
(x – 5)
=b.
5
6
17. A
B
C
D ,
=If then AD = BC.
Cross-Multiplication-Rule
Proportions
Example A. Solve for x.
3
x
5
2
=a. cross multiply
6 = 5x
= x5
6
2
3
(x + 2)
(x – 5)
=b. cross multiply
18. A
B
C
D ,
=If then AD = BC.
Cross-Multiplication-Rule
Proportions
Example A. Solve for x.
3
x
5
2
=a. cross multiply
6 = 5x
= x
2
3
(x + 2)
(x – 5)
=b. cross multiply
2(x – 5) = 3(x + 2)
5
6
19. A
B
C
D ,
=If then AD = BC.
Cross-Multiplication-Rule
Proportions
Example A. Solve for x.
3
x
5
2
=a. cross multiply
6 = 5x
= x
2
3
(x + 2)
(x – 5)
=b. cross multiply
2(x – 5) = 3(x + 2)
2x – 10 = 3x + 6
5
6
20. A
B
C
D ,
=If then AD = BC.
Cross-Multiplication-Rule
Proportions
Example A. Solve for x.
3
x
5
2
=a. cross multiply
6 = 5x
= x
2
3
(x + 2)
(x – 5)
=b. cross multiply
2(x – 5) = 3(x + 2)
2x – 10 = 3x + 6
–10 – 6 = 3x – 2x
5
6
21. A
B
C
D ,
=If then AD = BC.
Cross-Multiplication-Rule
Proportions
Example A. Solve for x.
3
x
5
2
=a. cross multiply
6 = 5x
= x
2
3
(x + 2)
(x – 5)
=b. cross multiply
2(x – 5) = 3(x + 2)
2x – 10 = 3x + 6
–10 – 6 = 3x – 2x
–16 = x
5
6
22. Proportions
When setting up a proportional equation for a word problem,
the two quantities for the same type of measurement must
occupy the same position in the fractions
23. Proportions
When setting up a proportional equation for a word problem,
the two quantities for the same type of measurement must
occupy the same position in the fractions i.e. both must be in
numerator, or both must be in the denominator.
24. Proportions
Example B. A recipe calls for 3 eggs and 4 cups of flour.
How many eggs are needed if we have 10 cups of flour?
When setting up a proportional equation for a word problem,
the two quantities for the same type of measurement must
occupy the same position in the fractions i.e. both must be in
numerator, or both must be in the denominator.
25. Proportions
Example B. A recipe calls for 3 eggs and 4 cups of flour. How
many eggs are needed if we have 10 cups of flour?
Let x be the number of eggs needed.
When setting up a proportional equation for a word problem,
the two quantities for the same type of measurement must
occupy the same position in the fractions i.e. both must be in
numerator, or both must be in the denominator.
26. Proportions
Example B. A recipe calls for 3 eggs and 4 cups of flour.
How many eggs are needed if we have 10 cups of flour?
Let x be the number of eggs needed. Write the two types of
measurements, the number of eggs and the number of cups
of flour, in a column as shown.
When setting up a proportional equation for a word problem,
the two quantities for the same type of measurement must
occupy the same position in the fractions i.e. both must be in
numerator, or both must be in the denominator.
27. Proportions
Example B. A recipe calls for 3 eggs and 4 cups of flour.
How many eggs are needed if we have 10 cups of flour?
Let x be the number of eggs needed. Write the two types of
measurements, the number of eggs and the number of cups
of flour, in a column as shown.
number of eggs
cups of flour
When setting up a proportional equation for a word problem,
the two quantities for the same type of measurement must
occupy the same position in the fractions i.e. both must be in
numerator, or both must be in the denominator.
28. Proportions
Example B. A recipe calls for 3 eggs and 4 cups of flour.
How many eggs are needed if we have 10 cups of flour?
Let x be the number of eggs needed. Write the two types of
measurements, the number of eggs and the number of cups
of flour, in a column as shown. Then write down the ratios.
number of eggs
cups of flour
When setting up a proportional equation for a word problem,
the two quantities for the same type of measurement must
occupy the same position in the fractions i.e. both must be in
numerator, or both must be in the denominator.
29. Proportions
Example B. A recipe calls for 3 eggs and 4 cups of flour.
How many eggs are needed if we have 10 cups of flour?
Let x be the number of eggs needed. Write the two types of
measurements, the number of eggs and the number of cups
of flour, in a column as shown. Then write down the ratios.
number of eggs
cups of flour
x
10
When setting up a proportional equation for a word problem,
the two quantities for the same type of measurement must
occupy the same position in the fractions i.e. both must be in
numerator, or both must be in the denominator.
30. Proportions
Example B. A recipe calls for 3 eggs and 4 cups of flour.
How many eggs are needed if we have 10 cups of flour?
Let x be the number of eggs needed. Write the two types of
measurements, the number of eggs and the number of cups
of flour, in a column as shown. Then write down the ratios.
number of eggs
cups of flour
3
4
x
10
When setting up a proportional equation for a word problem,
the two quantities for the same type of measurement must
occupy the same position in the fractions i.e. both must be in
numerator, or both must be in the denominator.
31. Proportions
Example B. A recipe calls for 3 eggs and 4 cups of flour.
How many eggs are needed if we have 10 cups of flour?
Let x be the number of eggs needed. Write the two types of
measurements, the number of eggs and the number of cups
of flour, in a column as shown. Then write down the ratios.
number of eggs
cups of flour
3
4
x
10
=
Set them equal, we get
x
10
3
4
When setting up a proportional equation for a word problem,
the two quantities for the same type of measurement must
occupy the same position in the fractions i.e. both must be in
numerator, or both must be in the denominator.
32. Proportions
Example B. A recipe calls for 3 eggs and 4 cups of flour.
How many eggs are needed if we have 10 cups of flour?
Let x be the number of eggs needed. Write the two types of
measurements, the number of eggs and the number of cups
of flour, in a column as shown. Then write down the ratios.
number of eggs
cups of flour
3
4
x
10
=
Set them equal, we get
x
10
3
4
cross multiply
4x = 30
When setting up a proportional equation for a word problem,
the two quantities for the same type of measurement must
occupy the same position in the fractions i.e. both must be in
numerator, or both must be in the denominator.
33. Proportions
Example B. A recipe calls for 3 eggs and 4 cups of flour.
How many eggs are needed if we have 10 cups of flour?
Let x be the number of eggs needed. Write the two types of
measurements, the number of eggs and the number of cups
of flour, in a column as shown. Then write down the ratios.
number of eggs
cups of flour
3
4
x
10
=
Set them equal, we get
x
10
3
4
cross multiply
4x = 30
x =
30
4
When setting up a proportional equation for a word problem,
the two quantities for the same type of measurement must
occupy the same position in the fractions i.e. both must be in
numerator, or both must be in the denominator.
34. Proportions
Example B. A recipe calls for 3 eggs and 4 cups of flour.
How many eggs are needed if we have 10 cups of flour?
Let x be the number of eggs needed. Write the two types of
measurements, the number of eggs and the number of cups
of flour, in a column as shown. Then write down the ratios.
number of eggs
cups of flour
3
4
x
10
=
Set them equal, we get
x
10
3
4
cross multiply
4x = 30
x =
30
4
= 7½ We need 7½ eggs.
When setting up a proportional equation for a word problem,
the two quantities for the same type of measurement must
occupy the same position in the fractions i.e. both must be in
numerator, or both must be in the denominator.
35. Example C. On a map, 4 inches corresponds to 21 miles in
real distance. What is the real distance between two points
if they are 14 inches apart on the map?
Proportions
36. Example C. On a map, 4 inches corresponds to 21 miles in
real distance. What is the real distance between two points
if they are 14 inches apart on the map?
Proportions
Let x be the number of real distance in miles and write the
two types of measurements as
37. Example C. On a map, 4 inches corresponds to 21 miles in
real distance. What is the real distance between two points
if they are 14 inches apart on the map?
Proportions
Let x be the number of real distance in miles and write the
two types of measurements as
miles
inches
38. Example C. On a map, 4 inches corresponds to 21 miles in
real distance. What is the real distance between two points
if they are 14 inches apart on the map?
Proportions
Let x be the number of real distance in miles and write the
two types of measurements as
miles
inches
x
14
39. Example C. On a map, 4 inches corresponds to 21 miles in
real distance. What is the real distance between two points
if they are 14 inches apart on the map?
Proportions
Let x be the number of real distance in miles and write the
two types of measurements as
miles
inches
21
4
x
14
40. Example C. On a map, 4 inches corresponds to 21 miles in
real distance. What is the real distance between two points
if they are 14 inches apart on the map?
Proportions
Let x be the number of real distance in miles and write the
two types of measurements as
miles
inches
21
4
x
14
Set them equal, we get
21
4
x
14
=
41. Example C. On a map, 4 inches corresponds to 21 miles in
real distance. What is the real distance between two points
if they are 14 inches apart on the map?
Proportions
Let x be the number of real distance in miles and write the
two types of measurements as
miles
inches
21
4
x
14
Set them equal, we get
21
4
x
14
= cross multiply
4x = 294
42. Example C. On a map, 4 inches corresponds to 21 miles in
real distance. What is the real distance between two points
if they are 14 inches apart on the map?
Proportions
Let x be the number of real distance in miles and write the
two types of measurements as
miles
inches
21
4
x
14
Set them equal, we get
21
4
x
14
= cross multiply
4x = 294
x =
294
4
43. Example C. On a map, 4 inches corresponds to 21 miles in
real distance. What is the real distance between two points
if they are 14 inches apart on the map?
Proportions
Let x be the number of real distance in miles and write the
two types of measurements as
miles
inches
21
4
x
14
Set them equal, we get
21
4
x
14
= cross multiply
4x = 294
x =
294
4
= 73½
So 4 inches corresponds to 73½ miles.
45. Proportions
Given the following cats, which cat on the right is of the same
shape as the one on the left?
A.
B.
Geometric proportion gives us the sense of “similarity”, that is,
objects that are of the same shape but different sizes.
46. Proportions
Given the following cats, which cat on the right is of the same
shape as the one on the left? The answer of course is A.
A.
B.
Geometric proportion gives us the sense of “similarity”, that is,
objects that are of the same shape but different sizes.
47. Proportions
Given the following cats, which cat on the right is of the same
shape as the one on the left? The answer of course is A.
A.
B.
This sense of similarity is due to the fact that
“all corresponding linear measurements are in proportion”.
Geometric proportion gives us the sense of “similarity”, that is,
objects that are of the same shape but different sizes.
50. Proportions
B.
For example, given the following similar cats, let’s identify
some corresponding points as shown. Suppose the following
measurements are known.
6
10 12
51. Proportions
B.
For example, given the following similar cats, let’s identify
some corresponding points as shown. Suppose the following
measurements are known.
6
10 12
3
52. Proportions
B.
For example, given the following similar cats, let’s identify
some corresponding points as shown. Suppose the following
measurements are known. Then we may find the distances
y and z by proportion.
6
10 12
3
y z
53. Proportions
B.
For example, given the following similar cats, let’s identify
some corresponding points as shown. Suppose the following
measurements are known. Then we may find the distances
y and z by proportion. Since the distances between the tips
of the ears is 6 : 3 or 2 : 1 ratio,
6
10 12
3
y z
54. Proportions
B.
For example, given the following similar cats, let’s identify
some corresponding points as shown. Suppose the following
measurements are known. Then we may find the distances
y and z by proportion. Since the distances between the tips
of the ears is 6 : 3 or 2 : 1 ratio, we see that
y =
6
10 12
3
y z
55. Proportions
B.
For example, given the following similar cats, let’s identify
some corresponding points as shown. Suppose the following
measurements are known. Then we may find the distances
y and z by proportion. Since the distances between the tips
of the ears is 6 : 3 or 2 : 1 ratio, we see that
y = 10/5 = 2 and z = 12/2 = 6.
6
10 12
3
y z
56. Proportions
6
10 12
3
5 6
For example, given the following similar cats, let’s identify
some corresponding points as shown. Suppose the following
measurements are known. Then we may find the distances
y and z by proportion. Since the distances between the tips
of the ears is 6 : 3 or 2 : 1 ratio, we see that
y = 10/5 = 2 and z = 12/2 = 6.
58. Proportions
Definition. Two geometric objects are similar if all
corresponding linear measurements are in proportion.
B.
Hence if X, Y and Z are the measurements from the original
cat as shown,
X
Y Z
59. Proportions
Definition. Two geometric objects are similar if all
corresponding linear measurements are in proportion.
B.
Hence if X, Y and Z are the measurements from the original
cat as shown, and x, y and z are the corresponding
measurements in the similar copy,
X
Y Z
x
y z
60. Proportions
Definition. Two geometric objects are similar if all
corresponding linear measurements are in proportion.
B.
Hence if X, Y and Z are the measurements from the original
cat as shown, and x, y and z are the corresponding
measurements in the similar copy, then it must be that
X
Y Z
x
y z
X
x =
Y
y =
Z
z
61. Proportions
Definition. Two geometric objects are similar if all
corresponding linear measurements are in proportion.
B.
Hence if X, Y and Z are the measurements from the original
cat as shown, and x, y and z are the corresponding
measurements in the similar copy, then it must be that
X
Y Z
x
y z
X
x =
Y
y =
Z
z
or that X
Y =
x
y etc…
62. Proportions
Definition. Two geometric objects are similar if all
corresponding linear measurements are in proportion.
B.
Hence if X, Y and Z are the measurements from the original
cat as shown, and x, y and z are the corresponding
measurements in the similar copy, then it must be that
X
Y Z
x
y z
X
x =
Y
y =
Z
z
or that X
Y =
x
y etc…
This mathematical formulation of similarity is the basis for
biometric security software such as facial recognition systems.
63. Proportions
The reason that linear proportionality gives us the sense of
similarity is due to visual geometry, that is, the visual images
of an object at different distances form a “cone”.
64. Proportions
The reason that linear proportionality gives us the sense of
similarity is due to visual geometry, that is, the visual images
of an object at different distances form a “cone”.
65. Proportions
The reason that linear proportionality gives us the sense of
similarity is due to visual geometry, that is, the visual images
of an object at different distances form a “cone”.
One can easily demonstrate that in such projection, similar
images of two different sizes must preserve the ratio of two
corresponding distance measurements – as in the above
definition of similarity.
66. Proportions
The reason that linear proportionality gives us the sense of
similarity is due to visual geometry, that is, the visual images
of an object at different distances form a “cone”.
One can easily demonstrate that in such projection, similar
images of two different sizes must preserve the ratio of two
corresponding distance measurements – as in the above
definition of similarity.
Similar triangles
68. Proportions
In the definition of similarity, the measurements in proportion
must be linear. It's not true that the ratio of areas is
proportional to linear ratio of two similar objects.
69. Proportions
In the definition of similarity, the measurements in proportion
must be linear. It's not true that the ratio of areas is
proportional to linear ratio of two similar objects.
Because the ratio of the linear measurements is 1 : 2,
1
2
70. Proportions
In the definition of similarity, the measurements in proportion
must be linear. It's not true that the ratio of areas is
proportional to linear ratio of two similar objects.
Because the ratio of the linear measurements is 1 : 2,
we may conclude that the ratio of the area–measurements is
1 : (2)2 or 1 : 4 ratio.
1
2
linear ratio 1 : 2
area–ratio 1 : 4 = 22
:
71. Proportions
1
2
linear ratio 1 : 2
area–ratio 1 : 4 = 22
:
In the definition of similarity, the measurements in proportion
must be linear. It's not true that the ratio of areas is
proportional to linear ratio of two similar objects.
Because the ratio of the linear measurements is 1 : 2,
we may conclude that the ratio of the area–measurements is
1 : (2)2 or 1 : 4 ratio. So it would take 4 times as much paint to
paint the larger one.
72. Proportions
1
2
linear ratio 1 : 2
area–ratio 1 : 4 = 22
linear ratio 1 : r
area–ratio 1 : r2
1
r
: :
In the definition of similarity, the measurements in proportion
must be linear. It's not true that the ratio of areas is
proportional to linear ratio of two similar objects.
Because the ratio of the linear measurements is 1 : 2,
we may conclude that the ratio of the area–measurements is
1 : (2)2 or 1 : 4 ratio. So it would take 4 times as much paint to
paint the larger one. Given two similar objects with linear
ratio 1 : r then their area–ratio is 1 : r2.
73. Proportions
1
2
linear ratio 1 : 2
area–ratio 1 : 4 = 22
linear ratio 1 : r
area–ratio 1 : r2
1
r
: :
In the definition of similarity, the measurements in proportion
must be linear. It's not true that the ratio of areas is
proportional to linear ratio of two similar objects.
Because the ratio of the linear measurements is 1 : 2,
we may conclude that the ratio of the area–measurements is
1 : (2)2 or 1 : 4 ratio. So it would take 4 times as much paint to
paint the larger one. Given two similar objects with linear
ratio 1 : r then their area–ratio is 1 : r2. This is also true for
the surface areas.
74. Proportions
1
2
linear ratio 1 : 2
area–ratio 1 : 4 = 22
linear ratio 1 : r
area–ratio 1 : r2
1
r
: :
In the definition of similarity, the measurements in proportion
must be linear. It's not true that the ratio of areas is
proportional to linear ratio of two similar objects.
Because the ratio of the linear measurements is 1 : 2,
we may conclude that the ratio of the area–measurements is
1 : (2)2 or 1 : 4 ratio. So it would take 4 times as much paint to
paint the larger one. Given two similar objects with linear
ratio 1 : r then their area–ratio is 1 : r2. This is also true for
the surface areas i.e. two similar 3D solids of 1 : r linear
ratio have 1 : r2 as their surface–area ratio.
75. Proportions
The volume of an object is the measurement of the
3-dimensional space the object occupies.
76. Proportions
Because the ratio of the linear measurements is 1 : 2,
:
linear ratio 1 : 2
1
2
The volume of an object is the measurement of the
3-dimensional space the object occupies.
77. Proportions
Because the ratio of the linear measurements is 1 : 2,
we may conclude that the ratio of the volume–measurements
is 1 : (2)3 or 1 : 8 ratio.
:
linear ratio 1 : 2
volume–ratio 1 : 8 = 23
1
2
The volume of an object is the measurement of the
3-dimensional space the object occupies.
78. Proportions
Because the ratio of the linear measurements is 1 : 2,
we may conclude that the ratio of the volume–measurements
is 1 : (2)3 or 1 : 8 ratio. So the weight of large one is 8 times
as much as the small one (if they are made of the same
thing).*
:
linear ratio 1 : 2
volume–ratio 1 : 8 = 23
1
2
The volume of an object is the measurement of the
3-dimensional space the object occupies.
79. Proportions
Because the ratio of the linear measurements is 1 : 2,
we may conclude that the ratio of the volume–measurements
is 1 : (2)3 or 1 : 8 ratio. So the weight of large one is 8 times
as much as the small one (if they are made of the same
thing).* Given two similar objects with linear ratio 1 : r
then their volume–ratio is 1 : r3.
linear ratio 1 : r
volume–ratio 1 : r3
::
linear ratio 1 : 2
volume–ratio 1 : 8 = 23
1
2 1
r
The volume of an object is the measurement of the
3-dimensional space the object occupies.
82. The Golden Ratio
Proportions
An important ratio in arts and science is the golden ratio.
Cut a line segment of length 1 into two, a long one L and a
short one s such that the ratio of “1 to L” is the same as
“L to s”,
83. The Golden Ratio
Proportions
An important ratio in arts and science is the golden ratio.
Cut a line segment of length 1 into two, a long one L and a
short one s such that the ratio of “1 to L” is the same as
“L to s”,
1
84. The Golden Ratio
Proportions
Cut a line segment of length 1 into two, a long one L and a
short one s such that the ratio of “1 to L” is the same as
“L to s”,
1
L
An important ratio in arts and science is the golden ratio.
85. The Golden Ratio
Proportions
Cut a line segment of length 1 into two, a long one L and a
short one s such that the ratio of “1 to L” is the same as
“L to s”,
1
L
An important ratio in arts and science is the golden ratio.
1
L
86. The Golden Ratio
Proportions
Cut a line segment of length 1 into two, a long one L and a
short one s such that the ratio of “1 to L” is the same as
“L to s”,
1
L s
An important ratio in arts and science is the golden ratio.
L
s
1
L =
87. The Golden Ratio
Proportions
Cut a line segment of length 1 into two, a long one L and a
short one s such that the ratio of “1 to L” is the same as
“L to s”,
1
L s
An important ratio in arts and science is the golden ratio.
L
s
1
L =
The ratio1/L is the Golden Ratio.
88. The Golden Ratio
Proportions
L
(1 – L)
1
L
Cut a line segment of length 1 into two, a long one L and a
short one s such that the ratio of “1 to L” is the same as
“L to s”,
1
L s
An important ratio in arts and science is the golden ratio.
L
s
1
L =
Since s = 1 – L, we’ve
=
The ratio1/L is the Golden Ratio.
89. The Golden Ratio
Proportions
L
(1 – L)
1
L
cross multiply
Cut a line segment of length 1 into two, a long one L and a
short one s such that the ratio of “1 to L” is the same as
“L to s”,
1
L s
An important ratio in arts and science is the golden ratio.
L
s
1
L =
Since s = 1 – L, we’ve
=
1 – L
=
L2
The ratio1/L is the Golden Ratio.
90. The Golden Ratio
Proportions
L
(1 – L)
1
L
cross multiply
0 = L2 + L – 1
Cut a line segment of length 1 into two, a long one L and a
short one s such that the ratio of “1 to L” is the same as
“L to s”,
1
L s
An important ratio in arts and science is the golden ratio.
L
s
1
L =
Since s = 1 – L, we’ve
=
1 – L
=
L2
The ratio1/L is the Golden Ratio.
91. The Golden Ratio
Proportions
L
(1 – L)
1
L
cross multiply
0 = L2 + L – 1
Cut a line segment of length 1 into two, a long one L and a
short one s such that the ratio of “1 to L” is the same as
“L to s”,
1
L s
An important ratio in arts and science is the golden ratio.
L
s
1
L =
Since s = 1 – L, we’ve
=
1 – L
=
L2
We’ll see later that the answer for L is about 0.618.
http://en.wikipedia.org/wiki/Golden_ratio
http://en.wikipedia.org/wiki/Golden_spiral
The ratio1/L is the Golden Ratio.
92. Ex. A. Solve for x.
Proportions
3
2
2x + 1
3
=
21
x
3 =1.
–2x/7
–5
1½
3
=9.
2.
6
2=3.
x + 1
–57 =4. –2x + 1 6
=5. x + 1 23x =6. 5x + 1
3
4x =
8
2/3
7/5
x
14
=7.
9
4/3
3
2x/5
=8.
10. – x + 2
2
–4
3 =11.
3
2x + 1
1
x – 2
=12.
x – 5
2
2x – 3
3
=13.
3
2
x – 4
x – 1
=14.
–3
5
2x + 1
3x
=15.
3x + 2
2
2x + 1
3
=16. 3
2
2x + 1
x – 3
=17.
2
x
x + 1
3
=18.
x
2
x + 4
x
=19.
1
2
93. Ex. B. (Solve each problem. It’s easier if fractional proportions
are rewritten as proportions of integers.)
Proportions
20. If 4 cups of flour need 3 tsp of salt,
then 10 cups flour need how much of salt?
Different cookie recipes are given, find the missing amounts.
21. If 5 cups of flour need 7 cups of water
then 10 cups water need how much of flour?
23. If 2 ½ tsp of butter flour need 3/4 tsp of salt,
then 6 tsp of salt need how much of flour.
22. If 5 cups of flour need 7 cups of water
then 10 cups flour need how much of water?
24. If 1¼ inches equals 5 miles real distance, how many
miles is 5 inches on the map?
For the given map scales below, find the missing amounts.
25. If 2½ inches equals 140 miles real distance, how many
inches on the map correspond to the distance of 1,100 miles?
94. Proportions
A. B. C.
3
4
1
5
5
6
26. x
20 x
20
6
27.
x
3x + 4 2xx+2
15 x
x
2x – 3
28.
29. 30. 31.
32–37. Find the surface areas of each cat above if A’s surface
area is 2 ft2, B’s surfaces area is 7 ft2. and C’s is 280 ft2?
Ex. C. 26 – 31. Solve for x. Use the given A, B and C,.
38–43. Find the weight of each cat above if cat A is 20 lbs,
cat B is 8 lb. and cat C is 350 lb?