This document provides a toolbox of mathematical facts, formulas, and tricks to help with MATHCOUNTS coaching. It includes lists of prime numbers up to 100, common fractions and their decimal and percent equivalents, perfect squares and cubes, square roots, and formulas for perimeter, area, volume, slope, distance, and more geometric relationships.
This document provides information about the Beck Math Club, including its coaches, class format with beginner, intermediate and advanced levels, requirements for participation, objectives, meeting times, past results, website, calendar, classroom and outside competitions, details on various math competitions like Mathcounts, AMC, AIME, TMSCA, fees, discipline policy, snack program, and a call for parent involvement and volunteers.
This document contains various tables, charts, and graphs with numerical and statistical data. It includes information about hours of daylight by month, favorite rides by gender, test scores, stock market performance, lottery numbers, speed limits, geometric measurements, pet ownership statistics, roller coaster speeds, and conversation word counts. The data relates to topics like recreation, exams, business, gambling, transportation, shapes, animals, and interactions.
1. The document discusses various kinematics problems involving motion under uniform acceleration. It provides solutions using graphical, analytical and vector methods.
2. Methods include calculating time taken to cross a river based on velocities and angles, determining average and instantaneous velocities from distance-time graphs, resolving velocities into components, and finding the distance between particles moving with different initial velocities.
3. One problem involves three particles moving in a circle such that they are always at the vertices of an equilateral triangle, and calculates the distance traveled by one particle before they meet.
This document contains mathematical expressions, series expansions, and convergence tests. It provides series expansions for numerous functions around points including 0, π/2, and e. It also tests the convergence of series using tests such as the ratio test and provides convergence results. The document serves as a reference for mathematical formulas and series expansions.
Boas mathematical methods in the physical sciences 3ed instructors solutions...Praveen Prashant
This document contains mathematical expressions, series expansions, and convergence tests. It provides series expansions for various functions around points using Taylor series. It also tests the convergence of infinite series using tests like the limit comparison test, ratio test, and integral test. Several problems provide the interval of convergence for Taylor series expansions of different functions.
1. The document provides data on speed and time for a vehicle, as well as exercises involving ratios, percentages, fractions, and algebraic expressions.
2. It also contains information about variables that are related, such as area of a circle and radius, and examples of using linear equations to model real-world situations involving time, distance, and rate.
3. Additional sections cover graphs of linear and nonlinear functions, volumes and surface areas of geometric shapes, and modeling population changes between foxes and rabbits over time.
This document provides information about the Beck Math Club, including its coaches, class format with beginner, intermediate and advanced levels, requirements for participation, objectives, meeting times, past results, website, calendar, classroom and outside competitions, details on various math competitions like Mathcounts, AMC, AIME, TMSCA, fees, discipline policy, snack program, and a call for parent involvement and volunteers.
This document contains various tables, charts, and graphs with numerical and statistical data. It includes information about hours of daylight by month, favorite rides by gender, test scores, stock market performance, lottery numbers, speed limits, geometric measurements, pet ownership statistics, roller coaster speeds, and conversation word counts. The data relates to topics like recreation, exams, business, gambling, transportation, shapes, animals, and interactions.
1. The document discusses various kinematics problems involving motion under uniform acceleration. It provides solutions using graphical, analytical and vector methods.
2. Methods include calculating time taken to cross a river based on velocities and angles, determining average and instantaneous velocities from distance-time graphs, resolving velocities into components, and finding the distance between particles moving with different initial velocities.
3. One problem involves three particles moving in a circle such that they are always at the vertices of an equilateral triangle, and calculates the distance traveled by one particle before they meet.
This document contains mathematical expressions, series expansions, and convergence tests. It provides series expansions for numerous functions around points including 0, π/2, and e. It also tests the convergence of series using tests such as the ratio test and provides convergence results. The document serves as a reference for mathematical formulas and series expansions.
Boas mathematical methods in the physical sciences 3ed instructors solutions...Praveen Prashant
This document contains mathematical expressions, series expansions, and convergence tests. It provides series expansions for various functions around points using Taylor series. It also tests the convergence of infinite series using tests like the limit comparison test, ratio test, and integral test. Several problems provide the interval of convergence for Taylor series expansions of different functions.
1. The document provides data on speed and time for a vehicle, as well as exercises involving ratios, percentages, fractions, and algebraic expressions.
2. It also contains information about variables that are related, such as area of a circle and radius, and examples of using linear equations to model real-world situations involving time, distance, and rate.
3. Additional sections cover graphs of linear and nonlinear functions, volumes and surface areas of geometric shapes, and modeling population changes between foxes and rabbits over time.
Triangles can Equal Circles, it's all in hour you add them up.Leland Bartlett
This document outlines Leland Bartlett's proof that the area of a circle can be calculated by summing the areas of many triangles circumscribing the circle. It begins with area formulas and trigonometry lessons. Then, it shows the calculations for 4, 8, 16, 32, 1024, and 32768 triangles approximating a circle with diameter 6 and radius 3. The sums converge on the expected area and circumference values with very small differences. In the end, it is shown that the height of the triangles approximates the radius, proving the area formula. Leland Bartlett provides training in various topics like Microsoft Office and databases as a professional consultant and trainer.
The document discusses analyzing an investment opportunity to operate metro services in a city for 5 years. It provides estimated cash flows with normal distributions and costs of capital under different macroeconomic scenarios. The task is to estimate the maximum price that could be paid and assess viability. For the maximum price, simulations are used to model cash flows and calculate VPL at a 15% cost of capital. For viability, VPL is calculated for each scenario weighted by its probability to determine if the project is worthwhile.
1. The document contains sample questions and solutions for various topics in mathematics for Year 3 secondary school students in Malaysia. It covers topics like indices, standard form, financial mathematics, and more.
2. The questions range from single-step problems to multi-part questions involving concepts like indices, financial calculations involving interest, investments, loans, and currency conversions.
3. Detailed step-by-step workings are provided for most questions to demonstrate how to arrive at the solutions.
1. The document discusses geometric concepts such as lines, angles, and the Pythagorean theorem.
2. Equations and formulas are presented for calculating lengths of sides of right triangles based on the Pythagorean theorem.
3. Approximations of irrational numbers like the square root of 2 and pi are calculated through successive decimals.
1. The document discusses geometric concepts such as lines, angles, and the Pythagorean theorem.
2. Equations and formulas are presented for calculating lengths of sides of right triangles based on the Pythagorean theorem.
3. Approximations of irrational numbers like the square root of 2 and pi are calculated through successive decimals.
1. The document summarizes solutions to problems from chapter 3 of Engineering Electromagnetics by Hayt, Buck. It provides calculations and solutions for problems regarding electric field intensity D, electric flux, volume charge density, and more.
2. Key calculations include determining D at a point given point charges, determining electric flux through surfaces, calculating enclosed charge, and finding volume charge density using the divergence of D.
3. Solutions involve applying Gauss's law and knowing how to set up integrals in different coordinate systems to calculate relevant physical quantities for the electromagnetic problems.
Hand book of Howard Anton calculus exercises 8th editionPriSim
The document contains the table of contents for a calculus textbook. It lists 17 chapters covering topics such as functions, limits, derivatives, integrals, vector calculus, and applications of calculus. It also includes 6 appendices reviewing concepts in real numbers, trigonometry, coordinate planes, and polynomial equations.
1. This document contains a math test with 34 multiple choice questions covering exponents, scientific notation, averages, and other algebra topics.
2. The test examines skills like simplifying expressions with exponents, writing numbers in scientific and standard notation, identifying equivalent algebraic expressions, and calculating averages.
3. The questions require choosing the correct multiple choice answer that represents the outcome of performing the described mathematical operation or identifies the numerical result.
This document provides practice problems for simplifying multiplication expressions in algebra. It contains 21 practice problems asking the reader to simplify expressions involving multiplication of numbers and variables. It also contains an word problem asking the reader to write an expression and calculate the total amount spent buying notebooks, pens, and notepads at different prices. The answer key provides the simplified expressions or total amounts for each problem.
Physics Notes: Solved numerical of Physics first yearRam Chand
1. The document is a physics textbook covering solved numerical problems for the Sindh Textbook Board.
2. It was written by Dr. Ram Chand Raguel and covers topics like scalars and vectors, motion, statics, gravitation, and optics.
3. The author has visited research institutions in the US, Malaysia, Italy, and China and is a member of the American Association of Physics Teachers.
1. The document contains statistical data from surveys including percentages of respondents in different categories, correlation coefficients, and other numerical values.
2. Several key findings are reported, such as the percentage of respondents who were male versus female, the percentages who fell into different income brackets, and correlation coefficients for relationships between different variables.
3. Correlations were performed between variables like age and various attitudes, and significance values and correlation coefficients are reported for several comparisons.
8 dimension and properties table of equal leg angleChhay Teng
This document provides dimensional properties and specifications for equal leg angle steel beams of various sizes. It includes dimensions, cross-sectional area, weight, position of axes, surface area, and other mechanical properties. Sizes range from 20x20mm to 120x120mm beams with wall thicknesses of 3mm to 13mm.
This document discusses several mathematical concepts including palindromes, Fibonacci sequences, the golden ratio, Benford's law, and their appearances in nature, music, architecture, and other areas. Some key points:
- Palindromes like 121, 12321 appear when multiplying numbers by themselves.
- The Fibonacci sequence appears in patterns of growth like rabbit populations.
- The golden ratio of approximately 1.618 appears in spirals in nature and is used in design.
- Benford's law predicts the distribution of leading digits in many real-life datasets and can be used to detect fraud.
- Harmonic proportions and the golden ratio influence music, architecture, and aesthetics across disciplines
The document contains a multiplication table showing the results of multiplying single digit numbers 1-10 by other single digit numbers 1-10. It is arranged in a grid with the first number on the left representing the number being multiplied and the top number representing the number it is being multiplied by. The result of the multiplication is shown in each cell.
The European Bank for Reconstruction and Development (EBRD) belongs to a family of
multilateral development banks. As a development bank, our main mission is to help
businesses and economies thrive. Through our financial investment, business services and
involvement in high-level policy reform, we are well placed to promote entrepreneurship and
change lives. We operate in 38 countries of operation, helping them to transition to market
economies. We invest around €10billion a year into private and public sectors including
enterprises, financial institutions, venture, as well as public entities.
Everything we do pursues the goal of advancing the transition to open, market economies,
whilst fostering sustainable and inclusive growth. We operate in 38 economies that in one
shape or form are striving to achieve that transition. We invest around €10billion a year into
a mix of small, large private firms, local banks and microfinance institutions, venture capital
and local and national authorities. Our aim is to provide the right financing and a strong
valuable partnership to help these bodies grow and develop their own skills.
The Bank is unique in its outreach to small and medium-sized enterprises (SMEs). Not only
do we directly and indirectly finance €1.24 billion to over 200,000 SMEs a year, but we also
directly advise more than 2,400 small businesses a year.
We decided to extend our focus beyond SMEs into the early-stage business space. Here, the
Bank has begun investing in a number of young venture capital funds in the regions and
mobilising know-how through our Star Venture programme. Through the Star Venture
Programme, the EBRD aims at identifying high potential start-ups and to mobilise globally
sourced expertise to help these nascent firms to scale up rapidly. Star Venture leverages a
dedicated network of mentors and advisers to channel a whole range of bespoke advisory
services and industry best practices to start-ups, while also supporting accelerators in order
to benefit the wider entrepreneurial ecosystem.
1. The document contains various math exercises involving operations with fractions, percentages, algebraic expressions, and equations.
2. It provides example calculations and problems for students to work through involving topics like percentages, fractions, measurement conversions, profit/loss, rates of change, and algebraic expressions.
3. The exercises are presented in a workbook format with answers provided to check work.
This document contains an assignment sheet for a trigonometry class. It includes 9 assignments covering topics like trig functions, angle formulas, identities, and solving trig equations. There are also 3 tests scheduled. Each assignment has multiple problems or parts to complete. The document also provides introductory information on trigonometry concepts like radians, the unit circle, and right triangle trigonometry. It includes examples and practice problems for students to work through.
This document contains an assignment sheet for a trigonometry class. It includes 9 assignments covering topics like trig functions, angle formulas, trig identities, and solving trig equations. There are also 5 tests scheduled throughout the course. The assignments progress from introductory concepts like circle diagrams and function charts, to more advanced topics like trig identities and solving trig equations. Review sheets are assigned before each test.
11-1, 11-2 Perimeter & Area of Polygons.pptsmithj91
This document provides information and formulas for calculating the perimeter and area of various polygons. It includes formulas for finding the perimeter and area of rectangles, parallelograms, triangles, trapezoids, rhombi, kites, and regular polygons. The key points are that perimeter is found by adding all the sides of any polygon, and area formulas vary depending on the specific shape but generally involve multiplying base and height or base and apothem.
Triangles can Equal Circles, it's all in hour you add them up.Leland Bartlett
This document outlines Leland Bartlett's proof that the area of a circle can be calculated by summing the areas of many triangles circumscribing the circle. It begins with area formulas and trigonometry lessons. Then, it shows the calculations for 4, 8, 16, 32, 1024, and 32768 triangles approximating a circle with diameter 6 and radius 3. The sums converge on the expected area and circumference values with very small differences. In the end, it is shown that the height of the triangles approximates the radius, proving the area formula. Leland Bartlett provides training in various topics like Microsoft Office and databases as a professional consultant and trainer.
The document discusses analyzing an investment opportunity to operate metro services in a city for 5 years. It provides estimated cash flows with normal distributions and costs of capital under different macroeconomic scenarios. The task is to estimate the maximum price that could be paid and assess viability. For the maximum price, simulations are used to model cash flows and calculate VPL at a 15% cost of capital. For viability, VPL is calculated for each scenario weighted by its probability to determine if the project is worthwhile.
1. The document contains sample questions and solutions for various topics in mathematics for Year 3 secondary school students in Malaysia. It covers topics like indices, standard form, financial mathematics, and more.
2. The questions range from single-step problems to multi-part questions involving concepts like indices, financial calculations involving interest, investments, loans, and currency conversions.
3. Detailed step-by-step workings are provided for most questions to demonstrate how to arrive at the solutions.
1. The document discusses geometric concepts such as lines, angles, and the Pythagorean theorem.
2. Equations and formulas are presented for calculating lengths of sides of right triangles based on the Pythagorean theorem.
3. Approximations of irrational numbers like the square root of 2 and pi are calculated through successive decimals.
1. The document discusses geometric concepts such as lines, angles, and the Pythagorean theorem.
2. Equations and formulas are presented for calculating lengths of sides of right triangles based on the Pythagorean theorem.
3. Approximations of irrational numbers like the square root of 2 and pi are calculated through successive decimals.
1. The document summarizes solutions to problems from chapter 3 of Engineering Electromagnetics by Hayt, Buck. It provides calculations and solutions for problems regarding electric field intensity D, electric flux, volume charge density, and more.
2. Key calculations include determining D at a point given point charges, determining electric flux through surfaces, calculating enclosed charge, and finding volume charge density using the divergence of D.
3. Solutions involve applying Gauss's law and knowing how to set up integrals in different coordinate systems to calculate relevant physical quantities for the electromagnetic problems.
Hand book of Howard Anton calculus exercises 8th editionPriSim
The document contains the table of contents for a calculus textbook. It lists 17 chapters covering topics such as functions, limits, derivatives, integrals, vector calculus, and applications of calculus. It also includes 6 appendices reviewing concepts in real numbers, trigonometry, coordinate planes, and polynomial equations.
1. This document contains a math test with 34 multiple choice questions covering exponents, scientific notation, averages, and other algebra topics.
2. The test examines skills like simplifying expressions with exponents, writing numbers in scientific and standard notation, identifying equivalent algebraic expressions, and calculating averages.
3. The questions require choosing the correct multiple choice answer that represents the outcome of performing the described mathematical operation or identifies the numerical result.
This document provides practice problems for simplifying multiplication expressions in algebra. It contains 21 practice problems asking the reader to simplify expressions involving multiplication of numbers and variables. It also contains an word problem asking the reader to write an expression and calculate the total amount spent buying notebooks, pens, and notepads at different prices. The answer key provides the simplified expressions or total amounts for each problem.
Physics Notes: Solved numerical of Physics first yearRam Chand
1. The document is a physics textbook covering solved numerical problems for the Sindh Textbook Board.
2. It was written by Dr. Ram Chand Raguel and covers topics like scalars and vectors, motion, statics, gravitation, and optics.
3. The author has visited research institutions in the US, Malaysia, Italy, and China and is a member of the American Association of Physics Teachers.
1. The document contains statistical data from surveys including percentages of respondents in different categories, correlation coefficients, and other numerical values.
2. Several key findings are reported, such as the percentage of respondents who were male versus female, the percentages who fell into different income brackets, and correlation coefficients for relationships between different variables.
3. Correlations were performed between variables like age and various attitudes, and significance values and correlation coefficients are reported for several comparisons.
8 dimension and properties table of equal leg angleChhay Teng
This document provides dimensional properties and specifications for equal leg angle steel beams of various sizes. It includes dimensions, cross-sectional area, weight, position of axes, surface area, and other mechanical properties. Sizes range from 20x20mm to 120x120mm beams with wall thicknesses of 3mm to 13mm.
This document discusses several mathematical concepts including palindromes, Fibonacci sequences, the golden ratio, Benford's law, and their appearances in nature, music, architecture, and other areas. Some key points:
- Palindromes like 121, 12321 appear when multiplying numbers by themselves.
- The Fibonacci sequence appears in patterns of growth like rabbit populations.
- The golden ratio of approximately 1.618 appears in spirals in nature and is used in design.
- Benford's law predicts the distribution of leading digits in many real-life datasets and can be used to detect fraud.
- Harmonic proportions and the golden ratio influence music, architecture, and aesthetics across disciplines
The document contains a multiplication table showing the results of multiplying single digit numbers 1-10 by other single digit numbers 1-10. It is arranged in a grid with the first number on the left representing the number being multiplied and the top number representing the number it is being multiplied by. The result of the multiplication is shown in each cell.
The European Bank for Reconstruction and Development (EBRD) belongs to a family of
multilateral development banks. As a development bank, our main mission is to help
businesses and economies thrive. Through our financial investment, business services and
involvement in high-level policy reform, we are well placed to promote entrepreneurship and
change lives. We operate in 38 countries of operation, helping them to transition to market
economies. We invest around €10billion a year into private and public sectors including
enterprises, financial institutions, venture, as well as public entities.
Everything we do pursues the goal of advancing the transition to open, market economies,
whilst fostering sustainable and inclusive growth. We operate in 38 economies that in one
shape or form are striving to achieve that transition. We invest around €10billion a year into
a mix of small, large private firms, local banks and microfinance institutions, venture capital
and local and national authorities. Our aim is to provide the right financing and a strong
valuable partnership to help these bodies grow and develop their own skills.
The Bank is unique in its outreach to small and medium-sized enterprises (SMEs). Not only
do we directly and indirectly finance €1.24 billion to over 200,000 SMEs a year, but we also
directly advise more than 2,400 small businesses a year.
We decided to extend our focus beyond SMEs into the early-stage business space. Here, the
Bank has begun investing in a number of young venture capital funds in the regions and
mobilising know-how through our Star Venture programme. Through the Star Venture
Programme, the EBRD aims at identifying high potential start-ups and to mobilise globally
sourced expertise to help these nascent firms to scale up rapidly. Star Venture leverages a
dedicated network of mentors and advisers to channel a whole range of bespoke advisory
services and industry best practices to start-ups, while also supporting accelerators in order
to benefit the wider entrepreneurial ecosystem.
1. The document contains various math exercises involving operations with fractions, percentages, algebraic expressions, and equations.
2. It provides example calculations and problems for students to work through involving topics like percentages, fractions, measurement conversions, profit/loss, rates of change, and algebraic expressions.
3. The exercises are presented in a workbook format with answers provided to check work.
This document contains an assignment sheet for a trigonometry class. It includes 9 assignments covering topics like trig functions, angle formulas, identities, and solving trig equations. There are also 3 tests scheduled. Each assignment has multiple problems or parts to complete. The document also provides introductory information on trigonometry concepts like radians, the unit circle, and right triangle trigonometry. It includes examples and practice problems for students to work through.
This document contains an assignment sheet for a trigonometry class. It includes 9 assignments covering topics like trig functions, angle formulas, trig identities, and solving trig equations. There are also 5 tests scheduled throughout the course. The assignments progress from introductory concepts like circle diagrams and function charts, to more advanced topics like trig identities and solving trig equations. Review sheets are assigned before each test.
11-1, 11-2 Perimeter & Area of Polygons.pptsmithj91
This document provides information and formulas for calculating the perimeter and area of various polygons. It includes formulas for finding the perimeter and area of rectangles, parallelograms, triangles, trapezoids, rhombi, kites, and regular polygons. The key points are that perimeter is found by adding all the sides of any polygon, and area formulas vary depending on the specific shape but generally involve multiplying base and height or base and apothem.
3. IV. SQUARE ROOTS
1 =1 2 ≈ 1.414 3 ≈ 1.732 4 =2 5 ≈ 2.236
6 ≈ 2.449 7 ≈ 2.646 8 ≈ 2.828 9 =3 10 ≈ 3.162
V. FORMULAS
Perimeter: Volume:
Triangle p=a+b+c Cube V = s3
Square p = 4s Rectangular Prism V = lwh.
Rectangle p = 2l + 2w Cylinder V = πr2h
Circle (circumference) c = 2πr Cone V = (1/3)πr2h
c = πd Sphere V = (4/3)πr3
Pyramid V = (1/3)(area of base)h.
Area:
Rhombus A = (½)d1d2 Circle A = πr2
Square A = s2 Triangle A = (½)bh.a
Rectangle A = lw = bh Right Triangle A = (½)l1l2
Parallelogram A = bh Equilateral Triangle A = (¼) s2 3
Trapezoid A = (½)(b1 + b2)h.
Total Surface Area: Lateral Surface Area:
2
Cube T = 6s Rectangular Prism L = (2l + 2w)h
Rectangular Prism T = 2lw + 2lh + 2wh Cylinder L = 2πrh
Cylinder T = 2πr2 + 2πrh
Sphere T = 4πr2
Distance = Rate × Time
− y1 1y 6
1x 6
2
Slope of a Line with Endpoints (x1, y1) and (x2, y2): slope = m =
2 − x1
Distance Formula: distance between two points or length of segment with endpoints (x1, y1) and (x2, y2)
D= 1x 2 − x1 6 +1y
2
2 − y1 6 2
Midpoint Formula: midpoint of a line segment given two endpoints (x1, y1) and (x2, y2)
x + x y1 + y2
2
1 2
,
2
Circles:
x 12πr 6 , where x is the measure of the central angle of the arc
Length of an arc =
360
x 3πr 8 , where x is the measure of the central angle of the sector
Area of a sector =
360
2
MATHCOUNTS Coaching Kit 42
4. Combinations (number of groupings when the order of the items in the groups does not matter):
N!
Number of combinations = , where N = # of total items and R = # of items being chosen
R !( N − R)!
Permutations (number of groupings when the order of the items in the groups matters):
N!
Number of permutations = , where N = # of total items and R = # of items being chosen
( N − R)!
Length of a Diagonal of a Square = s 2
Length of a Diagonal of a Cube = s 3
Length of a Diagonal of a Rectangular Solid =x 2 + y 2 + z 2 , with dimensions x, y and z
N ( N − 3)
Number of Diagonals for a Convex Polygon with N Sides =
2
Sum of the Measures of the Interior Angles of a Regular Polygon with N Sides = (N − 2)180
Heron’s Formula:
For any triangle with side lengths a, b and c, Area = s( s − a )( s − b)( s − c) , where s = ½(a + b + c)
Pythagorean Theorem: (Can be used with all right triangles)
a2 + b2 = c2 , where a and b are the lengths of the legs and c is the length of the hypotenuse
Pythagorean Triples: Integer-length sides for right triangles form Pythagorean Triples – the largest
number must be on the hypotenuse. Memorizing the bold triples will also lead to other triples that are
multiples of the original.
3 4 5 5 12 13 7 24 25
6 8 10 10 24 26 8 15 17
9 12 15 15 36 39 9 40 41
Special Right Triangles:
45o – 45o – 90o 30o – 60o – 90o
hypotenuse = 2 (leg) = a 2 hypotenuse = 2(shorter leg) = 2b
hypotenuse c
leg = = longer leg = 3 (shorter leg) = b 3
2 2
longer leg hypotenuse
shorter leg = =
45°
3 2
c
a
30°
45° a c
a
60°
b
MATHCOUNTS Coaching Kit 43
5. a x
Geometric Mean: = therefore, x2 = ab and x = ab
x b
360
Regular Polygon: Measure of a central angle = , where n = number of sides of the polygon
n
360
Measure of vertex angle = 180 − , where n = number of sides of the polygon
n
Ratio of Two Similar Figures: If the ratio of the measures of corresponding side lengths is A:B,
then the ratio of the perimeters is A:B, the ratio of the areas is A 2 : B 2 and the ratio of the
volumes is A 3 : B 3 .
1 61 6
Difference of Two Squares: a 2 − b 2 = a − b a + b
Example: 12 2 − 9 2 = 112 − 96112 + 96 = 3 ⋅ 21 = 63
144 − 81 = 63
Determining the Greatest Common Factor (GCF): 5 Methods
1. Prime Factorization (Factor Tree) – Collect all common factors
2. Listing all Factors
3. Multiply the two numbers and divide by the Least Common Multiple (LCM)
Example: to find the GCF of 15 and 20, multiply 15 × 20 = 300,
then divide by the LCM, 60. The GCF is 5.
4. Divide the smaller number into the larger number. If there is a remainder, divide the
remainder into the divisor until there is no remainder left. The last divisor used is the GCF.
Example: 180 385 25 180 5 25
360 175 25 5 is the GCF of 180 and 385
25 5 0
5. Single Method for finding both the GCF and LCM
Put both numbers in a lattice. On the left, put ANY divisor of the two numbers and put the
quotients below the original numbers. Repeat until the quotients have no common factors
except 1 (relatively prime). Draw a “boot” around the left-most column and the bottom
row. Multiply the vertical divisors to get the GCF. Multiply the “boot” numbers (vertical
divisors and last-row quotients) to get the LCM.
40 140 40 140 40 140 The GCF is 2×10 = 20
2 20 70 2 20 70 2 20 70 The LCM is
10 10 2 7 2×10×2×7 = 280
VI. DEFINITIONS
Real Numbers: all rational and irrational numbers
Rational Numbers: numbers that can be written as a ratio of two integers
Irrational Numbers: non-repeating, non-terminating decimals; can’t be written as a ratio of two integers
(i.e. 7 , π )
MATHCOUNTS Coaching Kit 44
6. Integers: {…, -3, -2, -1, 0, 1, 2, 3, …}
Whole Numbers: {0, 1, 2, 3, …}
Natural Numbers: {1, 2, 3, 4, …}
Common Fraction: a fraction in lowest terms (Refer to “Forms of Answers” in the MATHCOUNTS
School Handbook for a complete definition.)
Equation of a Line:
A
Standard form: Ax + By = C with slope = −
B
Slope-intercept form: y = mx + b with slope = m and y-intercept = b
Regular Polygon: a convex polygon with all equal sides and all equal angles
1 1
Negative Exponents: x − n = and −n
= xn
xn x
Systems of Equations: x + y = 10 8 + y = 10 (8, 2) is the solution
x−y = 6 y = 2 of the system
2 x = 16
x = 8
Mean = Arithmetic Mean = Average
Mode = the number(s) occurring the most often; there may be more than one
Median = the middle number when written from least to greatest
If there is an even number of terms, the median is the average of the two middle terms.
Range = the difference between the greatest and least values
Measurements:
1 mile = 5280 feet
1 square foot = 144 square inches
1 square yard = 9 square feet
1 cubic yard = 27 cubic feet
VII. PATTERNS
Divisibility Rules:
Number is divisible by 2: last digit is 0,2,4,6 or 8
3: sum of digits is divisible by 3
4: two-digit number formed by the last two digits is divisible by 4
5: last digit is 0 or 5
6: number is divisible by both 2 and 3
8: three-digit number formed by the last 3 digits is divisible by 8
9: sum of digits is divisible by 9
10: last digit is 0
MATHCOUNTS Coaching Kit 45
7. Sum of the First N Odd Natural Numbers = N 2
Sum of the First N Even Natural Numbers = N 2 + N = N(N + 1)
N
Sum of an Arithmetic Sequence of Integers: × (first term + last term), where N = amount of
2
numbers/terms in the sequence
Find the digit in the units place of a particular power of a particular integer
Find the pattern of units digits: 71 ends in 7
72 ends in 9
(pattern repeats 73 ends in 3
every 4 exponents) 74 ends in 1
75 ends in 7
Divide 4 into the given exponent and compare the remainder with the first four exponents.
(a remainder of 0 matches with the exponent of 4)
Example: What is the units digit of 722?
22 ÷ 4 = 5 r. 2, so the units digit of 722 is the same as the units digit of 72, which is 9.
VIII. FACTORIALS (“n!” is read “n factorial”)
n! = (n)×(n −1)×(n − 2)×…×(2)×(1) Example: 5! = 5 × 4 × 3 × 2 × 1 = 120
0! = 1
1! = 1
2! = 2 Notice 6! 6 × 5 × 4 × 3 × 2 ×1
= = 30
3! = 6 4! 4 × 3 × 2 ×1
4! = 24
5! = 120
6! = 720
7! = 5040
IX. PASCAL’S TRIANGLE
Pascal’s Triangle Used for Probability:
Remember that the first row is row zero (0). Row 4 is 1 4 6 4 1. This can be used to
determine the different outcomes when flipping four coins.
1 4 6 4 1
way to get ways to get ways to get ways to get way to get
4 heads 0 tails 3 heads 1 tail 2 heads 2 tails 1 head 3 tails 0 heads 4 tails
For the Expansion of (a + b)n , use numbers in Pascal’s Triangle as coefficients.
1 (a + b)0 = 1
1 1 (a + b)1 = a + b
1 2 1 (a + b)2 = a2 + 2ab + b2
1 3 3 1 (a + b)3 = a3 + 3a2b + 3ab2 + b3
1 4 6 4 1 (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
1 5 10 10 5 1 (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5
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8. For 2n, add all the numbers in the nth row. (Remember the triangle starts with row 0.)
1 20 = 1
1 1 21 = 1 + 1 = 2
1 2 1 22 = 1 + 2 + 1 = 4
1 3 3 1 23 = 1 + 3 + 3 + 1 = 8
1 4 6 4 1 24 = 1 + 4 + 6 + 4 + 1 = 16
1 5 10 10 5 1 25 = 1 + 5 + 10 + 10 + 5 + 1 = 32
X. SQUARING A NUMBER WITH A UNITS DIGIT OF 5
(n5)2 = n × (n + 1) 2 5 , where n represents the block of digits before the units digit of 5
Examples:
(35)2 = 3×(3+1) 2 5 (125)2 = 12×(12+1) 2 5
= 3×(4) 2 5 = 12×(13) 2 5
=1225 =15625
= 1,225 = 15,625
XI. BASES
%DVH GHFLPDO ² RQO XVHV GLJLWV ²
Base 2 = binary – only uses digits 0 – 1
%DVH RFWDO ² RQO XVHV GLJLWV ²
Base 16 = hexadecimal – only uses digits 0 – 9, A – F (where A=10, B=11, …, F=15)
Changing from Base 10 to Another Base:
What is the base 2 representation of 125 (or “125 base 10” or “12510”)?
We know 125 = 1(102) + 2(101) + 5(100) = 100 + 20 + 5, but what is it equal to in base 2?
12510 = ?(2n) + ?(2n-1) + … + ?(20)
The largest power of 2 in 125 is 64 = 26, so we now know our base 2 number will be:
?(26) + ?(25) + ?(24) + ?(23) + ?(22) + ?(21)+ ?(20) and it will have 7 digits of 1’s and/or 0’s.
Since there is one 64, we have: 1(26) + ?(25) + ?(24) + ?(23) + ?(22) + ?(21)+ ?(20)
We now have 125 – 64 = 61 left over, which is one 32 = 25 and 29 left over, so we have:
1(26) + 1(25) + ?(24) + ?(23) + ?(22) + ?(21)+ ?(20)
In the left-over 29, there is one 16 = 24, with 13 left over, so we have:
1(26) + 1(25) + 1 (24) + ?(23) + ?(22) + ?(21)+ ?(20)
In the left-over 13, there is one 8 = 23, with 5 left over, so we have:
1(26) + 1(25) + 1(24) + 1(23) + ?(22) + ?(21)+ ?(20)
In the left-over 5, there is one 4 = 22, with 1 left over, so we have:
1(26) + 1(25) + 1(24) + 1(23) + 1(22) + ?(21)+ ?(20)
In the left-over 1, there is no 2 = 21, so we still have 1 left over, and our expression is:
1(26) + 1(25) + 1(24) + 1(23) + 1(22) + 0(21)+ ?(20)
The left-over 1 is one 20, so we finally have:
1(26) + 1(25) + 1(24) + 1(23) + 1(22) + 0(21)+ 1(20) = 11111012
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9. Now try What is the base 3 representation of 105?
The largest power of 3 in 105 is 81 = 34, so we now know our base 3 number will be:
?(34) + ?(33) + ?(32) + ?(31)+ ?(30) and will have 5 digits of 2’s, 1’s, and/or 0’s.
Since there is one 81, we have: 1(34) + ?(33) + ?(32) + ?(31)+ ?(30)
In the left-over 105 – 81 = 24, there is no 27 = 33, so we still have 24 and the expression:
1(34) + 0(33) + ?(32) + ?(31)+ ?(30)
In the left-over 24, there are two 9’s (or 32’s), with 6 left over, so we have:
1(34) + 0(33) + 2(32) + ?(31)+ ?(30)
In the left-over 6, there are two 3’s (or 31’s), with 0 left over, so we have:
1(34) + 0(33) + 2(32) + 2(31)+ ?(30)
Since there is nothing left over, we have no 1’s (or 30’s), so our final expression is:
1(34) + 0(33) + 2(32) + 2(31)+ 0(30) = 102203
The following is another fun algorithm for converting base 10 numbers to other bases:
12510 = ?2 10510 = ?3 12510 = ?16
1 r.1 1 r.0 7 r.13(D)
2 3 r.1 3 3 r.2 16 125
2 7 r.1 3 11 r.2
2 15 r.1 3 35 r.0 12510 = 7D16
2 31 r.0 3 105
2 62 r.1
Start
here 2 125 10510 = 102203
12510 = 11111012 xyzn = (x × n2) + (y × n1) + (z × n0)
Notice: Everything in bold shows the first division operation. The first remainder will be the last digit
in the base n representation, and the quotient is then divided again by the desired base. The process is
repeated until a quotient is reached that is less than the desired base. At that time, the final quotient
and remainders are read downward.
XII. FACTORS
Determining the Number of Factors of a Number: First find the prime factorization (include the 1 if a
factor is to the first power). Increase each exponent by 1 and multiply these new numbers together.
Example: How many factors does 300 have?
The prime factorization of 300 is 22 × 31 × 52 . Increase each of the exponents by 1 and multiply
these new values: (2+1) × (1+1) × (2+1) = 3 × 2 × 3 = 18. So 300 has 18 factors.
Finding the Sum of the Factors of a Number:
Example: What is the sum of the factors of 10,500?
(From the prime factorization 22 × 31 × 53 × 71, we know 10,500 has 3 × 2 × 4 × 2 = 48 factors.)
The sum of these 48 factors can be calculated from the prime factorization, too:
(20 + 21 + 22)(30 + 31)(50 + 51 + 52 + 53)(70 + 71) = 7 × 4 × 156 × 8 = 34,944.
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