MPA Math Refresher 2022

Quick Reference Guide

A list of short reminders about key math topics.

Where to Find This Page

https://wilcoxen.maxwell.insightworks.com/pages/mathref

A. General Advice

  1. Tips for Doing Math Reliably
  2. Key principles: (1) use a calculator and do not do mental math; (2) write out all steps, especially for algebra; (3) use scale units or scientific notation for large numbers; (4) check your work.

  3. Order of Operations, Khan Academy
  4. Evaluate expressions in this order: (1) Parentheses, (2) Exponents, (3) Multiplication and Division (left to right), (4) Addition and Subtraction (left to right): PEMDAS. Calculators will do this correctly if you enter the full calculation at once.

  5. Working with Percentages
  6. Use the decimal version of a percentage in calculations but use the percentage itself in written reports. Examples: (1) 5% interest on $250 = 0.05*$250 = $12.5; (2) a 20% decrease from an initial budget of $1 million is a reduction of 0.2*$1 million or $200,000 dollars; (3) 100 new people added to a workforce of 500 is a 20% increase.

B. Working with Large Numbers

  1. Scale Units for Large Numbers
  2. Report large numbers using scale units like `k` for thousands (`10^3`), `M` for millions (`10^6`) or `B` for billions (`10^9`) rather than writing out the full number. Examples:

    123,000 `->` 123 k
    34,500,000 `->` 34.5 M
    5,670,000,000 `->` 5.67 B
  3. Scientific Notation
  4. An alternative to using scale units is to use scientific notation. It can be easier during calculations but is sometimes harder to visualize. Examples:

    123,000 `->` 1.23 E5 or `1.23 \times 10^5`
    34,500,000 `->` 3.45 E7 or `3.45 \times 10^7`
  5. Engineering Notation
  6. An alternative to scientific notation that uses powers of ten that are multiples of 3: `10^3`, `10^6`, `10^9`, etc. If your calculator supports it, it's a convenient bridge between the two approaches above. Examples:

    123,000 `->` 123 E3 = 123 k
    34,500,000 `->` 34.5 E6 = 34.5 M
  7. Rounding
  8. In general, when reporting results, round numbers to 3-4 digits as appropriate for the problem. For intermediate steps, keep at least one more digit or avoid rounding. Examples:

    1.7142857 `->` 1.714
    25,467,235.16 `->` 25.5 M

C. Units

  1. Working with Units: Dimensional Analysis, Khan Academy
  2. It's often very useful to keep track of units in an analysis by carrying them through the calculation as though they were algebraic variables. For example if `v` is 60 miles/hour, and `t` is 2 hours, , then `vt` will be 120 and the units will be the units of `v` times the units of `t`: (miles/hour)*(hour) = miles.

D. Fractions and Ratios

  1. Multiplying Fractions
  2. Multiply two fractions by multiplying their numerators and denominators:

    `(x/y)*(a/b)=(x*a)/(y*b)` `->`
    `(2/3)*(4/5)=(2*4)/(3*5)=8/15`
  3. Multiplying a Fraction by a Number or Variable
  4. Multiplying a fraction by a number is the same as multiplying it by the number divided by 1:

    `x*(a/b)=(x/1)*(a/b)=(x*a)/(1*b)=(x*a)/b` `->`
    `4*(6/7)=(4*6)/7=24/7`
  5. Dividing Fractions, Khan Academy
  6. Dividing by a fraction is the same as multiplying by its reciprocal:

    `x/((a / b))=x*(b/a)` `->` `2/((4 / 5))=2*(5/4)=10/4=5/2`
  7. Simplifying Fractions
  8. Cancel out multiplicative terms that are are common to the numerator and denominator:

    `x^2/{x*y}={x*x}/{x*y}=(x/x) * (x/y)=x/y` `->` `42/4=(2*3*7)/(2*2)=(3*7)/2=21/2`
  9. Fractions and Ratios
  10. A ratio can be expressed as a fraction and vice versa. Use the fraction form or its decimal equivalent for calculations and the ratio form for written reports:

    `x:y = x/y` `->` `1:3 = 1/3`

E. Summation Notation

  1. Summation Notation, Khan Academy
  2. A capital Greek sigma, `\Sigma`, is a concise way to write a sum:

    `\sum_(n=1)^3 x^n = x^1+x^2+x^3` `->`
    `\sum_(n=0)^2 2^n = 2^0+2^1+2^2=7`

F. Exponents

  1. Basic Exponents, Khan Academy
  2. Anything raised to the zeroth power is 1 and 1 raised to any power is 1:

    `x^0=1` `->`
    `2^0=1`
         
    `1^n=1` `->`
    `1^5=1`
  3. Negative Exponents, Khan Academy
  4. A variable raised to a negative power is the same as the reciprocal of the variable raised to a positive power:

     `x^{-a}=1/x^a` `->`
    `2^{-2}=1/2^2=1/4`
  5. Exponents of Products and Quotients
  6. A product raised to a power is equal to the product of each of the terms raised to the power and quotients (fractions) raised to a power are equal to a fraction with the numerator and denominator each raised to the power:

    `(xy)^a=x^a*y^a` `->`
    `(2*3)^2=2^2*3^2=36`
         
    `(x/y)^a=x^a/y^a` `->`
    `(2/3)^2=2^2/3^2=4/9`
  7. Multiplying Exponential Terms with the Same Base
  8. If two exponential terms with the same base are multiplied, the result is the base raised to the sum of the exponents:

     `x^a*x^b=x^{a+b}` `->`
    `2^2*2^3=2^(2+3)=2^5=32`
  9. Exponential Term Raised to a Power
  10. If an exponential term is raised to a power, the result is the original base raised to the product of the exponents:

    `(x^a)^b=x^{a*b}` `->`
    `(2^2)^3=2^2*2^2*2^2=2^(2+2+2)=2^6=64`

G. Geometry

  1. Area of a Rectangle with Sides x and y
  2. Area`=xy`

    Example: `x=5 ft`, `y=6 ft` `->` Area`=30 ft^2`
  3. Area of a Right Triangle with Base b and Height h
  4. Area`=1/2 bh`

    Example: `b=5280 ft`, `h=200 ft` `->` Area`=528 k ft^2`
  5. Area of a Trapezoid with Bases `b_1` and `b_2` and height `h`
  6. Area`=((b_1+b_2))/2 h`

    Example: `b_1=10 m`, `b_2=6 m`, `h=3 m`
    `->` Area`=24 m^2`
  7. Area of a Circle with Radius r
  8. Area`=\pi r^2`

    Example: `r=5 m` `->` Area`=78.54 m^2`

H. Algebra

  1. Distributing a Single Term
  2. Distribute a term outside parentheses that aren't raised to a power by multiplying each of the terms in the parentheses by it:

    `a*(b+c)=a*b+a*c`  

    `->``2*(3+4)=2*3+2*4=14`    
  3. Factoring Out a Single Term, Khan Academy
  4. Factoring is the reverse of distributing:

    `a*b+a*c=a*(b+c)`
    `->``2*3+2*4=2*(3+4)=14`
  5. Distributing a Term in Parentheses
  6. Distribute a term in parentheses by multiplying each of the terms in the other parentheses by each term in the first parentheses:

    `(a+b)*(x+y)=a*x+a*y+b*x+b*y`
    `->``(2+3)*(4+5)=2*4+2*5+3*4+3*5=45`
  7. Factoring out a Term in an Expression Raised to a Power
  8. Factor out a common term inside parentheses that are raised to a power by: (1) factoring it out within the parentheses, and then (2) applying the exponent rule:

    `(a*b+a*c)^d=(a*(b+c))^d=a^d*(b+c)^d`
    `->``(2*3+2*4)^2=(2*(3+4))^2=2^2*(3+4)^2=196`
  9. Distributing a Term Across a Parenthesized Term Raised to a Power
  10. Distribute a multiplicative term outside parentheses that are raised to a power by first raising the term to one over the power and then multiplying each of the terms in the parentheses by it:

    `a*(b+c)^d=(a^{1/d}*b+a^{1/d}*c)^d`
    `->``4*(3+4)^2=(4^{1/2}*3+4^{1/2}*4)^2=(2*3+2*4)^2=196`

I. Graphing

Intercepts, Khan Academy

The X intercept of a curve is the value of X where the curve hits the X axis and Y=0. The Y intercept is the value of Y when the curve hits the Y axis and X=0. Examples:

    X intercept   Y intercept
`y=200-4x` `->`
`y=0, x = 50`   `x=0, y = 200`
`x+40=2y` `->` `y=0, x= -40`   `x=0, y=20`
Slope from Two Points, Khan Academy

The slope of a curve is the change between the two points in the Y variable, `\Delta Y`, divided by the change in the X variable, `\Delta X`. Variable `m` is often used for the slope.

`m=(\Delta Y)/(\Delta X)=(y_2-y_1)/(x_2-x_1)`
`->` If: point 1 has `x_1=2`, `y_1=5` and point 2 has `x_2=4`, `y_2=1`
  Then: `m=(1-5)/(4-2)=-2`
Slope from a Slope-Intercept Equation, Khan Academy

In an equation of the form `y=m*x+b`, the slope is `m` (and `b` is the Y intercept). Example:

`y=1.2x-30` `->`
`m=1.2` and `b=-30`
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Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 08/28/2022