The Maxwell School
Syracuse University
Syracuse University
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.
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.
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.
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 |
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` |
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 |
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 |
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.
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` |
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` |
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` |
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` |
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` |
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` |
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` |
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` |
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` |
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` |
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` |
Area`=xy`
| Example: | `x=5 ft`, `y=6 ft` | `->` | Area`=30 ft^2` |
Area`=1/2 bh`
| Example: | `b=5280 ft`, `h=200 ft` | `->` | Area`=528 k ft^2` |
Area`=((b_1+b_2))/2 h`
| Example: | `b_1=10 m`, `b_2=6 m`, `h=3 m` |
`->` | Area`=24 m^2` |
Area`=\pi r^2`
| Example: | `r=5 m` | `->` | Area`=78.54 m^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` |
Factoring is the reverse of distributing:
| `a*b+a*c=a*(b+c)` |
| `->``2*3+2*4=2*(3+4)=14` |
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` |
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` |
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` |
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` |
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` |
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` |