Skills Beta

Math you need: the one-page sheet

Fractions, Decimals and Percents

A part of a whole is the part divided by the whole, a decimal from 0 to 1, and a percent is that decimal times 100. Pick the whole from the words of the question, and compare groups of different sizes with decimals or percents, never raw counts.

A part of a whole is the part divided by the whole: $\frac{\text{part}}{\text{whole}}$ . The result is a decimal from 0 to 1.
A percent is a decimal times 100: $0.375 = 37.5\%$ . To go back, divide by 100.
The whole is the group the question says "of": "what part of the sport players" tells you to divide by the number of sport players.
Compare groups of different sizes with decimals or percents, not counts.

that number goes on the bottom of the fraction that number goes on top the decimal is between 0 and 1; times 100 gives the percent

percent: A percent is a part out of 100: the decimal form of a part of a whole, multiplied by 100. For example, 0.75 is 75%.
part of a whole: A part of a whole is the count in one group divided by the count in the whole group it belongs to, giving a number from 0 to 1.

Formulas, Order of Operations and Square Roots

Evaluate a formula by substituting values, then following the order of operations: grouped terms first (brackets, fraction bars, roots), then powers, then multiplication and division, then addition and subtraction. ∑ adds a list of terms, and a square root covers everything under it, so type brackets on a calculator.

Order of operations: brackets (and anything grouped by a fraction bar or root) first, then powers and roots, then multiplication and division, then addition and subtraction.
A square root $\sqrt{a}$ is the nonnegative number whose square is $a$ : $\sqrt{0.25} = 0.5$ . Squares are never negative: $(-3)^2 = 9$ .
Summation: $\sum x_i$ says add every value; $\sum (x_i - 5)^2$ says subtract 5 from each value, square each result, then add.
Show the work the way the exam asks: the formula, the numbers substituted, then the answer.

substitute the numbers for the letters first work out what is inside them before anything else every reader of the formula gets the same answer type it on a calculator with brackets around everything it covers

order of operations: The agreed order for evaluating an expression: grouped terms (brackets, fraction bars, roots) first, then powers, then multiplication and division, then addition and subtraction.
square root: The square root of a nonnegative number <math class="stat-m"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>, written <math class="stat-m"><semantics><mrow><msqrt><mi>a</mi></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{a}</annotation></semantics></math>, is the nonnegative number whose square is <math class="stat-m"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>; for example <math class="stat-m"><semantics><mrow><msqrt><mn>0.25</mn></msqrt><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">\sqrt{0.25} = 0.5</annotation></semantics></math>.
summation: The symbol <math class="stat-m"><semantics><mrow><mo largeop="true" movablelimits="false">∑</mo></mrow><annotation encoding="application/x-tex">\sum</annotation></semantics></math> stands for "add up": <math class="stat-m"><semantics><mrow><mo largeop="true" movablelimits="false">∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\sum x_i</annotation></semantics></math> is the sum of all the values <math class="stat-m"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_1, x_2, \ldots, x_n</annotation></semantics></math>.

Inequalities in Words

Translate words into inequalities before computing: at least is ≥ , at most and no more than are ≤ , more than is > , fewer than is < . For counts, change strict inequalities to include the next whole number ( X < 3 is X ≤ 2 ), and list the included values to check.

At least $k$ : $X \ge k$ . At most $k$ and no more than $k$ : $X \le k$ . More than $k$ : $X > k$ . Fewer than $k$ : $X < k$ . No fewer than $k$ : $X \ge k$ .
For counts, $X < k$ is $X \le k - 1$ and $X > k$ is $X \ge k + 1$ : "fewer than 3" is 0, 1 or 2.
"At least one" is every count except 0.
For continuous measurements, whether the exact boundary is included makes almost no difference.

"at least 3" is X ≥ 3 , "fewer than 3" is X < 3 "at least", "at most", "no more than" and "inclusive" include it; "more than" and "fewer than" do not X < 3 becomes X ≤ 2 , and X > 4 becomes X ≥ 5 the slip "fewer than 3 is 3 or fewer" shows up at once

at least: Phrases that set a boundary on a count or measurement: at least k (<math class="stat-m"><semantics><mrow><mo>≥</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\ge k</annotation></semantics></math>), at most k or no more than k (<math class="stat-m"><semantics><mrow><mo>≤</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\le k</annotation></semantics></math>), more than k (<math class="stat-m"><semantics><mrow><mo>></mo><mi>k</mi></mrow><annotation encoding="application/x-tex">> k</annotation></semantics></math>), fewer than k (<math class="stat-m"><semantics><mrow><mo><</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">< k</annotation></semantics></math>), no fewer than k (<math class="stat-m"><semantics><mrow><mo>≥</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\ge k</annotation></semantics></math>). For counts, fewer than k is the same as at most <math class="stat-m"><semantics><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k - 1</annotation></semantics></math>.

Factorials and Combinations

A factorial n! multiplies n by each whole number below it and counts the orders of n items ( 0! = 1 ). A combination _nC_k = (n!)/(k! (n − k)!) counts the ways to choose k of n when order does not matter; it equals _nC_(n − k) .

A factorial $n! = n \times (n - 1) \times \cdots \times 2 \times 1$ counts the orders of n different items; $0! = 1$ .
A combination ${}_nC_k$ $= \binom{n}{k}$ $= \frac{n!}{k!\,(n - k)!}$ counts the ways to choose k of n items when order does not matter.
Ordered choices overcount: $10 \times 9 \times 8 = 720$ counts each set of 3 six times, so ${}_{10}C_3 = 120$ .
On a calculator or in Bluebook, type ${}_{10}C_3$ as 10C3 or nCr(10, 3).

there are n choices for the first place, n − 1 for the next, and so on: n! orders n×(n − 1)×⋯ for k places, but each set of k appears k! times _nC_k = (n!)/(k! (n − k)!) counts the sets of k _nC_k = _nC_(n − k) , and _nC_0 = _nC_n = 1

factorial: For a whole number n, <math class="stat-m"><semantics><mrow><mi>n</mi><mo>!</mo><mo>=</mo><mi>n</mi><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mo>⋯</mo><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n! = n \times (n - 1) \times \cdots \times 2 \times 1</annotation></semantics></math>, the number of ways to put n different items in order. By definition <math class="stat-m"><semantics><mrow><mn>0</mn><mo>!</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0! = 1</annotation></semantics></math>.
combination: The number of ways to choose k items from n when order does not matter: <math class="stat-m"><semantics><mrow><msub><mrow></mrow><mi>n</mi></msub><msub><mi>C</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">{}_nC_k</annotation></semantics></math><wbr><math class="stat-m"><semantics><mrow><mo form="infix">=</mo><mrow><mo>(</mo><mfrac linethickness="0"><mi>n</mi><mi>k</mi></mfrac><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">= \binom{n}{k}</annotation></semantics></math><wbr><math class="stat-m"><semantics><mrow><mo form="infix">=</mo><mfrac><mrow><mi>n</mi><mo>!</mo></mrow><mrow><mi>k</mi><mo>!</mo><mspace width="0.1667em"></mspace><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">= \frac{n!}{k!\,(n - k)!}</annotation></semantics></math>, read "n choose k".

Lines: Slope and Intercept

A line y = a + bx has slope b , the change in y for each one-unit increase in x (rise over run), and y-intercept a , the value of y when x = 0 . Interpret both in context with units, and the intercept only when x = 0 makes sense.

A line is $y = a + bx$ . The slope $b$ is the change in y for each one-unit increase in x; its units are y units per x unit.
Slope from two points: $b$ $= \frac{y_2 - y_1}{x_2 - x_1}$ , rise over run, subtracting in the same order on top and bottom.
The y-intercept $a$ is the value of y when $x = 0$ . Interpret it only when $x = 0$ makes sense.
In context: "For each additional hour, the cost increases by 3 dollars."

a is the y-intercept and b is the slope y changes by b : up if b > 0 , down if b < 0 y equals a , which has meaning only if x = 0 makes sense in context slope = rise over run, then solve for a from one point

slope: The slope of a line <math class="stat-m"><semantics><mrow><mi>y</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">y = a + bx</annotation></semantics></math> is <math class="stat-m"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>, the change in y for each one-unit increase in x. From two points, <math class="stat-m"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math><wbr><math class="stat-m"><semantics><mrow><mo form="infix">=</mo><mfrac><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">= \frac{y_2 - y_1}{x_2 - x_1}</annotation></semantics></math>.
y-intercept: The y-intercept of a line <math class="stat-m"><semantics><mrow><mi>y</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">y = a + bx</annotation></semantics></math> is <math class="stat-m"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>, the value of y when <math class="stat-m"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x = 0</annotation></semantics></math>.