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Numbers and operations — Formula Sheet

5 formulas from Math Foundations, each with what it means, when to use it and the trap it sets.

Order of operations

() \to x^{n} \to \times \div \to + -

What it says: Brackets first, then powers, then multiply and divide left to right, then add and subtract left to right.
When to reach for it: Any expression with more than one operation in it.
Watch out: Multiplication and division share a rank, and so do addition and subtraction — you work left to right within each rank, not multiplication before division.

8 - 2 \times 3^{2} = 8 - 2 \times 9 = 8 - 18 = - 10

The power resolves before the multiplication, and the multiplication before the subtraction.

Signs when multiplying and dividing

(-) (-) = +, (-) (+) = -

What it says: An even number of minus signs gives a plus; an odd number gives a minus.
When to reach for it: Any product or quotient where at least one factor is negative.
Watch out: This rule is for multiplying and dividing only. Adding two negatives keeps them negative: −3 + (−4) = −7.

(- 2) (- 3) (- 4) = - 24

Three minus signs — an odd count — so the result is negative.

Adding fractions

\frac{a}{b} + \frac{c}{d} = \frac{a d + c b}{b d}

What it says: Put both fractions over the same bottom, then add only the tops.
When to reach for it: Two fractions with different denominators that need to become one.
Watch out: Never add the denominators. 1/2 + 1/3 is 5/6, not 2/5 — the denominator names the size of the piece, and adding pieces does not change their size.

\frac{1}{2} + \frac{1}{3} = \frac{3 + 2}{6} = \frac{5}{6}

Six is the shared denominator; only the numerators are summed.

Dividing fractions

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

What it says: Dividing by a fraction is multiplying by it upside down.
When to reach for it: A fraction divided by a fraction, or a fraction stacked inside a fraction.
Watch out: Flip the second fraction only. Flipping both gives you the reciprocal of the answer.

\frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2} = \frac{9}{8}

The divisor is inverted; the dividend is untouched.

Percentage change

change = \frac{new - old}{old} \times 100%

What it says: How much it moved, measured against where it started.
When to reach for it: Any "increased by / decreased by / what percent" question.
Watch out: The denominator is always the original value. A rise from 40 to 50 is +25%, but the fall back from 50 to 40 is −20% — the two are not the same percentage.

\frac{50 - 40}{40} \times 100% = 25%

Measured against 40, the starting value.

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