POP

E1.6 The Four Operations

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01 — Explanation

What order do I carry out operations in?

BIDMAS/BODMAS: Brackets, Indices, Division and Multiplication (left to right), then Addition and Subtraction (left to right). On multi-step non-calculator questions, scan the whole expression first to spot hidden priority operations before doing anything else.

02 — Explanation

How do I add and subtract fractions?

Find a common denominator first:

\frac{a}{c}\pm\frac{b}{d} = \frac{ad \pm bc}{cd}

\frac{3}{4}+\frac{1}{6} = \frac{9+2}{12}=\frac{11}{12}

03 — Explanation

How do I multiply and divide fractions?

\text{Multiply: } \frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd} \qquad \text{Divide: } \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}

Cancel common factors before multiplying to keep numbers small: $\dfrac{5}{6}\times\dfrac{3}{10}=\dfrac{1}{4}$ . Division always becomes multiplication by the reciprocal ("keep, change, flip"): $\dfrac{2}{5}\div\dfrac{4}{15}=\dfrac{2}{5}\times\dfrac{15}{4}=\dfrac{3}{2}$ .

04 — Explanation

How do I handle mixed numbers safely, especially with subtraction?

Convert every mixed number to an improper fraction before combining them — especially if subtraction could push a part negative.

Examiner Tip: Splitting $3\frac{1}{2} - 5\frac{2}{3}$ into whole-number and fraction parts separately, i.e. $(3-5)+(\frac{1}{2}-\frac{2}{3})$ , is a common shortcut that goes wrong with sign errors. Convert to improper fractions first, every time.

Worked Example

Without a calculator, work out $2\dfrac{1}{4} + 1\dfrac{5}{6} \div \dfrac{11}{12}$ , giving your answer as a mixed number.

BIDMAS: division happens before addition, so evaluate $1\frac{5}{6}\div\frac{11}{12}$ first.
Convert to an improper fraction: $1\frac{5}{6} = \frac{11}{6}$ .
Keep–change–flip: $\frac{11}{6}\div\frac{11}{12} = \frac{11}{6}\times\frac{12}{11} = \frac{12}{6} = 2$ .
Now add: $2\frac{1}{4} + 2 = 4\frac{1}{4}$ .

05 — Explanation

How do the four operations work with negative numbers?

Operation	Rule	Example
Adding a negative	Same as subtracting	$5 + (-3) = 5 - 3 = 2$
Subtracting a negative	Same as adding	$5 - (-3) = 5 + 3 = 8$
Multiplying/dividing, same signs	Result is positive	$(-4)\times(-2)=8$
Multiplying/dividing, different signs	Result is negative	$(-4)\times2=-8$

Examiner Tip: "Two minuses make a plus" only applies when the signs sit next to each other ( $-\times-$ or $-\,-\,$ ). It does not mean every negative answer is wrong — $(-4)\times2$ is correctly $-8$ , not $8$ .

06 — Explanation

How do I apply the four operations to a real-life context, like temperature change?

Treat "rise" as addition and "fall" as subtraction of a positive amount, keeping careful track of the sign of the running total throughout.

Worked Example

The temperature at midnight is $-6°\text{C}$ . It rises by $4°\text{C}$ by 6 a.m., then falls by $9°\text{C}$ by midday. Find the temperature at midday.

Start: $-6$ .
Rise of $4$ : $-6+4=-2$ .
Fall of $9$ : $-2-9=-11$ .
The temperature at midday is $-11°\text{C}$ .

07 — Explanation

How do brackets change the order in which I combine mixed operations?

Anything inside a bracket must be fully resolved into a single value before it interacts with anything outside it, even if that means working out an addition or subtraction "out of the usual order" because it happens to sit inside brackets. Scan an entire expression for bracketed groups before starting any arithmetic, so you don't accidentally apply BIDMAS term-by-term across a bracket boundary.