Carry out the four operations on integers, fractions and decimals, apply the order of operations, and round answers sensibly

What this dot point is asking

SEAB wants you to add, subtract, multiply and divide whole numbers (including negative numbers), fractions and decimals, to apply the correct order of operations, and to round your answers to a sensible degree of accuracy. These skills sit underneath almost every other topic, so being fast and accurate here saves time everywhere else.

The answer

The four operations and negative numbers

The four operations are addition, subtraction, multiplication and division. With negative numbers, two rules cover most cases:

• Adding a negative is the same as subtracting: $5 + (-3) = 5 - 3 = 2$.
• Multiplying or dividing two numbers with the same sign gives a positive; with different signs gives a negative. So $(-4) \times (-3) = 12$ but $(-4) \times 3 = -12$.

Order of operations

When an expression mixes operations, work in this order: Brackets, then Indices (powers), then Division and Multiplication (left to right), then Addition and Subtraction (left to right). A useful memory aid is BIDMAS.

For example, $20 - 6 \div 2 = 20 - 3 = 17$, because the division happens before the subtraction.

Fractions

To add or subtract fractions, rewrite them over a common denominator, then combine the numerators:

To multiply, multiply numerators and denominators: $\dfrac{2}{3} \times \dfrac{3}{4} = \dfrac{6}{12} = \dfrac{1}{2}$. To divide, multiply by the reciprocal (flip the second fraction): $\dfrac{2}{3} \div \dfrac{4}{5} = \dfrac{2}{3} \times \dfrac{5}{4} = \dfrac{10}{12} = \dfrac{5}{6}$.

Decimals and rounding

Decimals are fractions written with a decimal point. To round, look at the first digit you are cutting off: if it is $5$ or more, round up; otherwise round down. For example, $3.764$ rounded to $1$ decimal place is $3.8$, because the next digit is $6$. Rounded to $2$ decimal places it is $3.76$.

A sensible rule for money is to round to $2$ decimal places (the nearest cent), and for many measurements $3$ significant figures is enough unless the question says otherwise.

Examples in context

Example 1. A shopping bill. A pen costs \1.20$and a notebook costs$\$2.50$. You buy $3$ pens and $2$ notebooks. The cost is 3 \times 1.20 + 2 \times 2.50 = 3.60 + 5.00 = \8.60$. The multiplications happen before the addition, exactly as BIDMAS requires, which is why writing the expression first prevents errors.

Example 2. Sharing a length. A ribbon $\dfrac{3}{4}$ metre long is cut into pieces each $\dfrac{1}{8}$ metre long. The number of pieces is $\dfrac{3}{4} \div \dfrac{1}{8} = \dfrac{3}{4} \times \dfrac{8}{1} = \dfrac{24}{4} = 6$. Dividing by a fraction gives a larger number, which makes sense because the pieces are small.

Try this

• Cue. Work out $7 + 2 \times (10 - 4)$. Brackets give $6$, then $2 \times 6 = 12$, then $7 + 12 = 19$.
• Cue. Evaluate $\dfrac{5}{6} - \dfrac{1}{3}$. Common denominator $6$: $\dfrac{5}{6} - \dfrac{2}{6} = \dfrac{3}{6} = \dfrac{1}{2}$.
• Cue. Round $48.3719$ to $2$ decimal places. The third decimal is $1$, so round down to $48.37$.