Topic 2 of 7

Arithmetic with negative fractions and decimals

Step 9 · Proportional reasoning topics

Add, subtract, multiply and divide positive and negative numbers, including fractions and decimals, using the number line and the sign rules.

Adding and subtracting signed numbers

On a number line, adding a positive number moves right and adding a negative number moves left. Subtracting a number is the same as adding its opposite.

−2 + 5: start at −2 and move 5 to the right, landing on 3.

Example: Different signs

What is −7 + 3?

The signs differ, so take the smaller size from the larger: 7 − 3 = 4.
The −7 is bigger in size, so the answer is negative.

Answer−4

Example: Subtracting a negative

What is 5 − (−2)?

Subtracting −2 is adding 2.
5 + 2 = 7.

Answer7

Multiplying and dividing signed numbers

Multiply or divide the sizes, then decide the sign: same signs give a positive answer, different signs a negative one.

Signs	Example	Answer
both positive	3 × 4	12
both negative	−3 × (−4)	12
one of each	3 × (−4)	−12
one of each	−12 ÷ 3	−4

Example: Dividing

What is −24 ÷ (−6)?

24 ÷ 6 = 4.
The signs match, so the answer is positive.

Answer4

Fractions and decimals with signs

Negative fractions follow the same rules. $-\frac{3}{4}$ , $\frac{-3}{4}$ and $\frac{3}{-4}$ are all the same number.

Example: Adding negative fractions

What is $-\frac{1}{2} + \frac{1}{3}$ ?

In sixths: $-\frac{3}{6} + \frac{2}{6}$ .
−3 + 2 = −1.

Answer $-\frac{1}{6}$

Example: Multiplying fractions

What is $-\frac{2}{3} \times \left(-\frac{9}{4}\right)$ ?

The signs match, so the answer is positive.
$\frac{2 \times 9}{3 \times 4} = \frac{18}{12} = \frac{3}{2}$

Answer $\frac{3}{2}$ , or $1\frac{1}{2}$

Example: Decimals

What is −2.5 × 0.4?

2.5 × 0.4 = 1.
The signs differ, so the answer is negative.

Answer−1

Try it yourself

Work each one on paper first, then open the answer to check.

What is −9 + 4?
Show answer
−5
What is −3 − 6?
Show answer
−9
What is 2 − (−7)?
Show answer
9
What is −8 × (−1.5)?
Show answer
12
What is −45 ÷ 9?
Show answer
−5
What is $\frac{3}{4} - \left(-\frac{1}{2}\right)$ ?
Show answer
$\frac{5}{4}$ , or $1\frac{1}{4}$
Write $-\frac{7}{8}$ as a decimal.
Show answer
−0.875