Signed fraction/decimal calculations | Year 8

01 / LEARN

Understand the idea.

Fractions keep their meaning

Fractions can be added only after their pieces have the same size. Use a common denominator, keep each numerator's sign, then add or subtract the signed numerators. A minus sign before a fraction belongs to its whole value.

For multiplication, multiply numerators and denominators and apply the sign rule. For division, multiply by the reciprocal of the nonzero divisor. Cancel common factors in products; do not cancel separate terms across an addition sign.

Decimals and negative amounts

For decimal addition and subtraction, align equal place values; trailing zeros do not change the number. Decide the direction before calculating: subtracting a positive amount from a negative balance takes it farther below zero.

For decimal multiplication or division, determine the sign and calculate the size. Multiplying both numbers in a division by the same nonzero power of ten removes decimal places without changing the quotient: $1.2 \div 0.3 = 12 \div 3$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Add unlike signed fractions

Calculate $- \frac{3}{4} + \frac{5}{6}$ .

Worked solution

1
Choose equal-sized parts.
$\begin{aligned} - \frac{3}{4} & = - \frac{9}{12} \\ \frac{5}{6} & = \frac{10}{12} \end{aligned}$
Twelve is a common multiple of four and six.
2
Add the signed numerators.
$- \frac{9}{12} + \frac{10}{12} = \frac{- 9 + 10}{12}$
The denominator stays twelve because the parts have not changed size.
3
Simplify the result.
$\frac{- 9 + 10}{12} = \frac{1}{12}$
Ten positive twelfths outweigh nine negative twelfths by one twelfth.

Answer

$\frac{1}{12}$

EXAMPLE 02

Combine decimal operations

Calculate $- 1.2 \div 0.3 - (- 0.5)$ .

Worked solution

1
Calculate the division first.
$\begin{aligned} - 1.2 \div 0.3 & = - 12 \div 3 \\ = - 4 \end{aligned}$
Scaling both dividend and divisor by ten preserves the quotient.
2
Reverse the subtraction of a negative.
$- 4 - (- 0.5) = - 4 + 0.5$
Subtracting a negative amount is equivalent to adding its positive opposite.
3
Add the signed decimal values.
$- 4.0 + 0.5 = - 3.5$
Moving half a unit toward zero leaves a negative value.

Answer

-3.5

03 / YOUR TURN

Try it yourself.

Check your understanding

Calculate $- \frac{1}{2} - \frac{1}{4}$ .

Show answer & reasoning

$- \frac{3}{4}$

Write the half as two quarters: $- \frac{1}{2} - \frac{1}{4} = - \frac{2}{4} - \frac{1}{4} = - \frac{3}{4}$ . Subtracting another positive quarter moves farther below zero.

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Calculate $- \frac{2}{3} + \frac{1}{4}$ .
2 marks
Calculate $\frac{1}{2} - (- \frac{3}{5})$ .
2 marks
Calculate $(- \frac{5}{6}) (- \frac{9}{10})$ .
2 marks
Calculate $- \frac{3}{4} \div \frac{1}{8}$ .
2 marks
Calculate $- 2.4 + 0.75$ .
2 marks
The temperature is -1.5°C. It falls by 2.75°C, then rises by 0.5°C. Find the final temperature.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Calculate $- \frac{2}{3} + \frac{1}{4}$ .
2 marks
$- \frac{8}{12} + \frac{3}{12} = - \frac{5}{12}$ .
$- \frac{5}{12}$
Calculate $\frac{1}{2} - (- \frac{3}{5})$ .
2 marks
$\frac{1}{2} + \frac{3}{5} = \frac{5}{10} + \frac{6}{10} = \frac{11}{10}$ .
$\frac{11}{10}$
Calculate $(- \frac{5}{6}) (- \frac{9}{10})$ .
2 marks
The product is positive.
$\frac{45}{60} = \frac{3}{4}$ .
$\frac{3}{4}$
Calculate $- \frac{3}{4} \div \frac{1}{8}$ .
2 marks
$- \frac{3}{4} \times 8 = - 6$ .
-6
Calculate $- 2.4 + 0.75$ .
2 marks
$- 2.40 + 0.75 = - 1.65$ .
-1.65
The temperature is -1.5°C. It falls by 2.75°C, then rises by 0.5°C. Find the final temperature.
3 marks
$- 1.5 - 2.75 + 0.5 = - 4.25 + 0.5 = - 3.75$ .
-3.75°C

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Four operations with integers; signed fraction/decimal calculations; order of operations; terminating and recurring decimals; rational and irrational numbers; estimating square roots and recognising pi; choosing efficient exact/decimal representations.

Official curriculum source ↗

In this lesson

Understand the idea.

Fractions keep their meaning

Decimals and negative amounts

See the method in action.

Add unlike signed fractions

Choose equal-sized parts.

Add the signed numerators.

Simplify the result.

Combine decimal operations

Calculate the division first.

Reverse the subtraction of a negative.

Add the signed decimal values.

Try it yourself.

Check your understanding

Keep practising