Cambridge IGCSE Maths (0580) · Non-calculator · 1.4, 1.6

Fractions without a calculator

Fractions come up on their own and inside almost every other topic: probability, ratio, algebra and exact answers. On a non-calculator paper you need to add, subtract, multiply and divide them, including mixed numbers, by hand.

Core and Extended
Paper 1 (Core) and Paper 2 (Extended)
8 questions
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The method

Change mixed numbers to improper fractions first: \(2\tfrac{3}{4} = \tfrac{11}{4}\).
Adding and subtracting: write both fractions over a common denominator (the lowest common multiple is easiest), then add or subtract the numerators.
Multiplying: cancel common factors across the fractions first, then multiply the numerators and the denominators.
Dividing: keep the first fraction, turn the second upside down and multiply.
A fraction of an amount: divide by the denominator, then multiply by the numerator.
Extended: to write a recurring decimal as a fraction, call it \(x\), multiply by a power of 10 so the repeating part lines up, and subtract.
Give the answer in its simplest form, and as a mixed number if the question started with mixed numbers.

Worked examples

Example 1 Core non-calculator · medium

Work out \(2\tfrac{3}{4} + 1\tfrac{5}{6}\). Give your answer as a mixed number.

\(2\tfrac{3}{4} = \tfrac{11}{4}\) and \(1\tfrac{5}{6} = \tfrac{11}{6}\)
\(\tfrac{11}{4} + \tfrac{11}{6} = \tfrac{33}{12} + \tfrac{22}{12} = \tfrac{55}{12}\)
\(\tfrac{55}{12} = 4\tfrac{7}{12}\)

Answer: \(4\tfrac{7}{12}\)

Example 2 Core non-calculator · medium

Work out \(3\tfrac{1}{3} \div 1\tfrac{1}{9}\).

\(3\tfrac{1}{3} = \tfrac{10}{3}\) and \(1\tfrac{1}{9} = \tfrac{10}{9}\)
\(\tfrac{10}{3} \div \tfrac{10}{9} = \tfrac{10}{3} \times \tfrac{9}{10} = \tfrac{9}{3} = 3\)

Answer: \(3\)

Common slips

Adding the denominators: \(\tfrac{1}{3} + \tfrac{1}{4}\) is not \(\tfrac{2}{7}\).
Trying to work with the whole numbers and fractions of mixed numbers separately when multiplying or dividing.
Turning the wrong fraction upside down when dividing.
Leaving the answer unsimplified, or showing no working when the question asks you to show every step.

Practice questions

No calculator. Write down every step, then open the worked answer.

Question 1 Core non-calculator · easy

Work out \(\tfrac{3}{8} + \tfrac{5}{12}\).

Show the worked answer

The lowest common multiple of \(8\) and \(12\) is \(24\).
\(\tfrac{9}{24} + \tfrac{10}{24} = \tfrac{19}{24}\)

Answer: \(\tfrac{19}{24}\)

Question 2 Core non-calculator · easy

Find \(\tfrac{3}{5}\) of \(85\).

Show the worked answer

\(85 \div 5 = 17\)
\(17 \times 3 = 51\)

Answer: \(51\)

Question 3 Core non-calculator · medium

Work out \(4\tfrac{1}{2} - 2\tfrac{2}{3}\). Give your answer as a mixed number.

Show the worked answer

\(\tfrac{9}{2} - \tfrac{8}{3} = \tfrac{27}{6} - \tfrac{16}{6} = \tfrac{11}{6}\)
\(\tfrac{11}{6} = 1\tfrac{5}{6}\)

Answer: \(1\tfrac{5}{6}\)

Question 4 Core non-calculator · medium

Work out \(2\tfrac{2}{5} \times 1\tfrac{7}{8}\). Give your answer as a mixed number.

Show the worked answer

\(\tfrac{12}{5} \times \tfrac{15}{8}\)
Cancel first: \(\tfrac{12}{8} = \tfrac{3}{2}\) and \(\tfrac{15}{5} = 3\), so this is \(\tfrac{3 \times 3}{2} = \tfrac{9}{2}\).
\(\tfrac{9}{2} = 4\tfrac{1}{2}\)

Answer: \(4\tfrac{1}{2}\)

Question 5 Core non-calculator · medium

Work out \(1\tfrac{3}{4} \div 2\tfrac{1}{3}\).

Show the worked answer

\(\tfrac{7}{4} \div \tfrac{7}{3} = \tfrac{7}{4} \times \tfrac{3}{7}\)
\(= \tfrac{3}{4}\)

Answer: \(\tfrac{3}{4}\)

Question 6 Extended non-calculator · medium

Show that the recurring decimal \(0.\dot{2}\dot{7}\) (that is, \(0.272727\ldots\)) is equal to \(\tfrac{3}{11}\).

Show the worked answer

Let \(x = 0.272727\ldots\). Then \(100x = 27.272727\ldots\)
Subtract: \(99x = 27\), so \(x = \tfrac{27}{99}\).
Divide top and bottom by \(9\): \(\tfrac{3}{11}\).

Answer: \(\tfrac{3}{11}\)

Question 7 Extended non-calculator · hard

Write \(0.1\dot{3}\) (that is, \(0.1333\ldots\)) as a fraction in its simplest form.

Show the worked answer

Let \(x = 0.1333\ldots\). Then \(10x = 1.333\ldots\) and \(100x = 13.333\ldots\)
Subtract: \(90x = 12\), so \(x = \tfrac{12}{90}\).
Divide top and bottom by \(6\): \(\tfrac{2}{15}\).

Answer: \(\tfrac{2}{15}\)

Question 8 Core non-calculator · hard

A water tank is \(\tfrac{2}{3}\) full. After \(18\) litres are used, it is \(\tfrac{5}{12}\) full.
Work out how many litres the tank holds when it is full.

Show the worked answer

\(\tfrac{2}{3} - \tfrac{5}{12} = \tfrac{8}{12} - \tfrac{5}{12} = \tfrac{3}{12} = \tfrac{1}{4}\)
So \(\tfrac{1}{4}\) of the tank is \(18\) litres.
\(18 \times 4 = 72\)

Answer: \(72\) litres

Every example and question on this page is original, written for IGCSE Math Revision rather than taken from Cambridge papers, and each answer was re-solved independently before publishing. The syllabus is summarised in our own words: check the official 0580 syllabus.

Keep going

Fractions, Decimals and Percentages questions for 0580: Core and Extended, each labelled calculator or non-calculator
The Four Operations questions for 0580: Core and Extended, each labelled calculator or non-calculator
Number practice for 0580, then My mistakes and a Skill Builder on what you missed
Fractions, decimals and percentages notes and the 0580 formula sheet
0580 past papers: time yourself on a specimen Paper 1 or Paper 2
All six non-calculator methods

Questions students ask

Which 0580 papers test fractions without a calculator?

Fractions is on both tiers (syllabus 1.4, 1.6), so it can come up on Paper 1 (Core) and Paper 2 (Extended), the two non-calculator papers, as well as on the calculator papers.

Are these Cambridge questions?

No. The method is explained in our own words, and every example and question was written for IGCSE Math Revision and checked independently before publishing. For real papers, use the Cambridge copies linked from our 0580 past papers page.