Module 2 of 3Lesson 1 of 13 · Unit 12 · Fractions I: equivalent fractions and comparing
Mathematics Form 1 · Term 2 · Unit 12

🔢 Fractions I: equivalent fractions and comparing

NumberIGCSE path ✓JC ✓7Nf.017Nf.06
💡
Why this matters

When a cake is cut into 8 slices and you get 2, or it is cut into 4 slices and you get 1, you get exactly the same amount of cake. Fractions are how we describe sharing fairly, and equivalent fractions are the secret that lets us compare any two shares.

🎯 By the end of this unit

  • I can explain what the top and bottom numbers of a fraction mean.
  • I can make equivalent fractions and write a fraction in its simplest form.
  • I can change between improper fractions and mixed numbers.
  • I can compare and order fractions using a common denominator.
  • I can find a fraction of an amount.
📖 Key words

numerator
the top number: how many equal parts we have
denominator
the bottom number: how many equal parts the whole is cut into
equivalent fractions
fractions that look different but are the same size, such as \(\dfrac{1}{2}\) and \(\dfrac{3}{6}\)
simplest form
a fraction whose numerator and denominator have no common factor except 1
improper fraction
a fraction whose numerator is bigger than its denominator, such as \(\dfrac{7}{4}\)
mixed number
a whole number and a fraction together, such as \(1\dfrac{3}{4}\)

1 Learn it

What a fraction means

A fraction describes part of a whole that has been cut into equal parts.

The denominator (bottom) tells you how many equal parts the whole is cut into.

The numerator (top) tells you how many of those parts you have.

3 parts shaded5 equal parts in the whole bar35
Three fifths of the bar is shaded.
⭐ Key ideaThe parts must be equal. A bar cut into 5 unequal pieces does not show fifths.

Equivalent fractions: same amount, different names

Look at the fraction wall. One half is exactly the same length as two quarters, and as four eighths.

To make an equivalent fraction, multiply the numerator and the denominator by the same number.

Cutting every piece into 2 doubles the number of pieces, and doubles the number you have, so the amount stays the same.

  1. \(\dfrac{1}{2} = \dfrac{1 \times 2}{2 \times 2}\)
  2. \(\dfrac{1 \times 2}{2 \times 2} = \dfrac{2}{4}\)
1 whole1 half2 quarters4 eighths
The dashed line shows that 1 half, 2 quarters and 4 eighths are all the same length.
⭐ Key ideaWhatever you do to the bottom, do to the top. Multiply (or divide) both by the same number.

Simplest form

To simplify a fraction, divide the numerator and denominator by a common factor.

Keep going until the only common factor is 1. Dividing by the highest common factor (HCF) does it in one step.

  1. \(\dfrac{12}{18}\): the HCF of 12 and 18 is 6
  2. \(\dfrac{12 \div 6}{18 \div 6} = \dfrac{2}{3}\)
⭐ Key ideaSimplifying does not change the size of the fraction. It just uses bigger, fewer pieces.

Improper fractions and mixed numbers

\(\dfrac{7}{4}\) means 7 quarters. Four quarters make 1 whole, so 7 quarters is 1 whole and 3 quarters left over.

To change an improper fraction to a mixed number, divide the numerator by the denominator. The remainder stays as the fraction part.

To go back, multiply the whole number by the denominator and add the numerator.

  1. \(7 \div 4 = 1\) remainder \(3\)
  2. \(\dfrac{7}{4} = 1\dfrac{3}{4}\)
1 whole3 quarters
Seven quarters fill one whole bar and three quarters of another.

Comparing and ordering fractions

Fractions with the same denominator are easy to compare: just compare the numerators.

Otherwise, change them to equivalent fractions with a common denominator (a common multiple of the denominators, usually the LCM).

Fraction× to make twelfthsIn twelfths
\(\dfrac{3}{4}\)× 3\(\dfrac{9}{12}\)
\(\dfrac{2}{3}\)× 4\(\dfrac{8}{12}\)
\(\dfrac{5}{6}\)× 2\(\dfrac{10}{12}\)
⭐ Key ideaSame denominator → compare numerators. The bigger numerator is the bigger fraction.

A fraction of an amount

To find \(\dfrac{3}{5}\) of P40, first find one fifth by dividing by the denominator, then multiply by the numerator.

  1. One fifth: \(40 \div 5 = 8\)
  2. Three fifths: \(8 \times 3 = 24\)
  3. So \(\dfrac{3}{5}\) of P40 is P24
⭐ Key ideaDivide by the bottom, multiply by the top.

2 Watch it work

EXAMPLE 1

Making an equivalent fraction

Fill in the missing number: \(\dfrac{3}{5} = \dfrac{\square}{20}\)
  1. The denominator changed from 5 to 20
  2. \(20 \div 5 = 4\), so the denominator was multiplied by 4
  3. Do the same to the numerator: \(3 \times 4 = 12\)
Answer \(\dfrac{3}{5} = \dfrac{12}{20}\)
EXAMPLE 2

Simplifying

Write \(\dfrac{24}{36}\) in its simplest form.
  1. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  2. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  3. The HCF is 12
  4. \(\dfrac{24 \div 12}{36 \div 12} = \dfrac{2}{3}\)
Answer \(\dfrac{2}{3}\)
EXAMPLE 3

Mixed number to improper fraction

Write \(3\dfrac{2}{5}\) as an improper fraction.
  1. Each whole is 5 fifths
  2. 3 wholes: \(3 \times 5 = 15\) fifths
  3. Add the 2 fifths: \(15 + 2 = 17\) fifths
Answer \(3\dfrac{2}{5} = \dfrac{17}{5}\)
EXAMPLE 4

Ordering fractions

Put \(\dfrac{3}{4}\), \(\dfrac{2}{3}\) and \(\dfrac{5}{6}\) in order, smallest first.
  1. The LCM of 4, 3 and 6 is 12
  2. \(\dfrac{3}{4} = \dfrac{9}{12}\)
  3. \(\dfrac{2}{3} = \dfrac{8}{12}\)
  4. \(\dfrac{5}{6} = \dfrac{10}{12}\)
  5. Order the numerators: 8, 9, 10
Answer \(\dfrac{2}{3}\), \(\dfrac{3}{4}\), \(\dfrac{5}{6}\)
EXAMPLE 5

A fraction of an amount

Kagiso earns P1 200 a month. She saves \(\dfrac{2}{5}\) of it. How much does she save?
  1. One fifth: \(1200 \div 5 = 240\)
  2. Two fifths: \(240 \times 2 = 480\)
Answer P480

3 Your turn

Try each question in your book first. Then tap the button to check every step.

1🟢 Warm-up
What fraction of this bar is shaded?
👀 Show the answer
Step by step
  1. The bar has 8 equal parts, so the denominator is 8
  2. 5 parts are shaded, so the numerator is 5
Answer \(\dfrac{5}{8}\)
2🟢 Warm-up
Fill in the missing number: \(\dfrac{1}{3} = \dfrac{\square}{12}\)
👀 Show the answer
Step by step
  1. \(12 \div 3 = 4\), so multiply by 4
  2. \(1 \times 4 = 4\)
Answer \(\dfrac{4}{12}\)
3🟢 Warm-up
Write \(\dfrac{10}{15}\) in its simplest form.
👀 Show the answer
Step by step
  1. The HCF of 10 and 15 is 5
  2. \(\dfrac{10 \div 5}{15 \div 5} = \dfrac{2}{3}\)
Answer \(\dfrac{2}{3}\)
4🟢 Warm-up
Find \(\dfrac{1}{4}\) of 28 cattle.
👀 Show the answer
Step by step
  1. Divide by the denominator: \(28 \div 4 = 7\)
Answer 7 cattle
5🔵 Core
Write \(\dfrac{23}{6}\) as a mixed number.
👀 Show the answer
Step by step
  1. \(23 \div 6 = 3\) remainder \(5\)
  2. 3 wholes and 5 sixths left over
Answer \(3\dfrac{5}{6}\)
6🔵 Core
Write \(4\dfrac{3}{8}\) as an improper fraction.
👀 Show the answer
Step by step
  1. \(4 \times 8 = 32\) eighths
  2. \(32 + 3 = 35\) eighths
Answer \(\dfrac{35}{8}\)
7🔵 Core
Which is bigger, \(\dfrac{5}{8}\) or \(\dfrac{3}{5}\)? Show how you know.
👀 Show the answer
Step by step
  1. The LCM of 8 and 5 is 40
  2. \(\dfrac{5}{8} = \dfrac{25}{40}\)
  3. \(\dfrac{3}{5} = \dfrac{24}{40}\)
  4. 25 is bigger than 24
Answer \(\dfrac{5}{8}\) is bigger
8🔵 Core
Write these in order, smallest first: \(\dfrac{7}{10}\), \(\dfrac{3}{5}\), \(\dfrac{3}{4}\)
👀 Show the answer
Step by step
  1. The LCM of 10, 5 and 4 is 20
  2. \(\dfrac{7}{10} = \dfrac{14}{20}\)
  3. \(\dfrac{3}{5} = \dfrac{12}{20}\)
  4. \(\dfrac{3}{4} = \dfrac{15}{20}\)
  5. Order the numerators: 12, 14, 15
Answer \(\dfrac{3}{5}\), \(\dfrac{7}{10}\), \(\dfrac{3}{4}\)
9🔵 Core
A bag of mealie meal has a mass of 12.5 kg. Thato uses \(\dfrac{3}{5}\) of it in a month. What mass does she use?
👀 Show the answer
Step by step
  1. One fifth: \(12.5 \div 5 = 2.5\) kg
  2. Three fifths: \(2.5 \times 3 = 7.5\) kg
Answer 7.5 kg
10🔵 Core
In a class of 32 learners, \(\dfrac{3}{8}\) walk to school. How many learners do not walk to school?
👀 Show the answer
Step by step
  1. One eighth: \(32 \div 8 = 4\)
  2. Walkers: \(4 \times 3 = 12\)
  3. Not walking: \(32 – 12 = 20\)
Answer 20 learners
11🟠 Stretch
Find a fraction that lies between \(\dfrac{2}{3}\) and \(\dfrac{3}{4}\).
👀 Show the answer
Step by step
  1. The LCM of 3 and 4 is 12: \(\dfrac{8}{12}\) and \(\dfrac{9}{12}\)
  2. There is no whole number between 8 and 9, so use twenty-fourths
  3. \(\dfrac{8}{12} = \dfrac{16}{24}\)
  4. \(\dfrac{9}{12} = \dfrac{18}{24}\)
  5. \(\dfrac{17}{24}\) lies between them
Answer \(\dfrac{17}{24}\) (other answers are possible)
12🟠 Stretch
Neo says \(\dfrac{4}{9}\) is bigger than \(\dfrac{1}{2}\) because 4 and 9 are bigger numbers than 1 and 2. Is he right? Explain.
👀 Show the answer
Step by step
  1. Use a common denominator of 18
  2. \(\dfrac{4}{9} = \dfrac{8}{18}\)
  3. \(\dfrac{1}{2} = \dfrac{9}{18}\)
  4. 8 is less than 9
Answer No. \(\dfrac{4}{9}\) is less than \(\dfrac{1}{2}\). Big numbers do not mean a big fraction.
13🟠 Stretch
Lesego has a 1 500-litre water tank. On Monday it is \(\dfrac{4}{5}\) full. On Tuesday it is \(\dfrac{7}{10}\) full. How many litres of water were used between Monday and Tuesday?
👀 Show the answer
Step by step
  1. Monday: one fifth is \(1500 \div 5 = 300\) litres
  2. Four fifths: \(300 \times 4 = 1200\) litres
  3. Tuesday: one tenth is \(1500 \div 10 = 150\) litres
  4. Seven tenths: \(150 \times 7 = 1050\) litres
  5. Used: \(1200 – 1050 = 150\) litres
Answer 150 litres

4 Watch out!

✗ Wrong\(\dfrac{2}{3} = \dfrac{2 + 3}{3 + 3} = \dfrac{5}{6}\)
✓ Right\(\dfrac{2}{3} = \dfrac{2 \times 2}{3 \times 2} = \dfrac{4}{6}\)

Adding the same number to the top and bottom changes the size of the fraction. Only multiplying or dividing both by the same number keeps it equivalent.

✗ Wrong\(\dfrac{1}{8}\) is bigger than \(\dfrac{1}{5}\) because 8 is bigger than 5.
✓ Right\(\dfrac{1}{5}\) is bigger than \(\dfrac{1}{8}\).

A bigger denominator means the whole is cut into more pieces, so each piece is smaller.

✗ Wrong\(\dfrac{17}{5} = 3\dfrac{1}{5}\)
✓ Right\(\dfrac{17}{5} = 3\dfrac{2}{5}\)

\(17 \div 5 = 3\) remainder 2, because \(3 \times 5 = 15\) and \(17 – 15 = 2\). The remainder becomes the numerator.

5 In your exams

🇧🇼 Junior Certificate (JCE)

In JCE Paper 1 (no calculator), you are often asked to simplify a fraction, find a missing number in equivalent fractions, or pick the largest fraction from four choices. In Paper 2, fractions of amounts appear inside money and measurement problems. Always show the common denominator you used.

🎓 Later in IGCSE

IGCSE Mathematics expects you to simplify fractions without a calculator and to write answers in their simplest form. Ordering fractions, decimals and percentages together is a common Paper 1 question, and fractions of amounts appear in almost every paper.

🏆 Challenge
A water tank is \(\dfrac{3}{4}\) full. After 60 litres are used, it is \(\dfrac{1}{2}\) full. How many litres does the full tank hold?
👀 Reveal the solution
Step by step
  1. The 60 litres used is the difference between the two fractions
  2. \(\dfrac{3}{4} – \dfrac{1}{2} = \dfrac{3}{4} – \dfrac{2}{4}\)
  3. \(\dfrac{3}{4} – \dfrac{2}{4} = \dfrac{1}{4}\) of the tank
  4. So \(\dfrac{1}{4}\) of the tank is 60 litres
  5. The full tank: \(60 \times 4 = 240\) litres
Answer 240 litres
📝 Remember

  • Denominator = how many equal parts in the whole. Numerator = how many parts you have.
  • Equivalent fractions: multiply or divide the top and bottom by the same number.
  • Simplest form: divide the top and bottom by their HCF.
  • To compare fractions, give them a common denominator, then compare the numerators.
  • Fraction of an amount: divide by the bottom, multiply by the top.

PreviousNext