What the top and bottom of a fraction actually mean, how to build equivalent fractions, and how to simplify to lowest terms using the HCF — with a bar diagram, worked examples, a worksheet with memo and a quiz.
A fraction is simply a way of describing part of a whole. The denominator (bottom) tells you how many equal pieces the whole was cut into; the numerator (top) tells you how many of those pieces you have. Everything else about fractions grows out of that one idea.
1Equivalent fractions
Different-looking fractions can describe exactly the same amount. If you cut a cake into 4 pieces and take 2, you have the same amount as cutting it into 2 and taking 1. We call these equivalent fractions.
- 1Multiply top and bottom by 2: 23 = 46.Both parts must be multiplied by the same number.
- 2By 3: 23 = 69. By 5: 23 = 1015.Any multiplier works.
- 3So 46, 69 and 1015 all equal 23.They describe the same amount, just cut into more pieces.
- 1Ask what the bottom was multiplied by: 5 × 4 = 20.Compare the two denominators.
- 2Do the same to the top: 3 × 4 = 12.Whatever you do to the bottom you must do to the top.
- 3So 35 = 1220.
2Simplifying to lowest terms
A fraction is in lowest terms (or 'simplest form') when the only number that divides into both the top and the bottom is 1. To get there, divide both by their HCF.
- 1Find the HCF of 18 and 24: 18 = 2×32, 24 = 23×3 → HCF = 2 × 3 = 6.Use the prime-factor method from the Whole Numbers lessons.
- 2Divide both parts by 6: 18 ÷ 6 = 3 and 24 ÷ 6 = 4.Dividing top and bottom by the same number keeps the value.
- 31824 = 34.3 and 4 share no factor except 1, so this is lowest terms.
If you cannot spot the HCF at once, just keep dividing by any common factor you can see (2, 3, 5 …) until nothing divides both. You will land on the same answer — it just takes a few more steps.
Never add or subtract to simplify. 68 does not become 57 by taking 1 off each part. Simplifying only ever uses division by a common factor: 68 = 34.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Fill in the missing number: 27 = ?/21
Show answer ▾Hide answer ▴
7 × 3 = 21, so 2 × 3 = 6 → 621 - 2Fill in the missing number: 34 = 15/?
Show answer ▾Hide answer ▴
3 × 5 = 15, so 4 × 5 = 20 - 3Write two fractions equivalent to 34.
Show answer ▾Hide answer ▴
e.g. 68 and 912 - 4Simplify 1218.
Show answer ▾Hide answer ▴
HCF = 6 → 23 - 5Simplify 4560.
Show answer ▾Hide answer ▴
HCF = 15 → 34 - 6Simplify 2436.
Show answer ▾Hide answer ▴
HCF = 12 → 23 - 7Is 79 in lowest terms? Give a reason.
Show answer ▾Hide answer ▴
Yes — 7 and 9 share no common factor except 1.
Quick Quiz
4 quick questions on what you just read. Take it when you feel ready.
Now practise it
Download Grade 8 past papers and worksheets on this topic.