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Mathematics

Equivalent Fractions Practice Questions, Naming the Factor Every Time

Eleven questions on missing numerators, simplifying to lowest terms and ordering unlike fractions, each answer naming the factor or denominator it used.

The short answer

Two fractions are equivalent when the same number has multiplied or divided both the numerator and the denominator. 3/4 = 15/20 because both parts were multiplied by 5. Simplifying runs it backwards: divide both parts by their highest common factor, so 18/24 becomes 3/4 by dividing by 6.

The questions

  1. Question 1Foundation1 marks

    Fill in the missing number: 3/4 = ?/20.

    Show the working

    15

    1. Compare the denominators: 4 × 5 = 20.
    2. The multiplier is 5, and it must be applied to both parts.
    3. 3 × 5 = 15.
    4. 3/4 = 15/20.
  2. Question 2Foundation1 marks

    Fill in the missing number: 2/5 = 14/?.

    Show the working

    35

    1. Compare the numerators: 2 × 7 = 14.
    2. The multiplier is 7.
    3. 5 × 7 = 35.
    4. 2/5 = 14/35.
  3. Question 3Foundation2 marks

    Simplify 18/24 to its lowest terms.

    Show the working

    3/4

    1. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
    2. The highest common factor is 6.
    3. 18 ÷ 6 = 3 and 24 ÷ 6 = 4.
    4. 3/4, and 3 and 4 share no factor above 1, so it is finished.
  4. Question 4Foundation2 marks

    Simplify 45/72 to its lowest terms.

    Show the working

    5/8

    1. Both numbers have digit sums divisible by 9: 4 + 5 = 9 and 7 + 2 = 9.
    2. The highest common factor is 9.
    3. 45 ÷ 9 = 5 and 72 ÷ 9 = 8.
    4. 5/8 — nothing divides both 5 and 8.
  5. Question 5Core3 marks

    Simplify 84/126 to its lowest terms.

    Show the working

    2/3

    1. 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7.
    2. Shared factors: 2, 3 and 7, so the HCF is 42.
    3. 84 ÷ 42 = 2 and 126 ÷ 42 = 3.
    4. 2/3. Dividing by 2 first, then 3, then 7 reaches the same answer in three steps instead of one.
  6. Question 6Core3 marks

    Which of 6/10, 9/15, 12/18 and 15/25 is not equal to 3/5?

    Show the working

    12/18

    1. 6/10 divides by 2 to give 3/5.
    2. 9/15 divides by 3 to give 3/5.
    3. 15/25 divides by 5 to give 3/5.
    4. 12/18 divides by 6 to give 2/3, which is larger than 3/5.
    5. So 12/18 is the odd one out.
  7. Question 7Core3 marks

    Write 7/8 and 5/6 with a common denominator, and say which is larger.

    Show the working

    21/24 and 20/24, so 7/8 is larger

    1. 8 = 2³ and 6 = 2 × 3, so the LCM is 2³ × 3 = 24.
    2. 7/8: multiply both parts by 3 to get 21/24.
    3. 5/6: multiply both parts by 4 to get 20/24.
    4. 21 twenty-fourths beats 20 twenty-fourths, so 7/8 is larger — by 1/24.
  8. Question 8Stretch4 marks

    Put these in order, smallest first: 5/8, 2/3, 7/12, 3/4.

    Show the working

    7/12, 5/8, 2/3, 3/4

    1. Denominators 8, 3, 12 and 4 have LCM 24.
    2. 5/8 = 15/24 (× 3), 2/3 = 16/24 (× 8), 7/12 = 14/24 (× 2), 3/4 = 18/24 (× 6).
    3. Order the numerators: 14, 15, 16, 18.
    4. So the order is 7/12, 5/8, 2/3, 3/4.
  9. Question 9Stretch4 marks

    Put these in order, smallest first: 4/7, 5/9, 3/5, 11/20.

    Show the working

    11/20, 5/9, 4/7, 3/5

    1. The LCM of 7, 9, 5 and 20 is 1,260, which is a lot of arithmetic for four comparisons.
    2. Divide each out instead: 4 ÷ 7 = 0.571, 5 ÷ 9 = 0.556, 3 ÷ 5 = 0.6, 11 ÷ 20 = 0.55.
    3. Three decimal places is enough to separate all four.
    4. Order: 0.55, 0.556, 0.571, 0.6, so 11/20, 5/9, 4/7, 3/5.
  10. Question 10Stretch3 marks

    A fraction simplifies to 4/9 and its denominator is 63. What is its numerator?

    Show the working

    28

    1. The denominator went from 9 to 63, so the multiplier is 63 ÷ 9 = 7.
    2. The same multiplier applies to the numerator.
    3. 4 × 7 = 28.
    4. Check: 28/63 divides by 7 to give 4/9.
  11. Question 11Stretch4 marks

    Explain why there is no whole number n for which n/12 equals 5/8.

    Show the working

    Because n would be 7.5

    1. If n/12 = 5/8, then n = 5 × 12 ÷ 8.
    2. 5 × 12 = 60, and 60 ÷ 8 = 7.5, which is not a whole number.
    3. The reason is that 5/8 is already in lowest terms, so any equivalent fraction must have a denominator that is a multiple of 8.
    4. 12 is not a multiple of 8, so no equivalent fraction has denominator 12.

Where these go wrong

  • Adding the same number to the numerator and denominator instead of multiplying, so 3/4 becomes 4/5.
  • Simplifying by a common factor that is not the highest and stopping there, leaving 18/24 as 9/12.
  • Applying the multiplier to only one part of the fraction when filling in a missing number.
  • Comparing fractions by numerator alone, so 7/12 is judged larger than 5/8.
  • Assuming any two denominators can be matched, and forcing a fraction in lowest terms into a denominator that is not a multiple of its own.

One value, many names

3/4, 6/8, 15/20 and 75/100 are four names for the same amount. Nothing has been added to a fraction to turn one into another — the pieces have been cut smaller and there are proportionally more of them, which is why both numbers move by the same multiplier.

That is the test for every answer on this page. If you cannot name the number that multiplied or divided both parts, the step is not an equivalence. Adding 1 to the top and 1 to the bottom turns 3/4 into 4/5, which is a different and larger fraction.

Lowest terms means dividing by the highest common factor

Simplifying 18/24 by 2 gives 9/12, which is correct but not finished. Dividing by the HCF, which is 6, reaches 3/4 in one step. Both routes arrive at the same place, and pupils who go in small steps are not wrong — they just have more chances to stop early.

The way to know you have finished is that the numerator and denominator share no factor above 1. For 84/126 the HCF is 42, so the answer is 2/3, and you can stop the moment you notice that 2 and 3 have nothing in common.

Comparing needs one denominator, not four opinions

Fractions cannot be compared while they name different sizes of piece. To order 5/8, 2/3, 7/12 and 3/4, put them all over 24 — the LCM of 8, 3, 12 and 4 — giving 15/24, 16/24, 14/24 and 18/24, and the order reads off directly.

Where the denominators share nothing useful, converting to decimals is the faster honest method rather than a cheat. 4/7, 5/9, 3/5 and 11/20 have an LCM of 1,260; dividing each out to three decimal places takes far less time and answers the same question.

Common questions

Does a fraction always have to be given in its lowest terms?

Where a question says "simplify" or "give your answer in its simplest form", yes, and marks are lost without it. Elsewhere an unsimplified answer is not wrong, but it is harder to compare with anything else, so the habit of finishing is worth having by default.

Is there a quick way to spot the highest common factor?

Check the small tests first: both even means 2, digit sums divisible by 3 means 3, both ending in 0 or 5 means 5. If nothing obvious appears, write both numbers as products of primes and multiply what they share — for 84 and 126 that gives 2 × 3 × 7 = 42 straight away.

Why is my child comparing fractions by the bottom number?

Because with whole numbers bigger digits mean more, and 1/8 having smaller value than 1/3 contradicts everything learned so far. Cutting the same cake into 8 and into 3 usually settles it in one demonstration: more pieces means smaller pieces. After that, the common-denominator method has something to attach to.

Can I use decimals to compare fractions instead?

Yes, and for awkward denominators it is the sensible method — 4/7 against 5/9 is far quicker as 0.571 against 0.556 than over a denominator of 63. Just keep enough decimal places to separate the values, and be aware that a question asking you to show a common denominator wants that method specifically.

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