Mathematics • Year 7 • Unit 1 • Lesson 7
Equivalent & Simplifying — Mixed Challenge
Bring Lesson 7 together: build equivalents, simplify fully, cross-multiply to check equivalence, cross-cancel before multiplying, and find a classic add-instead-of-divide mistake. End with an open digit puzzle.
1. Mixed problems — choose the right idea
Each question uses a different part of Lesson 7. Decide which idea applies before you start writing. Show your working. 2 marks each
1.1 Find the missing number: 3/4 = ?/24.
1.2 Simplify 56/72 fully.
1.3 Are 5/8 and 15/24 equivalent? Use cross-multiplication to check, then state your conclusion.
1.4 Use cross-cancellation to evaluate 5/6 × 3/10 in lowest terms.
1.5 Simplify 64/80 fully.
1.6 A test was marked out of 40. Maya scored 28. Aisha scored 21 out of 30 on a different test. Use cross-multiplication to decide whose score is higher.
2. Find the mistake
Another Year 7 student has tried to simplify 8/12 fully. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — simplify 8/12 fully:
Line 1: Both top and bottom are even, so divide both by 2.
Line 2: 8 ÷ 2 = 4 → new top = 4
Line 3: 12 − 2 = 10 → new bottom = 10
Line 4: New fraction: 4/10
Line 5: Check HCF(4, 10) = 2 → divide again: 4/10 = 2/5. Answer: 2/5.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Whatever you do to the top, do the same to the bottom — and "the same" means the same operation, not just the same number.3. Open-ended challenge — build a family of equivalents
This question has more than one correct answer. Show three that work and explain your method. 4 marks
3.1 Start with the fraction 3/5.
(i) Write down three different equivalent fractions to 3/5, each with a different denominator. Show how you got each one.
(ii) Among all the fractions you could ever write (with whole-number numerator and denominator) that are equivalent to 3/5, what is the simplest form? Justify your answer.
Bonus: How many equivalent fractions to 3/5 are there in total? Explain.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 3/4 = ?/24
Bottom 4 → 24 is × 6. Top: 3 × 6 = 18. So 3/4 = 18/24.
1.2 — Simplify 56/72
HCF(56, 72) = 8. 56 ÷ 8 = 7, 72 ÷ 8 = 9. So 7/9. Check HCF(7, 9) = 1 ✓.
1.3 — 5/8 vs 15/24
Cross-multiply: 5 × 24 = 120 and 8 × 15 = 120. Both equal 120, so yes, equivalent. (Also: 15/24 simplifies to 5/8.)
1.4 — 5/6 × 3/10 with cross-cancellation
Look diagonally. 5 (top-left) and 10 (bottom-right) share factor 5: 5 ÷ 5 = 1, 10 ÷ 5 = 2. 3 (top-right) and 6 (bottom-left) share factor 3: 3 ÷ 3 = 1, 6 ÷ 3 = 2.
After cancelling: 1/2 × 1/2 = 1/4.
1.5 — Simplify 64/80
HCF(64, 80) = 16. 64 ÷ 16 = 4, 80 ÷ 16 = 5. So 4/5. Check HCF(4, 5) = 1 ✓.
1.6 — Whose score is higher?
Compare 28/40 with 21/30. Cross-multiply: 28 × 30 = 840 and 40 × 21 = 840. Both equal 840, so the scores are equal (both simplify to 7/10). Neither is higher.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) "Divide both by 2" means use the same operation (division) on both. The student subtracted 2 from the bottom instead of dividing by 2. That breaks the equivalent-fractions rule.
(c) Corrected working:
Line 3 (fixed): 12 ÷ 2 = 6 → new bottom = 6.
Line 4 (fixed): New fraction = 4/6.
Line 5 (fixed): HCF(4, 6) = 2 → divide again: 4 ÷ 2 = 2, 6 ÷ 2 = 3. Answer: 2/3.
Quick sanity check: 8/12 with HCF = 4 gives 2/3 in one step. ✓
3 — Equivalent-fractions challenge (sample solution)
(i) Three equivalents to 3/5: multiply top and bottom by 2 → 6/10; by 3 → 9/15; by 4 → 12/20. (Cross-check each: 3 × 10 = 30 = 5 × 6 ✓, etc.)
(ii) Simplest form: HCF(3, 5) = 1, so 3/5 is already in its simplest form. There's no smaller equivalent.
Bonus: There are infinitely many equivalent fractions, because we can multiply 3 and 5 by any whole number (2, 3, 4, 5, …) and get a new equivalent each time.
Marking: 2 marks for three valid equivalents with clear working; 1 mark for identifying 3/5 as simplest form with HCF = 1 justification; 1 mark for the bonus (infinitely many).