Mathematics • Year 7 • Unit 1 • Lesson 8
Add & Subtract Fractions — Mixed Challenge
Pull every Lesson 8 idea together: same-denominator combine, find-the-LCD combine, mixed-number arithmetic with borrowing, simplifying answers, and spotting the classic "add the denominators" error.
1. Mixed problems — choose the right idea
Each question uses a different part of Lesson 8. Decide which idea applies before you start writing. Show your working. 2 marks each
1.1 5/12 + 1/12 + 3/12 = ?
1.2 7/10 − 1/4. Simplify your answer.
1.3 1 1/4 + 2 1/2. Write the answer as a mixed number in simplest form.
1.4 5/6 + 3/8. Find the LCD, convert, add, and simplify if possible.
1.5 3 1/3 − 1 2/3. The fraction part will need borrowing — show all steps.
1.6 A bag of marbles holds 5/8 red marbles and 1/4 blue marbles (the rest are green). What fraction is green?
2. Find the mistake
Another Year 7 student has tried to calculate 2/3 + 1/4. 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 — calculate 2/3 + 1/4:
Line 1: Denominators are different (3 and 4) — need to find the LCD.
Line 2: LCD = 12.
Line 3: Convert: 2/3 = (2 × 4)/(3 × 4) = 8/12. 1/4 = (1 × 3)/(4 × 3) = 3/12.
Line 4: Add: 8/12 + 3/12 = 11/24.
Line 5: HCF(11, 24) = 1, so the final answer is 11/24.
(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? At the add step, the denominator stays the same — you only add the tops.3. Open-ended challenge — fraction targets
This question has more than one correct answer. Show one that works and explain. 4 marks
3.1 Find two unit fractions (fractions of the form 1/n where n is a whole number greater than 1) that add to give 5/6.
For example: 1/__ + 1/__ = 5/6. Find a valid pair, show the LCD step that confirms the sum, and then check the answer.
Bonus: Is there a way to write 1 as the sum of two different unit fractions? Explain why or why not.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 5/12 + 1/12 + 3/12
Same denominator: add tops. 5 + 1 + 3 = 9. Keep 12. = 9/12 = 3/4 (HCF = 3).
1.2 — 7/10 − 1/4
LCD(10, 4) = 20. 7/10 = 14/20, 1/4 = 5/20. 14/20 − 5/20 = 9/20. HCF(9, 20) = 1, already simplified.
1.3 — 1 1/4 + 2 1/2
Wholes: 1 + 2 = 3. Fractions: 1/4 + 1/2 = 1/4 + 2/4 = 3/4. Combined: 3 3/4.
1.4 — 5/6 + 3/8
LCD(6, 8) = 24. 5/6 = 20/24, 3/8 = 9/24. Sum: 20/24 + 9/24 = 29/24. As a mixed number: 1 5/24. (HCF(5, 24) = 1, already simplified.)
1.5 — 3 1/3 − 1 2/3
Fractions: 1/3 − 2/3 = −1/3 (negative). Borrow 1 from the 3: 3 1/3 = 2 + 3/3 + 1/3 = 2 4/3.
Subtract: 2 4/3 − 1 2/3 = (2 − 1) + (4/3 − 2/3) = 1 + 2/3 = 1 2/3.
1.6 — Marble bag
Red + blue: 5/8 + 1/4 = 5/8 + 2/8 = 7/8. Green = 1 − 7/8 = 8/8 − 7/8 = 1/8.
2 — Find the mistake
(a) The mistake is on Line 4.
(b) When you add fractions with the same denominator, you add the tops only and keep the denominator the same. The student wrote 24 in the denominator (adding 12 + 12), which breaks the rule.
(c) Corrected working:
Line 4 (fixed): 8/12 + 3/12 = 11/12.
Line 5 (fixed): HCF(11, 12) = 1, so the final answer is 11/12.
Sanity check: 2/3 + 1/4 should be a bit less than 1 — and 11/12 is just under 1. ✓
3 — Unit fraction puzzle (sample solution)
Try 1/2 + 1/? = 5/6. Subtract: 1/? = 5/6 − 1/2 = 5/6 − 3/6 = 2/6 = 1/3. So 1/2 + 1/3 = 5/6. Check: LCD = 6, 1/2 = 3/6, 1/3 = 2/6. Sum: 3/6 + 2/6 = 5/6 ✓.
Bonus: Can 1 be written as the sum of two different unit fractions? Try 1/2 + 1/? = 1. Then 1/? = 1/2, so ? = 2. Both are 1/2 — they're not different. Try 1/3 + 1/? = 1. Then 1/? = 2/3, but 2/3 is not a unit fraction. In general, if 1/a + 1/b = 1 then b = a/(a−1), and for a ≥ 3 this isn't a whole number. So no — you cannot write 1 as the sum of two different unit fractions.
Marking: 2 marks for finding 1/2 + 1/3 = 5/6 with full working; 1 mark for the LCD-check; 1 mark for the bonus reasoning.