Mathematics • Year 8 • Unit 1 • Lesson 1
FDP — Mixed Challenge
Pull together everything from Lesson 1: switching between fractions, decimals and percentages; simplifying; and comparing values written in different forms. Six mixed problems, one "find the mistake", and one open-ended challenge.
1. Mixed problems — choose the right move
Each question uses a different combination of ideas from Lesson 1. Decide which move applies before you start writing. Show your working. 3 marks each
1.1 Convert 7/20 to a decimal AND a percentage.
1.2 Convert 0.125 to a percentage AND to a simplified fraction.
1.3 Simplify the fraction 24/40 to its lowest terms, then write it as a decimal AND a percentage.
1.4 Which is bigger: 3/8 or 40%? Show your reasoning.
1.5 Order these from smallest to largest: 0.42, 4/9, 41%, 2/5.
1.6 A test mark is recorded as 0.85. A second test mark is recorded as 17/20. Show that the two marks are exactly equal.
2. Find the mistake
Another student has tried to convert 3/4 to a percentage. 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 — write 3/4 as a percentage:
Line 1: 3/4 means 3 parts out of 4.
Line 2: To make it a percentage, I multiply 3/4 by 4 so the bottom is 100.
Line 3: 3 × 4 = 12 and 4 × 4 = 16, so 3/4 = 12/16.
Line 4: So 3/4 = 12%.
(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? To make the denominator 100, you need to multiply the bottom by something that turns it into 100 — and do the same to the top. What times 4 gives 100?3. Open-ended challenge — the same value, three ways
This question has more than one valid answer. 4 marks
3.1 Find three different fractions (with different numerators and denominators) that all equal 0.5. None of them are allowed to be 1/2 itself.
For each fraction you find:
(i) Write it down.
(ii) Show the working (or division) that confirms it equals 0.5.
(iii) Express it as a percentage too.
Bonus: Your three fractions must not simplify to the same look (e.g. 2/4 and 4/8 both clearly come from doubling — try to vary your approach).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 7/20
7 ÷ 20 = 0.35. Then 0.35 × 100 = 35%. So 7/20 = 0.35 = 35%.
1.2 — 0.125
0.125 × 100 = 12.5%. As a fraction: 0.125 = 125/1000. HCF of 125 and 1000 is 125, giving 1/8.
1.3 — 24/40
HCF of 24 and 40 is 8: 24 ÷ 8 = 3, 40 ÷ 8 = 5, so 24/40 = 3/5. As a decimal: 3 ÷ 5 = 0.6. As a percentage: 60%. So 24/40 = 3/5 = 0.6 = 60%.
1.4 — 3/8 or 40%?
3/8 = 3 ÷ 8 = 0.375 (= 37.5%). 40% = 0.40. Since 0.40 > 0.375, 40% is bigger (by 2.5 percentage points).
1.5 — Order 0.42, 4/9, 41%, 2/5
0.42 stays 0.42. 4/9 = 4 ÷ 9 ≈ 0.444. 41% = 0.41. 2/5 = 0.40.
Order: 2/5 < 41% < 0.42 < 4/9 (0.40 < 0.41 < 0.42 < 0.444).
1.6 — 0.85 vs 17/20
17 ÷ 20 = 0.85. Both marks are 0.85 (= 85%) — exactly equal. (Quick alternative: 17/20 × 5/5 = 85/100 = 0.85.)
2 — Find the mistake
(a) The mistake is on Line 2 (and the wrong number is then carried into Line 3).
(b) The student multiplied top and bottom by 4, giving a denominator of 16 — not 100. To turn the denominator into 100, they should have multiplied both top and bottom by 25 (because 4 × 25 = 100).
(c) Corrected working:
3/4 means 3 parts out of 4.
Multiply top and bottom by 25: 3 × 25 = 75 and 4 × 25 = 100, so 3/4 = 75/100.
So 3/4 = 75%. ✓
Sanity check: 1/4 = 25%, so 3/4 = 3 × 25% = 75%. Matches.
3 — Open-ended challenge (sample solution)
We need fractions where the top is exactly half the bottom. There are infinitely many.
Fraction 1: 3/6.
Working: 3 ÷ 6 = 0.5 ✓. As a percentage: 50%.
Fraction 2: 7/14.
Working: 7 ÷ 14 = 0.5 ✓. As a percentage: 50%.
Fraction 3: 50/100.
Working: 50 ÷ 100 = 0.5 ✓. As a percentage: 50%.
Other valid answers: 4/8, 5/10, 6/12, 8/16, 25/50, 100/200, or anything of the form n/(2n) where n ≥ 2.
Marking: 1 mark for each valid distinct fraction with correct working (up to 3 marks). 1 bonus mark for showing variety — e.g. picking 50/100 or another non-trivial example, not just consecutive doubles like 2/4, 4/8, 8/16.