Mathematics • Year 8 • Unit 1 • Lesson 2
Converting FDP — Mixed Challenge
Pull together every conversion from Lessons 1-2: fractions to decimals, decimals to percentages, simplifying, terminating vs recurring decimals, and ordering across forms.
1. Mixed problems — choose the right move
Each question uses a different combination of conversion ideas. Decide which move applies before you start writing. Show your working. 3 marks each
1.1 Convert 5/16 to a decimal AND a percentage. State whether it terminates or recurs.
1.2 Convert 1/6 to a recurring decimal AND a percentage (give percentage to 1 decimal place).
1.3 Convert 0.08 to a percentage AND to a simplified fraction.
1.4 Convert 175% to a decimal AND to a mixed number (a whole number plus a fraction).
1.5 Without doing any division, predict whether each fraction will terminate or recur: (a) 7/8, (b) 4/11, (c) 9/25, (d) 5/12. Explain your prediction in one sentence.
1.6 Order these from smallest to largest: 0.65, 2/3, 64%, 5/8.
2. Find the mistake
Another student has tried to convert 3% to a decimal. 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 — convert 3% to a decimal:
Line 1: "%" means "out of 100".
Line 2: To convert % → decimal, divide by 100.
Line 3: Dividing by 100 moves the decimal point one place to the left.
Line 4: So 3% = 0.3.
(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? Dividing by 10 moves the decimal one place left. Dividing by 100 moves it TWO. Lines 1 and 2 are correct — the error is about HOW FAR the decimal point moves.3. Open-ended challenge — terminate or recur?
This question has more than one valid answer. 4 marks
3.1 Find two fractions that terminate and two fractions that recur, all with the numerator 1 (i.e. of the form 1/n). Use four different denominators.
For each fraction:
(i) Write it down.
(ii) Convert it to a decimal (use recurring notation where needed).
(iii) State why it terminates or recurs in terms of the prime factors of the denominator.
Bonus: Find a denominator made from BOTH 2 and 5 (e.g. 20 or 40) — does it terminate or recur? Add it as a fifth example if you can.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 5/16
5 ÷ 16 = 0.3125. As a %: 31.25%. Terminates (16 = 2 × 2 × 2 × 2 — only 2s). So 5/16 = 0.3125 = 31.25%.
1.2 — 1/6
1 ÷ 6 = 0.1666... = 0.16̄. As a %: 0.166... × 100 ≈ 16.7% (to 1 dp). Recurs because 6 = 2 × 3 — the 3 makes it recur.
1.3 — 0.08
0.08 × 100 = 8%. As a fraction: 0.08 = 8/100. HCF = 4: 8 ÷ 4 = 2, 100 ÷ 4 = 25. So 2/25.
1.4 — 175%
175% = 175 ÷ 100 = 1.75. As a mixed number: 175/100 = 1 + 75/100 = 1 3/4 (since 75/100 simplifies to 3/4).
1.5 — Predict terminate or recur
(a) 7/8: 8 = 2³ — terminates (only 2s). (b) 4/11: 11 is prime, not 2 or 5 — recurs. (c) 9/25: 25 = 5² — terminates (only 5s). (d) 5/12: 12 = 2² × 3 — recurs (the 3 spoils it).
One-sentence rule: a fraction (in lowest terms) terminates only when its denominator's prime factors are all 2 or 5.
1.6 — Order 0.65, 2/3, 64%, 5/8
Convert all to decimals: 0.65 stays 0.65; 2/3 ≈ 0.667; 64% = 0.64; 5/8 = 0.625.
Order smallest → largest: 5/8 < 64% < 0.65 < 2/3 (0.625 < 0.64 < 0.65 < 0.667).
2 — Find the mistake
(a) The mistake is on Line 3 (and that wrong claim is then used on Line 4).
(b) Dividing by 100 moves the decimal point TWO places to the left, not one. Dividing by 10 moves it one place; dividing by 100 moves it two.
(c) Corrected working:
"%" means "out of 100".
To convert % → decimal, divide by 100.
Dividing by 100 moves the decimal point two places to the left.
So 3% = 0.03.
Quick sanity check: 3% should be very small. 0.3 is 30% — much too big. 0.03 is right.
3 — Open-ended challenge (sample solution)
We choose denominators whose prime factorisations are different.
Terminating 1: 1/4. 1 ÷ 4 = 0.25. Terminates because 4 = 2 × 2 — only 2s.
Terminating 2: 1/25. 1 ÷ 25 = 0.04. Terminates because 25 = 5 × 5 — only 5s.
Recurring 1: 1/3. 1 ÷ 3 = 0.3̄. Recurs because 3 is a prime other than 2 or 5.
Recurring 2: 1/9. 1 ÷ 9 = 0.1̄. Recurs because 9 = 3 × 3 — only the prime 3 appears.
Bonus: 1/20. 1 ÷ 20 = 0.05. Terminates, because 20 = 2 × 2 × 5 — only 2s and 5s, even though both appear.
Marking: 1 mark for each correct example with reasoning (up to 4). Bonus mark for correctly recognising that mixed 2s-and-5s denominators (e.g. 20, 40, 50) still terminate.