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This lesson covers moving freely between fractions, decimals and percentages (FDP), ordering sets that mix all three forms, and — for Higher tier [H] — converting a recurring decimal into an exact fraction using the algebraic method. Being able to switch form quickly is one of the most useful skills in OCR GCSE Mathematics (J560): it lets you compare quantities, simplify problems and check answers across the whole Number strand.
This lesson mainly builds AO1 fluency in conversions, with AO2 reasoning when you justify why a fraction terminates or recurs, and AO3 problem-solving in the algebraic proof for recurring decimals.
| Term | Meaning |
|---|---|
| Terminating decimal | A decimal that ends after a finite number of digits |
| Recurring decimal | A decimal in which a digit or block of digits repeats forever |
| FDP | Fraction–Decimal–Percentage (the three interchangeable forms) |
| Equivalent | Having the same value in a different form |
| Dot notation | Dots above the first and last repeating digits, e.g. 0.1˙8˙ |
| Simplest form | A fraction with numerator and denominator sharing no common factor >1 |
Fractions, decimals and percentages are three different costumes for the same underlying number. The score 21, the decimal 0.5 and the percentage 50% all describe "one half" — they are interchangeable, and being able to switch costume on demand is what lets you compare quantities given in different forms and check answers. The decimal is usually the most convenient "hub" form: almost every conversion is quickest if you go via the decimal.
The conversions form a simple cycle. Learn the routes and a few key equivalents and you can hop between any two forms.
| From → To | Method |
|---|---|
| Fraction → Decimal | Divide numerator by denominator |
| Decimal → Percentage | Multiply by 100 |
| Percentage → Decimal | Divide by 100 |
| Decimal → Fraction | Write over a power of 10, then simplify |
| Fraction → Percentage | ×100 (or convert to a decimal first) |
| Percentage → Fraction | Write over 100, then simplify |
Convert 87 to (a) a decimal and (b) a percentage.
(a) 7÷8=0.875. (b) 0.875×100=87.5%.
Answers: (a) 0.875, (b) 87.5%.
Convert 0.45 to a fraction in its simplest form.
0.45=10045. The HCF of 45 and 100 is 5, so 10045=209.
Answer: 209.
Convert 72% to a fraction in its simplest form.
72%=10072=2518 (dividing top and bottom by 4).
Answer: 2518.
Common error: Stopping at 5036. Always check whether the fraction simplifies further.
Convert (a) 0.06 to a percentage, and (b) 8% to a decimal.
(a) Decimal to percentage means ×100 (move the point two places right): 0.06×100=6%. (b) Percentage to decimal means ÷100 (move the point two places left): 8÷100=0.08.
Answers: (a) 6%, (b) 0.08.
Common error: Confusing 8% with 0.8. Per cent means "out of 100", so 8% is 1008=0.08, not 0.8 (which would be 80%).
These come up so often that knowing them by heart saves time on Paper 1.
| Fraction | Decimal | Percentage |
|---|---|---|
| 21 | 0.5 | 50% |
| 41 | 0.25 | 25% |
| 43 | 0.75 | 75% |
| 51 | 0.2 | 20% |
| 31 | 0.3˙ | 33.3˙% |
| 81 | 0.125 | 12.5% |
| 101 | 0.1 | 10% |
To order a set that mixes fractions, decimals and percentages, convert everything to decimals (a common form), compare, then write the answer back in the original forms.
Write in ascending order: 53, 0.58, 62%, 85.
Convert to decimals: 53=0.6, 0.58, 62%=0.62, 85=0.625.
In order: 0.58,0.6,0.62,0.625.
Answer: 0.58,53,62%,85.
OCR Exam Tip: Convert to decimals to at least enough places to separate the values. Here three decimal places are needed because 0.62 and 0.625 are close.
Write in descending order: 32, 0.66, 65%, 107.
Convert to decimals: 32=0.66˙≈0.667, 0.66, 65%=0.65, 107=0.7.
Largest first: 0.7,0.667,0.66,0.65.
Answer: 107,32,0.66,65%.
Common error: Rounding 32 to just 0.67 and then ranking it above 0.66 — correct here, but with closer values too-early rounding can flip the order. Carry an extra decimal place when values are near each other.
A fraction in its simplest form gives a terminating decimal exactly when its denominator has no prime factors other than 2 and 5 — because our base is 10=2×5. Any other prime factor (such as 3 or 7) forces a recurring decimal.
Without dividing, state whether each gives a terminating or recurring decimal: 407, 154, 509.
Factorise denominators: 40=23×5 (only 2s and 5s) → terminating; 15=3×5 (has a 3) → recurring; 50=2×52 → terminating.
Answer: 407 terminates, 154 recurs, 509 terminates.
This section is Higher tier only. The algebraic method is always expected. Let x be the recurring decimal, multiply by a power of 10 so that one full repeating block is shifted across, then subtract to eliminate the recurring tail.
Convert 0.4˙ to a fraction.
Let x=0.444… The block is one digit, so multiply by 10:
10x=4.444…
Subtract the original: 10x−x=4.444…−0.444…, so 9x=4 and x=94.
Answer: 94.
Convert 0.2˙7˙ to a fraction in its simplest form.
Let x=0.272727… The block "27" is two digits, so multiply by 100:
100x=27.272727…
Subtract: 100x−x=27.2727…−0.2727…, so 99x=27 and x=9927=113.
Answer: 113.
Common error: Multiplying by the wrong power of 10. Match the power to the length of the repeating block: one digit → ×10, two digits → ×100, three digits → ×1000.
Convert 0.416˙ to a fraction. (Here "41" does not repeat; only the "6" recurs.)
Let x=0.41666… The recurring block is one digit. Multiply by 10 and by 100 so that the recurring parts line up:
100x=41.6666…,10x=4.1666…
Subtract the smaller from the larger so the tails cancel: 100x−10x=41.66˙−4.16˙, so 90x=37.5.
x=9037.5=900375=125.
Check: 5÷12=0.416˙ ✓.
Convert 0.1˙08˙ to a fraction. (The block "108" repeats.)
Let x=0.108108… The block is three digits, so multiply by 1000:
1000x=108.108108…
Subtract: 1000x−x=108, so 999x=108 and x=999108=374 (dividing top and bottom by 27).
Answer: 374.
Use algebra to show that 0.9˙=1.
Let x=0.999… Then 10x=9.999… Subtracting: 9x=9, so x=1.
This surprising-but-true result confirms there is no "gap" between 0.9˙ and 1.
Write 0.625 as (i) a fraction in simplest form, (ii) a percentage; and order it against 95 and 61%.
(i) 0.625=1000625=85. (ii) 0.625×100=62.5%.
Order: 95=0.55˙≈0.556, 61%=0.61, 0.625. Ascending: 95,61%,0.625.
Specimen question modelled on the OCR J560 paper format: [H] Prove algebraically that the recurring decimal 0.21˙3˙ can be written as 22047.
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