Converting Fractions, Decimals and Percentages

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.

Key Vocabulary

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. $0.\dot{1}\dot{8}$
Simplest form	A fraction with numerator and denominator sharing no common factor $>1$

Converting Between the Three Forms

Fractions, decimals and percentages are three different costumes for the same underlying number. The score $\dfrac{1}{2}$ , 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	$\times 100$ (or convert to a decimal first)
Percentage → Fraction	Write over $100$ , then simplify

Worked Example 1

Convert $\dfrac{7}{8}$ to (a) a decimal and (b) a percentage.

(a) $7 \div 8 = 0.875$ . (b) $0.875 \times 100 = 87.5\%$ .

Answers: (a) $0.875$ , (b) $87.5\%$ .

Worked Example 2

Convert $0.45$ to a fraction in its simplest form.

$0.45 = \dfrac{45}{100}$ . The HCF of $45$ and $100$ is $5$ , so $\dfrac{45}{100} = \dfrac{9}{20}$ .

Answer: $\dfrac{9}{20}$ .

Worked Example 3

Convert $72\%$ to a fraction in its simplest form.

$72\% = \dfrac{72}{100} = \dfrac{18}{25}$ (dividing top and bottom by $4$ ).

Answer: $\dfrac{18}{25}$ .

Common error: Stopping at $\dfrac{36}{50}$ . Always check whether the fraction simplifies further.

Worked Example 3b

Convert (a) $0.06$ to a percentage, and (b) $8\%$ to a decimal.

(a) Decimal to percentage means $\times 100$ (move the point two places right): $0.06 \times 100 = 6\%$ . (b) Percentage to decimal means $\div 100$ (move the point two places left): $8 \div 100 = 0.08$ .

Answers: (a) $6\%$ , (b) $0.08$ .

Common error: Confusing $8\%$ with $0.8$ . Per cent means "out of $100$ ", so $8\%$ is $\dfrac{8}{100} = 0.08$ , not $0.8$ (which would be $80\%$ ).

Common Equivalents Worth Memorising

These come up so often that knowing them by heart saves time on Paper 1.

Fraction	Decimal	Percentage
$\dfrac{1}{2}$	$0.5$	$50\%$
$\dfrac{1}{4}$	$0.25$	$25\%$
$\dfrac{3}{4}$	$0.75$	$75\%$
$\dfrac{1}{5}$	$0.2$	$20\%$
$\dfrac{1}{3}$	$0.\dot{3}$	$33.\dot{3}\%$
$\dfrac{1}{8}$	$0.125$	$12.5\%$
$\dfrac{1}{10}$	$0.1$	$10\%$

Ordering Mixed FDP Sets

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.

Worked Example 4

Write in ascending order: $\dfrac{3}{5}$ , $0.58$ , $62\%$ , $\dfrac{5}{8}$ .

Convert to decimals: $\dfrac{3}{5} = 0.6$ , $0.58$ , $62\% = 0.62$ , $\dfrac{5}{8} = 0.625$ .

In order: $0.58, 0.6, 0.62, 0.625$ .

Answer: $0.58, \; \dfrac{3}{5}, \; 62\%, \; \dfrac{5}{8}$ .

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.

Worked Example 4b

Write in descending order: $\dfrac{2}{3}$ , $0.66$ , $65\%$ , $\dfrac{7}{10}$ .

Convert to decimals: $\dfrac{2}{3} = 0.6\dot{6} \approx 0.667$ , $0.66$ , $65\% = 0.65$ , $\dfrac{7}{10} = 0.7$ .

Largest first: $0.7, 0.667, 0.66, 0.65$ .

Answer: $\dfrac{7}{10}, \; \dfrac{2}{3}, \; 0.66, \; 65\%$ .

Common error: Rounding $\dfrac{2}{3}$ 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.

Terminating vs Recurring Decimals

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 $10 = 2 \times 5$ . Any other prime factor (such as $3$ or $7$ ) forces a recurring decimal.

Worked Example 5

Without dividing, state whether each gives a terminating or recurring decimal: $\dfrac{7}{40}$ , $\dfrac{4}{15}$ , $\dfrac{9}{50}$ .

Factorise denominators: $40 = 2^{3} \times 5$ (only $2$ s and $5$ s) → terminating; $15 = 3 \times 5$ (has a $3$ ) → recurring; $50 = 2 \times 5^{2}$ → terminating.

Answer: $\dfrac{7}{40}$ terminates, $\dfrac{4}{15}$ recurs, $\dfrac{9}{50}$ terminates.

Recurring Decimals to Fractions [H]

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.

Worked Example 6 [H]

Convert $0.\dot{4}$ to a fraction.

Let $x = 0.444\ldots$ The block is one digit, so multiply by $10$ :

$10x = 4.444\ldots$

Subtract the original: $10x - x = 4.444\ldots - 0.444\ldots$ , so $9x = 4$ and $x = \dfrac{4}{9}$ .

Answer: $\dfrac{4}{9}$ .

Worked Example 7 [H]

Convert $0.\dot{2}\dot{7}$ to a fraction in its simplest form.

Let $x = 0.272727\ldots$ The block " $27$ " is two digits, so multiply by $100$ :

$100x = 27.272727\ldots$

Subtract: $100x - x = 27.2727\ldots - 0.2727\ldots$ , so $99x = 27$ and $x = \dfrac{27}{99} = \dfrac{3}{11}$ .

Answer: $\dfrac{3}{11}$ .

Common error: Multiplying by the wrong power of $10$ . Match the power to the length of the repeating block: one digit → $\times 10$ , two digits → $\times 100$ , three digits → $\times 1000$ .

Extended Worked Examples

Worked Example 8 [H]: A non-recurring prefix

Convert $0.41\dot{6}$ to a fraction. (Here " $41$ " does not repeat; only the " $6$ " recurs.)

Let $x = 0.41666\ldots$ The recurring block is one digit. Multiply by $10$ and by $100$ so that the recurring parts line up:

$100x = 41.6666\ldots, \qquad 10x = 4.1666\ldots$

Subtract the smaller from the larger so the tails cancel: $100x - 10x = 41.6\dot{6} - 4.1\dot{6}$ , so $90x = 37.5$ .

$x = \dfrac{37.5}{90} = \dfrac{375}{900} = \dfrac{5}{12}.$

Check: $5 \div 12 = 0.41\dot{6}$ ✓.

Worked Example 9 [H]: A three-digit block

Convert $0.\dot{1}0\dot{8}$ to a fraction. (The block " $108$ " repeats.)

Let $x = 0.108108\ldots$ The block is three digits, so multiply by $1000$ :

$1000x = 108.108108\ldots$

Subtract: $1000x - x = 108$ , so $999x = 108$ and $x = \dfrac{108}{999} = \dfrac{4}{37}$ (dividing top and bottom by $27$ ).

Answer: $\dfrac{4}{37}$ .

Worked Example 10 [H]: Showing $0.\dot{9} = 1$

Use algebra to show that $0.\dot{9} = 1$ .

Let $x = 0.999\ldots$ Then $10x = 9.999\ldots$ Subtracting: $9x = 9$ , so $x = 1$ .

This surprising-but-true result confirms there is no "gap" between $0.\dot{9}$ and $1$ .

Worked Example 11: A full conversion chain

Write $0.625$ as (i) a fraction in simplest form, (ii) a percentage; and order it against $\dfrac{5}{9}$ and $61\%$ .

(i) $0.625 = \dfrac{625}{1000} = \dfrac{5}{8}$ . (ii) $0.625 \times 100 = 62.5\%$ .

Order: $\dfrac{5}{9} = 0.5\dot{5} \approx 0.556$ , $61\% = 0.61$ , $0.625$ . Ascending: $\dfrac{5}{9}, \; 61\%, \; 0.625$ .

Answering at different grade levels

Specimen question modelled on the OCR J560 paper format: [H] Prove algebraically that the recurring decimal $0.2\dot{1}\dot{3}$ can be written as $\dfrac{47}{220}$ .

Converting Fractions, Decimals and Percentages

Converting Fractions, Decimals and Percentages

Key Vocabulary

Converting Between the Three Forms

Worked Example 1

Worked Example 2

Worked Example 3

Worked Example 3b

Common Equivalents Worth Memorising

Ordering Mixed FDP Sets

Worked Example 4

Worked Example 4b

Terminating vs Recurring Decimals

Worked Example 5

Recurring Decimals to Fractions [H]

Worked Example 6 [H]

Worked Example 7 [H]

Extended Worked Examples

Worked Example 8 [H]: A non-recurring prefix

Worked Example 9 [H]: A three-digit block

Worked Example 10 [H]: Showing 0.9˙=10.\dot{9} = 10.9˙=1

Worked Example 11: A full conversion chain

Answering at different grade levels

More in Mathematics

Worked Example 10 [H]: Showing $0.\dot{9} = 1$