Converting Between Terminating Decimals and Fractions Revision Notes | GCSE AQA Higher Maths

1. Why This Conversion Matters

You have already worked with fractions and decimals separately. Now the exam expects you to move between them without hesitation — in the same question, sometimes in the same line of working.

GCSE Higher exam questions regularly present a value as a decimal and ask you to compare, order, or calculate with it alongside fractions. If you cannot convert fluently, you lose method marks even when your arithmetic is correct. This lesson gives you a reliable, repeatable method for both directions of conversion.

2. What Makes a Decimal Terminating?

A decimal terminates when it stops — there is a last digit after the decimal point. Compare $0.375$ (terminates after three decimal places) with $0.333...$ (the threes go on forever — that is a recurring decimal, which is a different topic).

The key insight is this: the position of the last digit after the decimal point tells you the denominator to use. One decimal place means tenths (denominator 10), two decimal places means hundredths (denominator 100), three decimal places means thousandths (denominator 1000).

3. Converting a Decimal to a Fraction

The method has two stages: write the decimal as a fraction over the correct power of ten, then simplify. That is it. The only skill you need beyond that is finding the HCF to simplify — which you have already practised.

Worked example: Decimal to Fraction: Converting 0.375

Write $0.375$ as a fraction in its simplest form.

1. $0.375$ has 3 decimal places — The last digit (5) is in the thousandths column, so the denominator will be 1000. [M1 — identifying correct denominator]
2. $0.375 = \frac{375}{1000}$ — Remove the decimal point and place the digits over 1000. The numerator is 375.
3. $375 = 3 \times 125 = 3 \times 5^3$ $1000 = 8 \times 125 = 2^3 \times 5^3$ \text{HCF} = 125 — Both share a factor of 125. Dividing by the HCF gives the simplest form. [M1 — correct simplification step]
4. $\frac{375 \div 125}{1000 \div 125} = \frac{3}{8}$ — The fraction is now fully simplified. Check: 3 and 8 share no common factors other than 1. [A1 — correct simplified fraction]

Answer: $0.375 = \frac{3}{8}$

Notice that the answer $\frac{3}{8}$ is one of The higher-tier specification examples. The denominator 8 is a power of 2 — which is exactly why this fraction produces a terminating decimal in the first place.

Worked example: Decimal to Fraction: Converting 3.5

Write $3.5$ as a fraction in its simplest form.

1. $3.5$ has 1 decimal place — The last digit (5) is in the tenths column, so the denominator is 10.
2. $3.5 = \frac{35}{10}$ — Write all digits (including the whole-number part) over 10. The numerator is 35. [M1]
3. $35 = 5 \times 7$ $10 = 5 \times 2$ \text{HCF} = 5 — Both share a factor of 5.
4. $\frac{35 \div 5}{10 \div 5} = \frac{7}{2}$ — The fraction is fully simplified. This is an improper fraction — its value is greater than 1, which matches the original decimal 3.5. [A1]

Answer: $3.5 = \frac{7}{2}$

Both $3.5 = \frac{7}{2}$ and $0.375 = \frac{3}{8}$ are listed directly in The higher-tier specification as expected equivalences. If you see either of these in an exam, you should be able to write the conversion immediately without working.

4. Converting a Fraction to a Decimal

Going the other way — fraction to decimal — uses division. You divide the numerator by the denominator. For fractions that produce terminating decimals, this division always ends cleanly with no remainder.

There is also a shortcut worth knowing: if you can scale the denominator to a power of 10 (10, 100, 1000,...) by multiplying top and bottom by the same number, you can read off the decimal directly. This is particularly useful In questions without a calculator.

Worked example: Fraction to Decimal: Converting 3/8 (Without a Calculator)

Write $\frac{3}{8}$ as a decimal without a calculator.

1. $8 \times 125 = 1000$ — Multiplying 8 by 125 gives 1000, which is a power of 10. This means we can use the equivalent fraction method. [M1]
2. $\frac{3}{8} = \frac{3 \times 125}{8 \times 125} = \frac{375}{1000}$ — Multiplying top and bottom by the same number keeps the fraction equivalent.
3. $\frac{375}{1000} = 0.375$ — A fraction over 1000 has three decimal places. Place the digits 375 so the last digit is in the thousandths column. [A1]

Answer: $\frac{3}{8} = 0.375$

Worked example: Fraction to Decimal: Converting 7/2 (Without a Calculator)

Write $\frac{7}{2}$ as a decimal without a calculator.

1. $2 \times 5 = 10$ — Multiplying 2 by 5 gives 10, so we scale both parts of the fraction by 5.
2. $\frac{7}{2} = \frac{7 \times 5}{2 \times 5} = \frac{35}{10}$ — The fraction is now expressed in tenths. [M1]
3. $\frac{35}{10} = 3.5$ — A fraction over 10 has one decimal place. The digits 35 with one decimal place give 3.5. [A1]

Answer: $\frac{7}{2} = 3.5$

5. The Division Method (With and Without a Calculator)

When scaling the denominator to a power of 10 is not straightforward — for example with $\frac{3}{16}$ — the reliable fallback is long division. On a calculator-allowed exam question, you press numerator $\div$ denominator and read the display. On a question without a calculator, you perform the division by hand.

Worked example: Fraction to Decimal: Converting 3/16 by Division

Write $\frac{3}{16}$ as a decimal. (This method works on both calculator-allowed questions and questions without a calculator.)

1. $3 \div 16$ — Numerator divided by denominator. Since 3 < 16, the result will be less than 1. Write 3 as 3.0000 to allow division to proceed.
2. $30 \div 16 = 1$ remainder $14$ — $16 \times 1 = 16$, and $30 - 16 = 14$. Write 1 in the tenths place.
3. $140 \div 16 = 8$ remainder $12$ — $16 \times 8 = 128$, and $140 - 128 = 12$. Write 8 in the hundredths place.
4. $120 \div 16 = 7$ remainder $8$ — $16 \times 7 = 112$, and $120 - 112 = 8$. Write 7 in the thousandths place.
5. $80 \div 16 = 5$ remainder $0$ — $16 \times 5 = 80$, remainder 0. The division terminates here. [A1]

Answer: $\frac{3}{16} = 0.1875$

Notice that the division ended with remainder zero — confirming this is a terminating decimal. If you were dividing and the remainders started repeating, that would signal a recurring decimal instead.

6. Working Interchangeably: Mixed Examples

The specification says you must work interchangeably with terminating decimals and their corresponding fractions. That means you should be comfortable converting in either direction, mid-question, without losing track of the value.

Here is a table of common terminating decimals and their fraction equivalents. These are worth committing to memory — they appear frequently across all three GCSE Higher exam papers.

Look at the final column. Every simplified denominator has only 2 and/or 5 as prime factors. This is not a coincidence — it is the precise condition that guarantees a fraction produces a terminating decimal. Denominators like 3, 6, 7, or 9 introduce other prime factors and produce recurring decimals.

7. Deliberate Practice

The following problems cover both directions of conversion and include values greater than one. Work through each one before checking the answer — the method is the same every time.

8. Exam Technique and Common Mistakes

This topic appears across all three GCSE Higher exam papers. In questions without a calculator, you must show all arithmetic working — a bare answer with no method shown risks scoring zero even if the answer is correct. On calculator-allowed exam questions, a calculator confirms your answer but the method steps still earn marks.