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G2.5 Fractions, Decimals and Percentages

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01 — Explanation

Why are fractions, decimals and percentages "the same thing"?

$\tfrac{1}{2}$ , $0.5$ and $50\%$ all describe the identical proportion of a whole — just in different notations. Exam questions constantly require you to switch between these forms within a single problem, so fluent conversion both ways is non-negotiable.

02 — Explanation

How do I convert between the three forms?

With a calculator allowed on both papers, each conversion reduces to one or two button presses:

From → To	Method	Example
Fraction → Decimal	numerator $\div$ denominator	$5 \div 8 = 0.625$
Decimal → Percentage	$\times 100$	$0.8 \times 100 = 80\%$
Percentage → Decimal	$\div 100$	$26 \div 100 = 0.26$
Percentage → Fraction	write over 100, simplify	$36\% = \tfrac{36}{100} = \tfrac{9}{25}$
Decimal → Fraction	use the place value of the last digit	$0.625 = \tfrac{625}{1000} = \tfrac{5}{8}$
Fraction → Percentage	convert to decimal, then $\times 100$	$\tfrac{7}{20} = 0.35 = 35\%$

Exam Tip

memorising $\tfrac{1}{4}=0.25=25\%$ , $\tfrac{1}{3}=0.\dot{3}=33.3\%$ , $\tfrac{1}{8}=0.125=12.5\%$ saves time on the easiest marks in the paper, which are usually clustered in the first ten questions.

03 — Explanation

How do I add or subtract fractions with different denominators?

Find a common denominator (the LCM is most efficient), convert each fraction, then add or subtract the numerators only.

\frac{4}{9} + \frac{1}{6} = \frac{8}{18} + \frac{3}{18} = \frac{11}{18}

Common Mistake

a "Show that..." question like this cannot be answered using a calculator's fraction button alone — you must display the common-denominator working, or no marks are awarded even for a correct answer.

04 — Explanation

How do I order a mixed list of fractions, decimals and percentages?

Convert everything to the same format — decimals are usually fastest — then compare directly.

Example: order $75\%, 0.72, 0.9, \tfrac{7}{10}$ from smallest. Converting: $0.75, 0.72, 0.9, 0.70$ . Ordered: $0.70 < 0.72 < 0.75 < 0.90$ , so the answer is $\tfrac{7}{10}, 0.72, 75\%, 0.9$ .

05 — Worked example

Worked exam-style question

There are 30 dogs in kennels. 12 are brown. (a) What fraction of the dogs are not brown, in simplest form? (b) Show that $\tfrac{4}{9}+\tfrac{1}{6}=\tfrac{11}{18}$ .

(a) Not brown: $30 - 12 = 18$ , so the fraction is $\tfrac{18}{30}$ . Dividing top and bottom by 6: $\tfrac{3}{5}$ .

(b) LCM of 9 and 6 is 18: $\tfrac{4}{9}=\tfrac{8}{18}$ and $\tfrac{1}{6}=\tfrac{3}{18}$ , so $\tfrac{8}{18}+\tfrac{3}{18}=\tfrac{11}{18}$ ✓.

Exam Tip

when a question starts with "Write..." or "Find...", a calculator answer alone is usually enough; when it starts with "Show that...", the examiner wants to see the intermediate working that proves the given answer, not just the answer itself. Spotting this difference in wording is often worth more than knowing the method.

Common Mistake

leaving a converted fraction unsimplified, e.g. writing $36\% = \tfrac{36}{100}$ and stopping there. Always check whether numerator and denominator share a common factor before finalising your answer — a correct but unsimplified fraction can still lose the accuracy mark.