Fractions, Decimals and Percentages

Exact answers use fractions, surds or multiples of $\pi$ rather than rounded decimals. Edexcel exam questions that say "give an exact answer" or "leave your answer in terms of $\pi$" require this.

Why fractions give exact answers:

$\frac{1}{3}=0.333\dots$ — as a decimal this never terminates. As a fraction it is exact. Calculations involving $\frac{1}{3}$ must stay as fractions to remain exact.

Multiples of π:

$\pi =3.14159\dots$ is irrational — its decimal expansion never terminates or repeats. An exact answer involving $\pi$ keeps it as a symbol.

Worked example — a circle has radius 5 cm. Give the circumference and area as exact values.

Circumference $=2\pi r=2\times \pi \times 5=10\pi$ cm

Area $=\pi r^2=\pi \times 5^2=25\pi$ cm²

Worked example — calculate $\frac{5}{6}+\frac{3}{4}$ exactly.

Common denominator is 12: $\frac{10}{12}+\frac{9}{12}=\frac{19}{12}=1\frac{7}{12}$ ✓

A terminating decimal has a finite number of digits after the decimal point: $0.5,0.125,3.75$.

A recurring decimal has one or more digits that repeat infinitely: $0.\dot{3}=0.333\dots$; $0.\dot{1}\dot{4}=0.141414\dots$

Which fractions terminate? A fraction $\frac{p}{q}$ in lowest terms terminates if and only if $q$ has no prime factors other than 2 and 5.

Fraction	Denominator factors	Terminating?	Decimal
$\frac{3}{8}$	$8=2^3$	Yes	0.375
$\frac{7}{20}$	$20=2^2\times 5$	Yes	0.35
$\frac{1}{3}$	$3$ (prime, not 2 or 5)	No	$0.\dot{3}$
$\frac{5}{12}$	$12=2^2\times 3$	No	$0.41\dot{6}$

Converting terminating decimals to fractions:

$0.375=\frac{375}{1000}=\frac{3}{8}$ (divide numerator and denominator by HCF = 125) ✓

The algebraic method: let $x$ equal the recurring decimal, multiply to shift one full cycle, then subtract.

Worked example — convert $0.\dot{7}$ to a fraction.

Let $x=0.777\dots$

$10x=7.777\dots$

Subtract: $10x-x=7$, so $9x=7$, giving $x=\frac{7}{9}$ ✓

Worked example — convert $0.2\dot{7}\dot{3}$ to a fraction (the repeating block is 73).

Let $x=0.2737373\dots$

$10x=2.737373\dots$

$1000x=273.7373\dots$

Subtract: $1000x-10x=271$, so $990x=271$, giving $x=\frac{271}{990}$ ✓

(This method also works in reverse — to convert a fraction to a recurring decimal, perform long division.)

Both fractions and percentages can be used to operate on a quantity — to find a part of a whole.

Fractions as operators:

$\frac{3}{5}$ of 240 $=\frac{3}{5}\times 240=\frac{720}{5}=144$

Percentages as operators — convert the percentage to a decimal multiplier:

Percentage	Decimal multiplier
20% of $x$	$0.2x$
7.5% of $x$	$0.075x$
135% of $x$	$1.35x$

Worked example — a shirt has an original price of £42. A 30% discount is applied. Find the sale price.

Discount amount: $30\%\text{of}\pounds 42=0.3\times 42=\pounds 12.60$

Sale price: $\pounds 42-\pounds 12.60=\pounds 29.40$ ✓

Note: 30% of £42 gives the discount amount, not the final price. Read the question carefully — it may ask for the discount, the final price, or the original price.

Ratio and fraction notation describe the same proportional relationship.

Converting a ratio to fractions:

If the ratio of girls to boys is $3:2$, then girls form $\frac{3}{5}$ of the total and boys form $\frac{2}{5}$ of the total.

Worked example — a drink is mixed in the ratio orange juice : lemonade $=2:3$. What fraction of the drink is lemonade?

Total parts: $2+3=5$. Lemonade is $\frac{3}{5}$ of the drink.

Worked example — a class of 30 students has a boys-to-girls ratio of $2:3$. How many girls are there?

Girls $=\frac{3}{5}\times 30=18$ ✓

Using fractions to split quantities in a given ratio:

Divide £240 in the ratio $5:3$:

Total parts: $5+3=8$. Each part $=\pounds 240÷8=\pounds 30$.

Shares: $5\times \pounds 30=\pounds 150$ and $3\times \pounds 30=\pounds 90$. Check: $150+90=240$ ✓

1. Rounding π when an exact answer is required

If a question says "give an exact answer" or "in terms of π", write $10\pi$ not $31.4\dots$. Rounding $\pi$ loses marks.

2. Confusing terminating with recurring

$0.666\dots$ is recurring — it is equal to $\frac{2}{3}$, not $0.7$. Likewise $0.125$ is terminating and equals $\frac{1}{8}$ exactly.

3. Percentage of — applying to the wrong value

"30% off the original price of £60" and "30% of the sale price" give different answers. The percentage applies to the base value specified in the question.

4. Ratio-to-fraction conversion

In ratio $3:2$, the fraction of the whole that is the first part is $\frac{3}{5}$, not $\frac{3}{2}$. Divide by the total number of parts (denominator = sum of all ratio parts).

Mistake	Correction
"Exact answer for area of circle with $r=4$: $50.27$ cm²"	Exact answer is $16\pi$ cm²
"$0.\dot{3}=\frac{1}{3.33}$"	$0.\dot{3}=\frac{1}{3}$ (use the algebraic subtraction method)
"Ratio $3:4$, so first part is $\frac{3}{4}$ of total"	First part is $\frac{3}{7}$ of total (denominator = sum of ratio parts)