Working interchangeably
A fraction, a decimal and a percentage are three ways of writing the same thing, and being able to move between them at will is what makes comparison questions quick. Convert everything to one form — decimals are usually easiest — before ordering or comparing.
Recurring decimals into fractionsHigher tier
A terminating decimal stops; a recurring decimal repeats forever, shown by a dot over the repeating digit or over the first and last digits of a repeating block. Both can be written exactly as fractions — which is why they are called rational numbers.
The algebraic method works every time. Call the decimal x, multiply by a power of 10 that shifts it by exactly one repeating block, then subtract. The repeating tails cancel, leaving something you can solve.
Worked example
Write 0.4272727… (with 27 recurring) as a fraction in its simplest form.
- Let x = 0.4272727…
- The repeating block is 27, which is two digits, and it starts after one non-repeating digit.
- Multiply by 10 to move past the non-repeating part: 10x = 4.272727…
- Multiply by 1000 to shift a further two digits, one whole block: 1000x = 427.272727…
- Subtract the first from the second, so the identical repeating tails cancel: 1000x − 10x = 427.2727… − 4.2727…, giving 990x = 423
- Solve: x = 423⁄990
- Simplify by dividing top and bottom by 9: 47⁄110
Answer: 47⁄110. Choose the two multipliers so the decimal parts are identical before subtracting — that is the whole trick.
Common mistake
Multiplying by the wrong power of 10 leaves different tails, so they do not cancel and the subtraction gives a decimal rather than a whole number. If the result of the subtraction is not a whole number, that is the error.
- Terminating decimal
- a decimal that stops
- Recurring decimal
- a decimal that repeats forever, shown by a dot over the repeating digit or over the first and last digits of a repeating block
- Rational number
- a number that can be written exactly as a fraction; every terminating and recurring decimal is rational
Fractions in ratio problems, and as operators
A fraction as an operator means using it to act on a quantity rather than to describe a part: finding ⅗ of 40 is multiplying, 40 × 3 ÷ 5 = 24. Percentages work identically — 15% of 60 is 0.15 × 60 = 9.
Multiplying by a fraction below 1 makes a quantity smaller and multiplying by a fraction above 1 makes it larger, which is worth holding on to as a check.
Working backwards is the harder direction and the more heavily examined one: if ⅔ of a number is 18, then the number is 18 ÷ ⅔ = 18 × 3⁄2 = 27. Dividing by a fraction is multiplying by its reciprocal.
Fractions in ratio problems come from reading a ratio as parts of a whole. If money is shared in the ratio 2 : 3, there are 5 parts in total, so one person gets 2⁄5 and the other 3⁄5. That conversion — from ratio to fraction of the total — is what turns most ratio questions into a single multiplication.
Worked example
In a club, the ratio of members who play chess to those who do not is 3 : 7. If 24 members play chess, how many members are there in total?
- Read the ratio as parts: 3 parts play, 7 do not, so 10 parts altogether.
- The players are 3⁄10 of the club.
- So 3⁄10 of the total is 24.
- Find one part: 24 ÷ 3 = 8 members per part.
- Multiply by the total number of parts: 8 × 10 = 80.
Answer: 80 members. Finding the value of one part first works for every ratio question and avoids setting up an equation.
Common mistake
A ratio of 3 : 7 does not mean 3⁄7 of the club plays chess — it means 3⁄10 does. Add the parts to get the denominator.
Before you practise
- Fraction to decimal: divide the top by the bottom. Decimal to percentage: multiply by 100.
- Convert everything to one form before comparing or ordering.
- A recurring decimal repeats forever and can always be written as a fraction.
- To convert one, set x equal to the decimal, multiply by two powers of 10 that leave identical tails, and subtract.
- If the subtraction does not give a whole number, the multipliers were wrong.
- A fraction or percentage as an operator is a multiplication: ⅗ of 40 is 40 × 3 ÷ 5.
- To work backwards, divide by the fraction — which means multiplying by its reciprocal.
- A ratio of a : b has a + b parts, so the first share is a⁄(a+b) of the whole.
- Find the value of one part first — it solves most ratio questions in one step.