M3 · Fractions, decimals and percentages | Edexcel GCSE Maths

Revise the key ideas

The numerator counts parts; the denominator tells how many equal parts make one whole. Equivalent fractions have the same value: 34 = 68. Multiply or divide numerator and denominator by the same non-zero number.
Three quarters of a whole
Enlarge diagram: Three quarters of a whole
Simplify a fraction by dividing by its HCF: 1824 = 34. A denominator cannot be zero.
An improper fraction has a numerator at least as large as its denominator. 114 = 234, because 11 ÷ 4 is 2 remainder 3. Convert mixed numbers to improper fractions before multiplication or division.
To convert a fraction to a decimal, divide numerator by denominator. 38 = 0.375. A percentage is a fraction out of 100, so 0.375 = 37.5%.
To convert a terminating decimal to a fraction, use its place value and simplify: 0.24 = 24100 = 625. 125% = 1.25 = 54; a percentage can exceed 100%.

For addition/subtraction, use a common denominator: 23 + 14 = 812 + 312 = 1112. Do not add denominators.
For multiplication, multiply numerators and denominators. Cancel common factors first if helpful: 35 × 1021 = 27.
For division, multiply by the reciprocal of the divisor: 34 ÷ 25 = 34 × 52 = 158 = 178.
Signs work as in ordinary multiplication: (−23) × 34 = −12. Put the sign outside the fraction to keep notation clear.

To find a fraction of an amount, divide by the denominator then multiply by the numerator: 35 of £80 = £48. This is the same as multiplying £80 by 35.
To find 15% of £60, use 0.15 × 60 = £9, or combine 10% (£6) and 5% (£3). Always keep units attached to the answer.
In a ratio 2 : 3, there are five parts in total. The first share is 25 of the total, not 23. The fraction comparing first share to second share is 23.
Keep exact fractions during calculations when an exact answer is requested. Rounding 13 to 0.33 too early loses accuracy.

A recurring decimal repeats forever. Write the repeating block with a bar, for example 0.27 = 0.272727…; this is different from the terminating decimal 0.27.
Worked example: Let x = 0.272727…. Then 100x = 27.272727…. Subtract x: 99x = 27, so x = 2799 = 311.
Worked example: For x = 0.16666…, 10x = 1.6666… and 100x = 16.6666…. Subtract to get 90x = 15, so x = 16. Align the repeating tails before subtracting.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.