FractionsEdexcel GCSE Maths: Revision notes

A fraction represents a part of a whole, written as a numerator over a denominator. A proper fraction has a numerator smaller than the denominator (e.g. 3/4); an improper fraction has a numerator equal to or larger than the denominator (e.g. 7/4); a mixed number combines a whole number and a proper fraction (e.g. 1 3/4).

To simplify a fraction, divide the numerator and denominator by their highest common factor (HCF).

• 12/18: HCF of 12 and 18 is 6, so 12/18 = 2/3
• Always check a simplified fraction cannot be reduced further

To convert between improper fractions and mixed numbers:

1. Improper → mixed: divide numerator by denominator; the quotient is the whole number, the remainder is the new numerator
2. Mixed → improper: multiply the whole number by the denominator, add the numerator, keep the same denominator

To add or subtract fractions, they must have the same denominator (a common denominator).

1. Find the lowest common multiple (LCM) of the denominators — this is the common denominator
2. Convert each fraction to an equivalent fraction with this denominator
3. Add or subtract the numerators, keeping the denominator the same
4. Simplify the answer

For mixed numbers, either convert to improper fractions first, or add/subtract the whole numbers and fractions separately (borrowing from the whole number if a subtraction of fractions would go negative).

Example: 2/3 + 1/4. Common denominator is 12: 8/12 + 3/12 = 11/12

Multiplying fractions: multiply the numerators together and the denominators together, then simplify.

• 2/3 × 3/5 = 6/15 = 2/5

You can also cancel common factors between any numerator and any denominator before multiplying to keep numbers smaller.

Dividing fractions: multiply by the reciprocal of the second fraction (flip it upside down).

• 2/3 ÷ 1/4 = 2/3 × 4/1 = 8/3 = 2 2/3

For mixed numbers, always convert to improper fractions first before multiplying or dividing.

Every fraction can be written as a decimal by dividing the numerator by the denominator.

• Terminating decimals end after a fixed number of digits (e.g. 1/4 = 0.25)
• Recurring decimals repeat a digit or group of digits forever (e.g. 1/3 = 0.333... = 0.3̇)

A fraction gives a terminating decimal only if, once fully simplified, its denominator's only prime factors are 2 and/or 5.

To convert a recurring decimal to a fraction, let x equal the decimal, multiply by a power of 10 to shift the repeating block, subtract to eliminate the recurring part, then solve for x.

A fraction can act as an operator meaning "of" — to find a fraction of a quantity, multiply the quantity by the fraction.

• Find 3/5 of 40: 40 × 3/5 = 24

Fractions also appear in ratio problems, where a ratio can be converted into fractions of a total to share an amount.

• Share £60 in the ratio 2:3:5 — total parts = 10, so the amounts are 2/10, 3/10, 5/10 of £60, giving £12, £18, £30