Fractions in ratios

Many OCR J560 ratio problems hinge on translating a ratio into the fractional share each part represents. This sits at the boundary of N (number) and R (ratio) and is examined on every paper.

Ratio → fraction

A ratio a : b means a parts to b parts. The total is (a + b) parts.

Each part is the fraction of the whole given by:

• First share = a / (a + b)
• Second share = b / (a + b)

Example. Share £60 in the ratio 2 : 3. Total parts = 5. Each part = 60 ÷ 5 = £12. Shares = £24 and £36. Equivalently: first share = 2/5 of 60 = £24; second share = 3/5 of 60 = £36.

Ratio of three (or more)

For 1 : 2 : 5, total = 8 parts. Fractions are 1/8, 2/8 (= 1/4), 5/8.

Fraction → ratio

If 3/7 of a class is boys, then 4/7 is girls. The boys-to-girls ratio is 3 : 4.

If 5/8 of a journey is by car and the rest by bus, the car-to-bus ratio is 5 : 3.

When you know one share

If A : B = 4 : 5 and A = 24, the parts equal 24 ÷ 4 = 6, so B = 5 × 6 = 30.

Combining ratios

If A : B = 2 : 3 and B : C = 4 : 5, scale to a common B: A : B = 8 : 12 (×4), B : C = 12 : 15 (×3). So A : B : C = 8 : 12 : 15.

Ratio of differences (common Higher trap)

"In a school, the ratio of boys to girls is 5 : 7. There are 24 more girls than boys. How many students are there?" Difference = 7 − 5 = 2 parts = 24, so 1 part = 12. Total = 12 parts × 12 = 144.

OCR mark scheme conventions

• M1 for finding the value of one part (e.g. total ÷ sum-of-parts).
• A1 for each correct share (cao for full marks).
• "Give your answer as a ratio in its simplest form" demands cancellation.
• A ratio answer should use colons (3 : 4), not "to" or "/".