M22 · Ratio and sharing | Edexcel GCSE Maths

Revise the key ideas

A ratio compares quantities in a specified order. Red : blue = 2 : 3 means two red parts for every three blue parts; reversing the order changes the ratio.
Simplify by dividing all parts by their HCF: 18 : 24 = 3 : 4. For three parts, 12 : 18 : 30 = 2 : 3 : 5.
Convert quantities to the same units before comparing: 50 cm : 2 m = 50 : 200 = 1 : 4.
A part-to-part ratio 2 : 3 gives part-to-whole fractions 25 and 35. The first part is 23 of the second part, but 25 of the total.
Ratio bar for sharing eighty-four pounds
Enlarge diagram: Ratio bar for sharing eighty-four pounds
To express one quantity as a fraction of another, use first quantitysecond quantity with matching units. The fraction can exceed 1.

Worked example: Share £84 in ratio 2 : 5. There are 7 parts; each is £12, so the shares are £24 and £60. Check both their total and their simplified ratio.
Worked example: A : B = 3 : 7 and B is 28. One part is 28 ÷ 7 = 4, so A = 12 and the total is 40. Do not divide 28 by all 10 parts.
Worked example: Two shares are in ratio 3 : 5 and differ by £18. Two parts represent £18, so one part is £9; the shares are £27 and £45.
For mixing, preserve the ratio when scaling quantities. A drink mixed concentrate : water = 1 : 4 needs 200 ml concentrate and 800 ml water for 1 litre of drink.
If a ratio describes concentrations or mixtures, be clear whether it is a mass ratio or volume ratio. Do not combine incompatible units silently.

Equivalent ratios have equal quotients: a : b = c : d means ab = cd where denominators are non-zero. This links ratios to proportional relationships.
If x : y = 2 : 5, then y = 52x. On a y-against-x graph this is a straight line through the origin.
To combine A : B = 2 : 3 and B : C = 4 : 5, make B match: 8 : 12 and 12 : 15, so A : B : C = 8 : 12 : 15.
Adding the same amount to both quantities generally changes their ratio. A 2 : 3 mixture does not stay in that ratio if you add 1 litre to each component.
After a change, work with actual quantities or parts before simplifying the new ratio. Multiplying both quantities by the same factor preserves the ratio; adding does not.

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.