Ratio, proportion and early algebra
Maths · Key Stage 2 · The School
Ratio compares parts
A ratio like 2:3 says that for every 2 of one thing there are 3 of the other. To share £20 in the ratio 2:3, first count the PARTS — 2 + 3 = 5 parts. One part is £20 ÷ 5 = £4. So the shares are £8 and £12. Finding the number of parts first is the step people skip, and skipping it is why ratio questions go wrong.
Proportion keeps it the same
Scaling a recipe is proportion: if 4 people need 200 g of rice, 6 people need 300 g, because everything is multiplied by the same amount. Find what one unit needs — 50 g per person — and multiply. This is the same reasoning as a map scale, a currency conversion and a best-value comparison in a shop.
A letter is just an unknown number
In algebra a letter stands for a number you do not know yet. `n + 5 = 12` says some number plus 5 makes 12, so n is 7. Solving means getting the letter alone by doing the same thing to both sides — subtract 5 from each side and you are left with n = 7. Nothing mysterious is happening; it is arithmetic with a gap in it.
Count the parts first: 2 + 3 = 5 parts. One part is £20 ÷ 5 = £4. So the shares are 2 × £4 = £8 and 3 × £4 = £12. Check: £8 + £12 = £20. The common error is reading 2:3 as two-thirds and answering £13.33 and £6.67 — which does not even total £20, and that check alone catches it. Always add the parts before you divide.