Rates and ratios

To simplify a ratio, divide both parts by the same number until they have no common factor left. For 6:8, the biggest number that divides both is 2, so 6 ÷ 2 = 3 and 8 ÷ 2 = 4, giving 6:8 = 3:4. When a bigger common factor exists, use it to finish in one step: 9:12 both divide by 3, straight to 3:4. Equal ratios describe the same proportion, just written with bigger or smaller numbers.

You can only compare amounts measured in the same unit. To simplify 50 cm to 2 m, convert first: 2 m = 200 cm, so the ratio is 50:200 = 1:4. Writing 50:2 would be wrong — it secretly compares centimetres with metres. Convert to the smaller unit, then simplify.

To keep a ratio the same, multiply (or divide) both parts by the same number. A blue-to-white paint mix is 2:3. To use 8 litres of blue, notice blue went from 2 to 8, which is ×4, so white is 3 × 4 = 12 litres. Scaling down works the same way: cordial to water 1:5 with 50 mL of water means water went 5 → 50 (×10), so cordial is 1 × 10 = 10 mL.

To split a total in a ratio, add the parts to find how many equal shares there are, then find the size of one share. To share $40 in the ratio 3:5, there are 3 + 5 = 8 parts; one part is 40 ÷ 8 = $5, so the shares are $15 and $25, which add back to $40. Three-part ratios work identically: paint mixed 2:3:5 in a 40 mL pot has 2 + 3 + 5 = 10 parts, each 4 mL, so the yellow (3 parts) is 12 mL. You can also work backwards: if the 4 parts of a 4:5 share are worth $20, one part is $5 and the whole 9 parts is $45.

A ratio compares the parts; the whole is their sum. If juniors to seniors is 3:2, the whole is 3 + 2 = 5 parts, so juniors are 3 out of 5, a fraction of 3/5 = 0.6. The classic mistake is reading 3:2 as “3 out of 2” — but 2 is the other part, not the total. Always add the parts to get the whole.

A unit rate tells you the amount per one unit; divide to find it. A car going 150 km in 3 hours travels 150 ÷ 3 = 50 km/h. Unit rates make value easy to compare: to find the best buy, reduce every option to the same “per” amount. 750 mL for $1.50 is 200 cents per litre, while 2 L for $3.60 is 180 cents per litre — so the bigger bottle is the better value.

Watch for these Year 8 ratio traps: order matters — 3:4 is not the same as 4:3, so match each number to the thing it counts. A ratio is part-to-part; to get a fraction of the whole you must add the parts first (3:2 → juniors are 3/5, not 3/2). Different units must be made the same before you simplify (50 cm : 2 m → 50 : 200). And for a best buy, compare a single shared unit (price per litre, cost per 100 g), not the raw price — the cheaper sticker can be the worse deal.