Proportional multi-term ratios

The idea

Two ratios with multiple terms are proportional, a : b : c : d :: p : q : r : s, when a/p = b/q = c/r = d/s, that is, every term is scaled by the same factor.

Puneet has only 2 red chillies

Viswanath's spice mix is 8 : 4 : 2 : 1 (coriander seeds : red chillies : toor dal : fenugreek). Puneet wants a mix that tastes the same, but he has only 2 red chillies, not 4.

How many spoons of coriander seeds should Puneet use?

• 2 spoons
• 4 spoons
• 6 spoons

Copy the recipe

Pick a factor and look at all four ingredients of Puneet's mix. Find the factor for each task. Then press Add 1 more red chilli to see what goes wrong.

Every term, the same factor

Two ratios with multiple terms are proportional, a : b : c : d :: p : q : r : s, when a/p = b/q = c/r = d/s, that is, every term is scaled by the same factor.

Puneet has half of the chillies, so every ingredient is halved: 8 : 4 : 2 : 1 :: 4 : 2 : 1 : 0.5

Check: 8 ÷ 4 = 2, 4 ÷ 2 = 2, 2 ÷ 1 = 2, 1 ÷ 0.5 = 2. All the quotients are the same.

If only one term is changed (one more chilli), its factor is different and the mix does not taste the same.

Notes

Two ratios with multiple terms are proportional, a : b : c : d :: p : q : r : s, when a/p = b/q = c/r = d/s, that is, every term is scaled by the same factor.

Check yourself

To make a special purple, red, blue and white paint are mixed in the ratio 2 : 3 : 5. Yasmin has 10 litres of white paint. How many litres of blue paint should she add?

Answer: 6 litres

White went from 5 to 10, so the factor is 2. Blue is 3 × 2 = 6 litres (red 2 × 2 = 4 litres).

Which ratio is proportional to 8 : 4 : 2 : 1?

Concrete for pillars is made from cement, sand and gravel in the ratio 1 : 1.5 : 3 (bags). With 3 bags of cement, how many bags of concrete can be made in all?

Answer: 16.5 bags

The factor is 3: cement 3, sand 4.5, gravel 9 (3 : 4.5 : 9). In all, 3 + 4.5 + 9 = 16.5 bags.

The ratio 6 : 9 : 15 is proportional to 2 : p : q. What is q?

Answer: 5

6 became 2, so every term is divided by 3: p = 9 ÷ 3 = 3 and q = 15 ÷ 3 = 5. So 6 : 9 : 15 :: 2 : 3 : 5.

• 12 : 8 : 4 : 2. The first term was multiplied by 1.5 but the others by 2. Every term needs the same factor.
• 16 : 8 : 4 : 2 — correct. Yes! Every term is multiplied by 2: 8 → 16, 4 → 8, 2 → 4, 1 → 2.
• 9 : 5 : 3 : 2. Here 1 was added to every term. Adding the same number is not scaling by the same factor.
• 10 : 6 : 4 : 3. Here 2 was added to every term. Adding the same number is not scaling by the same factor.