10.3 Ratios with More than Two Terms

Ratios are not stuck at two quantities. Mehul prepares a homemade chutney powder by grinding 6 spoons of roasted gram, 4 dried red chillies, 2 spoons of curry leaves, and 1 spoon of cumin. The ratio of gram : chillies : curry leaves : cumin is

6 : 4 : 2 : 1.

This ratio holds four terms. A ratio may carry as many terms as we wish, so long as every quantity changes by the same factor to keep the relationship proportional.

Now Farida finds only 2 dried red chillies in her pantry but wants the same flavour. Mehul used 4 chillies, so Farida has half that amount. To keep the taste identical, every ingredient must be halved:

6 : 4 : 2 : 1 \;\xrightarrow{\;\times\frac12\;}\; 3 : 2 : 1 : 0.5.

Both ratios are proportional, which we record as

6 : 4 : 2 : 1 \;::\; 3 : 2 : 1 : 0.5.

The Key Idea When two multi-term ratios are proportional, a : b : c : d \;::\; p : q : r : s, each pair of corresponding terms shares the same quotient: \frac{a}{p} = \frac{b}{q} = \frac{c}{r} = \frac{d}{s}.

Worked Example

A leaf-green paint is mixed Yellow : Blue : White = 3 : 2 : 5. Imran has 20 litres of white paint. How much yellow and blue should he add?

White corresponds to 5 parts. If 5 parts = 20 litres, then 1 part = 20 \div 5 = 4 litres. So \text{Yellow} = 3 \times 4 = 12 \text{ litres}, \qquad \text{Blue} = 2 \times 4 = 8 \text{ litres}. The paint uses 12 litres yellow, 8 litres blue and 20 litres white, a total of 12 + 8 + 20 = 40 litres.

Worked Example

A garden mortar is mixed cement : sand : gravel = 1 : 2 : 4. With 4 bags of cement, how many bags of mortar can we make?

We have 4 bags of cement, so multiply every term by 4: \text{cement} : \text{sand} : \text{gravel} = 4 : 8 : 16. Altogether that is 4 + 8 + 16 = 28 bags of mortar.

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