The idea

Combining and relating ratios

How to link two ratios that share a quantity into one three-part ratio, and how ratios connect to fractions and to the "1 : n" form.

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Ratio, proportion & rates of change · Ratio

How to link two ratios that share a quantity into one three-part ratio, and how ratios connect to fractions and to the "1 : n" form.

Often two ratios share a common quantity: "

a : b = 3 : 4

and

b : c = 6 : 5

". Both mention

b

, but they disagree about its size — one calls it

4

, the other

6

. You cannot just staple them into

3 : 4 : 5

, because in a valid three-part ratio the middle number has to mean the same amount of

b

in both halves.

The fix is to scale each ratio so the shared number matches. Make

b

the same in both by taking a common multiple of

4

and

6

, which is

12

a : b = 3 : 4 \;\xrightarrow{\times 3}\; 9 : 12, \qquad b : c = 6 : 5 \;\xrightarrow{\times 2}\; 12 : 10.

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