Technique

Join two ratios on their shared term

Arithmetic • Ratio and Proportion • Proportion

Statement

Two ratios

A : B = p : q

and

B : C = r : s

can be joined only when the shared term

B

carries the same number in each. Multiplying the first by

r

and the second by

q

makes

B

read

q r

in both, giving the continued ratio

A : B : C = p r : q r : q s

. Equivalently

\frac{A}{C} = \frac{p}{q} \times \frac{r}{s}

How to use it

1Scale each ratio by whatever the other's shared term is — or by the LCM of the two, which keeps the numbers small.
2For a long chain $A : B$ , $B : C$ , $C : D$ , $D : E$ , join one link at a time; only the links between the two terms asked for are needed, so a ratio outside them can be ignored entirely.
3When only the end-to-end ratio is wanted, multiply the fractions straight through: $\frac{B}{E} = \frac{B}{C} \cdot \frac{C}{D} \cdot \frac{D}{E}$ .

A : B = 2 : 3

and

B : C = 5 : 7

, find

A : B : C

Step-by-step

1
$A : B = 2 : 3 = 10 : 15$ and $B : C = 5 : 7 = 15 : 21$
Why: Scale each so the shared term $B$ reads $15$ , the LCM of $3$ and $5$ .
2
$A : B : C = 10 : 15 : 21$

Answer

10 : 15 : 21

Watch out

Never multiply the two ratios term by term as

2 \cdot 5 : 3 \cdot 7

— that is not a ratio of the same three quantities. Check the direction of each link before joining: a ratio given as

C : B

must be reversed before its

B

can be matched.

Builds on

k

Also known as

compound ratio, continued ratio, combine ratios, chain of ratios, A:B and B:C to A:B:C, ratio bridge, equalise the common term