Often you are told how two things compare, and separately how one of them compares to a third — and you want the single ratio linking all three. If A : B is known and B : C is known, they share the term B, and that shared term is the hinge you swing them together on to get one three-part ratio A : B : C.

Here's the catch, and the whole trick: B almost never means the same number in both ratios. In A : B = 2 : 3 the B is 3 parts; in B : C = 4 : 5 it's 4 parts. Before you can join them, you must make the shared term agree.

The method: make the shared term match

A ratio doesn't change when you scale every part by the same number, so you're free to rescale each ratio until the shared B reads the same in both. The cleanest common value is the lowest common multiple of the two B values.

Take A : B = 2 : 3 and B : C = 4 : 5. The two Bs are 3 and 4; their LCM is 12. Scale the first ratio by 4 and the second by 3 so both Bs become 12:

Now the two Bs agree, so they slot together into one ratio:

Step through the picture — the two strips are stretched until their B blocks line up, then joined:

Worked examples

• Shared term already matches. x : y = 3 : 5 and y : z = 5 : 2. Both ys are already 5, so no scaling is needed: x : y : z = 3 : 5 : 2.
• Scale one ratio. P : Q = 1 : 2 and Q : R = 6 : 7. Make Q match: LCM of 2 and 6 is 6, so scale the first by 3: 3 : 6. Then P : Q : R = 3 : 6 : 7.
• Then simplify. A : B = 4 : 6 and B : C = 9 : 12. LCM of 6 and 9 is 18: 12 : 18 and 18 : 24, giving 12 : 18 : 24 — which simplifies (divide by 6) to 2 : 3 : 4.

Why it works

A ratio is really a chain of equal fractions. A : B = 2 : 3 says \tfrac{A}{B} = \tfrac{2}{3}, and B : C = 4 : 5 says \tfrac{B}{C} = \tfrac{4}{5}. Rewriting each with the shared B = 12 makes the middle term literally the same quantity in both fractions, so the three parts can be read off a single scale. Scaling never changes a ratio — it just changes the units you count it in.