Two ratios with one amount in common
Ann, Ben and Cal have saved some money. The ratio of Ann's savings to Ben's is 2 : 3, and the ratio of Ben's savings to Cal's is 4 : 5. Ben's savings are in both ratios: they are 3 parts in the first ratio and 4 parts in the second.
Ben has only one amount of money, so it cannot be 3 parts of one size and also 4 parts of the same size. The parts in the two ratios are different sizes. Before the two ratios can be joined into one, Ben must be the same number of parts in both.
Ben's savings drawn twice, the same length both times. In the first ratio they are 3 parts, and in the second they are 4 parts, so a part is a different size in each ratio.
Make the shared amount match
Rewrite both ratios so that Ben is the same number of parts in each. That number must be a multiple of 3 and a multiple of 4, and the smallest such number is their Lowest Common Multiple (LCM), 12.
To turn Ben's 3 into 12, multiply both numbers in the first ratio by 4: 2 : 3 = 8 : 12. To turn Ben's 4 into 12, multiply both numbers in the second ratio by 3: 4 : 5 = 12 : 15. Each new ratio is equivalent to the one it came from, because both of its numbers were multiplied by the same number.
Cut each of Ben's 3 parts into 4, and each of his 4 parts into 3. Both bars are now 12 small parts, and the small parts are the same size in both.
Read straight through
Now Ben is 12 parts in both ratios, and a part is the same size in both. Ann has 8 of these parts, Ben has 12 and Cal has 15, so the three amounts can be written as one ratio: Ann : Ben : Cal = 8 : 12 : 15.
No number except 1 divides all of 8, 12 and 15, so this three-part ratio is already in its simplest form.
Questions often use letters for the amounts. Written that way, A : B = 2 : 3 and B : C = 4 : 5 give A : B : C = 8 : 12 : 15, with the same steps.
One size of part for all three: Ann has 8 parts, Ben has 12 and Cal has 15.
The usual mistakes
Joining the ratios as they stand. Writing 2 : 3 : 5 puts Ben's 3 parts from the first ratio next to Cal's 5 parts from the second, but those parts are different sizes. Ben must be matched first.
Adding Ben's two numbers. 3 + 4 = 7 is not a multiple of 3 or of 4. Ben's number must be a common multiple of both, and the lowest is 12.
Multiplying only Ben's number. Changing Ben's 3 to 12 but leaving Ann's 2 as it is gives 2 : 12, a different ratio. Every number in a ratio must be multiplied by the same number.
Leaving the second ratio behind. After matching Ben at 12, Cal is not still 5: the second ratio was multiplied by 3, so Cal is 5 × 3 = 15.