Combining ratios via a shared quantity drills

Combine two ratios or relationships that share a common quantity (a:b and b:c) into a single ratio, or use the combined relationship to split a total.

Free, timed SHSAT practice sets on combining ratios via a shared quantity— every question includes a full worked explanation, and there's no account required.

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How to combine two ratios

When two ratios share a quantity (a:b and b:c), make the shared quantity the same number in both ratios. Then join them.

  1. Find the least common multiple (LCM) of the two numbers for the shared quantity.
  2. Scale each ratio so the shared quantity equals that LCM.
  3. Write the three numbers as a:b:c.

Example: cats to dogs is \(3:4\) and dogs to birds is \(6:5\). Dogs are 4 and 6, and \(\text{LCM}(4, 6) = 12\). So \(3:4 = 9:12\) and \(6:5 = 12:10\). Cats to dogs to birds is \(9:12:10\).

To split a total, use the combined ratio. If there are 62 animals: \(9 + 12 + 10 = 31\) parts, one part is \(62 \div 31 = 2\), so there are \(9 \times 2 = 18\) cats.

Common trap: joining the raw numbers as \(3:4:5\). That is wrong, because the dogs are 4 in one ratio and 6 in the other.

Worked examples

Example 1.
In a group of students, the ratio of first graders to second graders is 6:5, and the ratio of second graders to third graders is 4:3. What is the combined ratio of first graders to second graders to third graders, in simplest form?
Show answer and explanation
Answer:
D

The shared term is second graders: scale 6:5 by 4 and 4:3 by 5 so both give second graders \(= 20\) (the LCM of 5 and 4). This gives 24:20:15, which is already in lowest terms. (Joining the original numbers without matching second graders gives 6:5:3 or 6:4:3. Writing 20 for second graders but swapping the factors on the outer terms (\(6 \times 5\) and \(3 \times 4\)) gives 30:20:12, which reduces to 15:10:6.)

Example 2.
In a group of 105 cookies, the ratio of chocolate cookies to oatmeal cookies is 2:3, and the ratio of oatmeal cookies to sugar cookies is 4:5. How many chocolate cookies are there?
Show answer and explanation
Answer:
D

Combining via the shared oatmeal cookies term gives chocolate cookies:oatmeal cookies:sugar cookies \(= 8:12:15\), so chocolate cookies \(= 105 \times 8 \div 35 = 24\). (Splitting with the unmatched numbers 2:3:5 gives 21. Finding the oatmeal cookies instead gives 36. Using only 2:3 and ignoring the sugar cookies, \(105 \times 2 \div 5\), gives 42.)