How to Combine Ratios
Combining ratios joins two ratios into one three-part ratio by matching the number they share. If a:b = 2:3 and b:c = 3:5, the shared 3 gives a:b:c = 2:3:5. If the shared numbers differ, scale them to their LCM first.
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Same Middle Number
- If the middle number is already the same, combine the ratios directly.
- Example: if and , then .
Different Middle Numbers
- If the middle numbers are different, make them the same first using the LCM.
- Example: and → make both have .
Writing the Combined Ratio
- Once the middle numbers match, write all three parts together as one ratio.
- Example: and combine to give .
What the Combined Ratio Means
- A combined ratio shows how three quantities compare at the same time.
- Example: means for every 24 of the first amount, there are 18 of the second and 45 of the third.
Practice Questions
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If and , what is ?
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is 3 in both ratios, so you can directly combine to get .
If and , what is ?
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Since is 7 in both ratios, you can directly combine them to get .
If and , what is ?
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To combine the ratios, first make the values equal by finding the LCM of 5 and 2, which is 10. Then scale each ratio: becomes , and becomes . So the combined ratio is .
In a bag, the ratio of blue balls to green balls is . The ratio of red balls to green balls is . What is the combined ratio of red:green:blue?
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Since green appears in both ratios, match that value first. The LCM of 4 and 2 is 4. Scale red:green to , and blue:green stays the same. Be careful with the order: red:green:blue = .
If and , what is ?
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To combine the ratios, match the value of since it appears in both ratios. The LCM of 6 and 4 is 12. Scale to and to . So the combined ratio is .
In a fruit blend, the ratio of apples to bananas is . The ratio of bananas to grapes is . If you used 25 apples, how many grapes are in the mix?
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Bananas appear in both ratios, so link them to make one full ratio: apples:bananas:grapes = . Since 25 apples represent the 5 parts for apples, divide 25 by 5 to find that one part equals 5. Grapes are 2 parts, so grapes.
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Interactive Activity
Combine two ratios into one using LCM
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To combine two ratios, first find the quantity they share. Make that shared quantity the same in both ratios by scaling, using the lowest common multiple when the values differ. Once the shared quantity matches, put the parts together to form a single three-part ratio.
If the shared quantity is already the same in both ratios, you do not need to scale anything. Simply put the numbers together to form the three-part ratio. For example, with a:b = 2:5 and b:c = 5:3, b is 5 in both, giving a:b:c = 2:5:3.
You make the shared quantity equal so the two ratios line up correctly. The shared quantity is the bridge between them, so it must describe the same amount in each ratio. Once it matches, the parts from both ratios fit together accurately into one three-part ratio.
Find the lowest common multiple of the two shared values. This is the smallest number both can divide into, so each ratio scales up to whole numbers. For example, the lowest common multiple of 5 and 2 is 10, so you make the shared quantity 10 in both ratios.
No. Multiplying every part of a ratio by the same number does not change the proportion it represents. The numbers get bigger, but the relationship stays the same. Scaling only makes the shared quantity match so the two ratios can be combined into one.
First find the value of one part by dividing a known amount by its number of parts. Then multiply that value by the parts you need. For example, if 8 parts of water is 24 buckets, one part is 3 buckets, so 15 parts of sand is 45 buckets.