Using Ratios ③ Compound Ratios (Renpi)・How to Connect Ratios
A ratio involving three or more quantities is called a “compound ratio.”
For example, when “A:B” and “B:C” are known separately, we learn how to create the compound ratio “A:B:C.” This is called combining ratios.
What You Will Learn
How to connect ratios:
When A:B ＝ 2:3 and B:C ＝ 4:5, the key is to make the numbers for the middle quantity, “B,” the same.
①Consider the LCM (least common multiple) of the numbers for B. The least common multiple of 3 and 4 is 12.
②Adjust A:B so that “B is 12.” Multiply both terms by 4 → A:B ＝ 8:12
③Adjust B:C so that “B is 12.” Multiply both terms by 3 → B:C ＝ 12:15
④Therefore, A:B:C ＝ 8:12:15
Example Problem
When A:B ＝ 3:4 and B:C ＝ 2:5, find A:B:C.
Solution:
①Make the numbers for “B” the same. The least common multiple of 4 and 2 is 4.
②A:B ＝ 3:4 (as is; B is 4)
③Multiply both terms of B:C ＝ 2:5 by 2 → B:C ＝ 4:10 (B has been made 4)
④A:B:C ＝ 3:4:10
When A:B ＝ 3:4 and B:C ＝ 2:5, find A:B:C.
Solution:
①Make the numbers for “B” the same. The least common multiple of 4 and 2 is 4.
②A:B ＝ 3:4 (as is; B is 4)
③Multiply both terms of B:C ＝ 2:5 by 2 → B:C ＝ 4:10 (B has been made 4)
④A:B:C ＝ 3:4:10
Key Points
Remember
- For a compound ratio, the trick is to make the numbers for the “shared quantity (such as B)” the same by using the least common multiple.
- When combining ratios, the rule is to “make the quantities in the shared part the same size.”
- For problems involving three quantities, first express them as a compound ratio, then think of it as a “sharing problem.”
Practice Problems
- When A:B ＝ 2:3 and B:C ＝ 6:5, find A:B:C.
- When A:B ＝ 3:2 and B:C ＝ 4:5, find A:B:C. Also, when A ＋ B ＋ C ＝ 114, find the values of A, B, and C.
- When P:Q ＝ 1:2 and Q:R ＝ 3:4, find P:Q:R. When P ＝ 30, find the values of Q and R.
Answers and Explanations
- Answer:A:B:C = 4:6:5
Explanation: A:B=2:3, B:C=6:5. Make the value of B the same. A:B=2:3 → ×2 → 4:6. B:C=6:5 (as is). Therefore, A:B:C=4:6:5. - Answer:A:B:C = 6:4:5. A=45.6、B=30.4、C=38
Explanation: A:B=3:2, B:C=4:5. Making B 4 gives A:B:C=6:4:5. The sum of the ratio parts is 15, and the total is 114, so 1 unit＝114÷15＝7.6. Therefore, A＝45.6、B＝30.4、C＝38. Check: 45.6＋30.4＋38＝114. - Answer:P:Q:R = 3:6:8. When P=30, Q=60、R=80
Explanation: P:Q=1:2, Q:R=3:4. The LCM of Q's numbers, 2 and 3, is 6. P:Q=1:2→×3→3:6. Q:R=3:4→×2→6:8. Therefore, P:Q:R=3:6:8. When P=30, 1 unit=30÷3=10. Q=10×6=60、R=10×8=80.