Multiple & Inverse Ratios

🧠

Quick recall

• A common middle term must match in both chained ratios.
• Inverse ratio reverses the terms; inverse proportion keeps the product constant.

On this page

See Ratio & Proportion for direct/inverse proportion, partnership, and age ratios. This page focuses on chaining and inverting ratios.

Core Rules

• Combining two ratios (multiple ratio): if $A:B=2:3$ and $B:C=4:5$, make B equal in both — $A:B=8:12$, $B:C=12:15$ → $A:B:C=8:12:15$
• Inverse ratio: the ratio of reciprocals — inverse of $a:b$ is $\dfrac{1}{a}:\dfrac{1}{b}=b:a$
• Used in inverse proportion problems: if speed doubles, time halves

Worked Examples

Example 1

Question: Find the inverse ratio of 3:5.

Step 1: Take reciprocals.

$$\frac{1}{3}:\frac{1}{5}$$

Step 2: Multiply both terms by 15.

$$5:3$$

Answer: 5:3.

Example 2

Question: A:B = 3:4 and B:C = 2:5. Find A:B:C.

Step 1: Match B at 4.

$$A:B=3:4,\qquad B:C=4:10$$

Step 2: Join the ratios.

$$A:B:C=3:4:10$$

Answer: 3:4:10, equivalently 6:8:20.

Example 3

Question: Six workers finish a job in 10 days. How many are needed for 5 days?

Step 1: Keep worker-days constant.

$$6\times10=M\times5$$

Step 2: Solve for the number of workers.

$$M=\frac{60}{5}=12$$

Answer: 12 workers.

Drill

Try each question before opening its answer.

Question 1

Find inverse ratio of 7:9.

• Show answer

  9:7

Question 2

If A:B=2:3, B:C=3:4, find A:B:C.

• Show answer

  2:3:4

Question 3

8 workers finish a job in 15 days. How many workers for 10 days?

• Show answer

  12 workers

Question 4

If A:B=5:6 and B:C=2:3, find A:B:C.

• Show answer

  10:12:18 = 5:6:9

Practice tracker

YYYY-MM-DD

• Attempted/Correct:
• Mistake pattern: