The average of x numbers is 15. The average of y numbers is 20. The average of all x+y numbers is 18. Find the ratio x:y.

Question

The average of x numbers is 15. The average of y numbers is 20. The average of all x+y numbers is 18. Find the ratio x:y.

1. A. 1:2
2. B. 2:1
3. C. 2:3 (Correct answer)
4. D. 3:2

Correct Answer

Option C — C

Detailed Solution & Explanation

The correct answer is Option C.

Key Points

• Use the weighted-average (alligation) relationship.
• Total of x numbers = 15x; total of y numbers = 20y.
• The combined average is 18:
• (15x + 20y) / (x + y) = 18
• 15x + 20y = 18x + 18y
• 2y = 3x
• x : y = 2 : 3.
• Quick check by alligation: the combined mean 18 is 3 away from 15 and 2 away from 20, giving the ratio of quantities as 2 : 3 (distances taken in cross fashion).

Exam Tip

• alligation gives ratios fast — ratio of the two groups = (distance of other mean from combined) : (distance of this mean from combined).

Additional Information

• Weighted average: (15x + 20y)/(x + y) = 18, which rearranges to 15x + 20y = 18x + 18y, so 2y = 3x and x : y = 2 : 3.
• Alligation gives the same answer in one step: the distances from the mean are 18 − 15 = 3 and 20 − 18 = 2, so the ratio is the inverse, 2 : 3.
• The group nearer the combined average always has the larger weight — a quick sanity check on any alligation answer.