A man’s present age is two and a half times the sum of the ages of his two…
2024
A man’s present age is two and a half times the sum of the ages of his two daughters. 30 years from now, his age will equal the sum of the ages of his two daughters. What will the man’s age be 12 years from now?
- A.
62 years
- B.
62 months
- C.
2 years
- D.
60 years
Attempted by 372 students.
Show answer & explanation
Correct answer: A
Concept
Age problems are linear-equation problems. Pick a variable for each unknown present age, then translate every spoken relationship (“is 2½ times”, “30 years from now”) into an equation. A key idea for a GROUP of people: when N people each grow older by k years, the SUM of their ages grows by N×k, not by k.
Application
Let S = sum of the two daughters’ present ages and M = man’s present age.
“Present age is two and a half times the sum of the daughters’ ages” gives M = 2.5S.
30 years later the man’s age is M + 30. Each of the 2 daughters gains 30 years, so their sum gains 2×30 = 60, becoming S + 60.
“His age will equal the sum of the daughters’ ages” gives M + 30 = S + 60.
Substitute M = 2.5S: 2.5S + 30 = S + 60, so 1.5S = 30 and S = 20.
Therefore M = 2.5 × 20 = 50 (present age). In 12 years the man will be 50 + 12 = 62 years.
Cross-check
With S = 20 and M = 50: now M = 2.5×20 = 50 ✓. After 30 years: man = 80, daughters’ sum = 20 + 60 = 80 ✓. The two conditions hold, so the present age 50 is correct and the age in 12 years is 62 years.