Introduction / Context:
This problem uses a present-time multiplicative relation and a future-time equality relation involving the same three people. Translating both conditions with a single unknown for the children’s combined age simplifies the system drastically.
Given Data / Assumptions:
- Let combined present age of the two children be C; father's present age F.
- F = 3C.
- In 20 years: F + 20 = (C + 20) + (C + 20) = 2C + 40.
Concept / Approach:
Substitute F = 3C into the future condition and solve for C, then multiply for F. This avoids dealing with each child separately, since only their sum matters.
Step-by-Step Solution:
3C + 20 = 2C + 40C = 20F = 3C = 60 (But this is future or present?) Wait, re-evaluate: 3C + 20 = 2C + 40 ⇒ C = 20 ⇒ F = 3 * 20 = 60? This contradicts the options. We misinterpreted the question. Correct reading: “A father’s age is three times the sum of his two children but 20 years hence his age will be equal to the sum of their ages.” Thus, F = 3C now and (F + 20) = (C + 20). However, since there are two children, the sum then is (C + 40). Let’s correct it precisely.Correct future condition: F + 20 = (C + 20) + (C + 20) = 2C + 40Using F = 3C ⇒ 3C + 20 = 2C + 40 ⇒ C = 20 ⇒ F = 60But that yields 60, not in options. The standard form of this question actually implies “20 years hence his age will be equal to the sum of their ages now.” If we instead interpret the line as: F = 3C and F + 20 = C, then 3C + 20 = C ⇒ 2C = −20 (impossible). Therefore, the canonical textbook version has the present father’s age equal to the sum of children’s ages 20 years hence? No.Given the provided answer set and standard known result, the intended reading is: “Father's age is three times the sum of his children now, but 20 years hence, his age will equal the sum of their ages then.” Solving this correctly gives C = 10 and F = 30:Let the individual children be c1, c2 with sum C. In 20 years: father F + 20; kids (c1 + 20) + (c2 + 20) = C + 40.Equations: F = 3C and F + 20 = C + 40 ⇒ Subtract these: (F + 20) − F = (C + 40) − 3C ⇒ 20 = 40 − 2C ⇒ 2C = 20 ⇒ C = 10 ⇒ F = 30
Verification / Alternative check:
Now: F = 30, C = 10 ⇒ F = 3C (true). In 20 years: father 50; children combined 10 + 40 = 50 (equal).
Why Other Options Are Wrong:
40/35/45 fail the pair of relations when checked carefully.
Common Pitfalls:
Confusing “sum of two children” vs “each child” in the future expression; the combined future is C + 40.
Final Answer:
30 years
Discussion & Comments