Ten years ago, a mother was four times as old as her daughter. Ten years from now, the mother will be twice as old as the daughter. What is the daughter's present age?
Verbal Reasoning
Problems on Ages
Difficulty: Medium
Choose an option
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A5 years
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B10 years
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C20 years
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D30 years
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ENone of these
Answer
Correct Answer: 20 years
Explanation
Introduction / Context:Age problems with “times as old” at two different times usually form a pair of linear equations in present ages. Solving gives the required current ages.
Given Data / Assumptions:
- 10 years ago: Mother = 4 × Daughter
- In 10 years: Mother = 2 × Daughter
Concept / Approach:Let present ages be M (mother) and D (daughter). Translate each time-based relation to an equation and solve simultaneously.
Step-by-Step Solution:
M − 10 = 4(D − 10) … (1)M + 10 = 2(D + 10) … (2)From (2): M = 2D + 20 − 10 = 2D + 10Plug into (1): 2D + 10 − 10 = 4D − 40 ⇒ 2D = 4D − 4040 = 2D ⇒ D = 20Then M = 2(20) + 10 = 50Verification / Alternative check:
10 years ago: M = 40, D = 10 ⇒ 40 = 4×10 ✓In 10 years: M = 60, D = 30 ⇒ 60 = 2×30 ✓Why Other Options Are Wrong:
- 5, 10, 30 do not satisfy both time-based multiplicative conditions.
- None of these: 20 works.
Common Pitfalls:Using 4 times and 2 times on present ages instead of the specified past/future ages; or making arithmetic slips when substituting between equations.
Final Answer:20 years