In a family, the combined age of husband, wife, and a son is 90 years. The husband is ten older. The difference in the age of the husband and wife is 1/3rd of the age of their son. After 10 years, the age of the son would be half the age of his father. What is the present age of his mother (in years)?
35
Note: The phrase "the husband is ten older" is incomplete in the question; the remaining three conditions are sufficient to solve uniquely.
Let husband = \(H\), wife = \(W\), son = \(S\).
Equations:
(i) \(H + W + S = 90\)
(ii) \(H - W = \dfrac{S}{3}\)
(iii) \(S + 10 = \dfrac{H + 10}{2} \Rightarrow H = 2S + 10\)
Substitute: From (ii), \(W = H - \dfrac{S}{3} = 2S + 10 - \dfrac{S}{3}\).
Plug into (i): \((2S+10) + \left(2S+10-\dfrac{S}{3}\right) + S = 90\)
\(\Rightarrow 5S - \dfrac{S}{3} + 20 = 90 \Rightarrow \dfrac{14S}{3} = 70 \Rightarrow S = 15\)
Hence \(H = 40\) and \(W = 40 - 5 = 35\).
The correct answer is 35 years.
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