The difference of the present ages of two friends Gopal and Manoj is 2 years. If the ratio of their ages after 18 years from now will be 21 : 22, then find the sum of their respective ages 10 years ago from now (in years).
30
Let Gopal's present age be \(G\) and Manoj's present age be \(M\), with \(M - G = 2\).
After 18 years, \(\dfrac{G + 18}{M + 18} = \dfrac{21}{22}\).
Cross-multiplying: \(22(G + 18) = 21(M + 18)\), so \(22G + 396 = 21M + 378\).
This gives \(22G - 21M = -18\). Substituting \(M = G + 2\): \(22G - 21(G + 2) = -18\).
\(22G - 21G - 42 = -18 \Rightarrow G = 24\), and \(M = 26\).
Ages 10 years ago: \(24 - 10 = 14\) and \(26 - 10 = 16\); their sum is \(14 + 16 = 30\).
Hence, the sum of their ages 10 years ago is 30 years.
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