Rakesh says to Mahesh, “I am as old as you were when I was one-third as old as you are.” If the sum of their ages is 60 years, find the present age of Mahesh (in years).
This type of age problem is best solved by translating each statement into an algebraic equation.
- Let Rakesh's present age be R and Mahesh's present age be M.
- From the second sentence ("the sum of their ages is 60"):
R + M = 60 --- (Equation 1)
- The first sentence ("I am as old as you were when I was one-third as old as you are") compares their ages now to a point in the past. The key principle is that the number of years ago is the same for both people.
(Rakesh's current age - Rakesh's past age) = (Mahesh's current age - Mahesh's past age)
- Substitute the parts of the statement into this structure:
R - (1/3)M = M - R
- Now, simplify this equation to find the relationship between their present ages:
2R = M + (1/3)M
2R = (4/3)M
R = (2/3)M --- (Equation 2)
- Now we have a simple system of two equations. Substitute the second equation into the first one:
(2/3)M + M = 60
- Combine the terms for M:
(5/3)M = 60
- Solve for M (Mahesh's age):
M = 60 × (3/5) = 12 × 3 = 36 years.
- If Mahesh is 36, then Rakesh is 60 - 36 = 24.
- Let's check the statement: "Rakesh (24) is as old as Mahesh was...". Mahesh was 24 years old 12 years ago (36 - 12).
- At that time (12 years ago), Rakesh was 24 - 12 = 12 years old.
- Is 12 "one-third as old as Mahesh is now (36)"? Yes, 12 = 36/3. The solution is correct.
Final Answer: The present age of Mahesh is 36 years.
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