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).
Step-by-step Solution:
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.