Question

Eight years ago, Rajesh was half as old as Shiv If the ratio of their ages after 4 years becomes 3:4, find the present age of Rajesh.

• A. 14 years
• B. 18 years
• C. 24 years
• D. 20 years
• E. 12 years

Answer 1

The Correct Option is A

Step 1: Understand the problem.
We are given two pieces of information:
- Eight years ago, Rajesh was half as old as Shiv.
- The ratio of their ages after 4 years becomes 3:4.
We are asked to find Rajesh's present age.

Step 2: Define variables for their current ages.
Let Rajesh's present age be \( R \) years and Shiv's present age be \( S \) years.

Step 3: Translate the first piece of information into an equation.
Eight years ago, Rajesh's age was \( R - 8 \) and Shiv's age was \( S - 8 \). According to the problem, Rajesh was half as old as Shiv:
\( R - 8 = \frac{1}{2} (S - 8) \)
Multiply both sides by 2 to eliminate the fraction:
\( 2(R - 8) = S - 8 \)
\( 2R - 16 = S - 8 \)
\( 2R - S = 8 \) ---- (1)

Step 4: Translate the second piece of information into an equation.
The ratio of their ages after 4 years is given as 3:4:
\( \frac{R + 4}{S + 4} = \frac{3}{4} \)
Cross-multiply to get rid of the fraction:
\( 4(R + 4) = 3(S + 4) \)
\( 4R + 16 = 3S + 12 \)
\( 4R - 3S = -4 \) ---- (2)

Step 5: Solve the system of equations.
We now have two equations:
- \( 2R - S = 8 \) ---- (1)
- \( 4R - 3S = -4 \) ---- (2)
First, multiply equation (1) by 3 to align the coefficients of \( S \):
\( 6R - 3S = 24 \) ---- (3)
Now subtract equation (2) from equation (3):
\( (6R - 3S) - (4R - 3S) = 24 - (-4) \)
\( 2R = 28 \)
\( R = 14 \)

Step 6: Conclusion.
Rajesh's present age is 14 years.

Final Answer:
The correct option is (A): 14 years.