Target-Equality under CI — Split ₹ 3903 so that A after 7 years equals B after 9 years (4% p.a.): Divide ₹ 3903 between A and B such that A's amount after 7 years at 4% compound equals B's amount after 9 years at the same rate. Find the two shares.
Aptitude
Compound Interest
Difficulty: Medium
Choose an option
Answer
Correct Answer: Rs. 2028, Rs. 1875
Explanation
Introduction / Context:When two parts of a sum will be left to grow for different durations at the same compound rate, equalizing their future amounts defines a proportional split today. This is a typical present-value balancing problem under compound interest.
Given Data / Assumptions:
- Total sum S = ₹ 3903
- Annual rate = 4% = 0.04
- A grows for 7 years; B grows for 9 years
- Condition: A's future amount = B's future amount
Concept / Approach:Let A's share be x; B's share is 3903 − x. Equality at maturity: x*(1.04)^7 = (3903 − x)*(1.04)^9. Divide both sides by (1.04)^7 to solve for x.
Step-by-Step Solution:
x = (3903 − x)*(1.04)^2 = (3903 − x)*1.0816.x + 1.0816x = 3903 * 1.0816 ⇒ 2.0816x = 4221.4848.x = 4221.4848 / 2.0816 = ₹ 2028 ⇒ B = 3903 − 2028 = ₹ 1875.Verification / Alternative check:
Forward check: 2028*(1.04)^7 equals 1875*(1.04)^9 (same value).Why Other Options Are Wrong:
- All other splits do not satisfy x*(1.04)^7 = (3903 − x)*(1.04)^9.
Common Pitfalls:
- Forgetting to divide out common growth before solving; mixing SI and CI.
Final Answer:Rs. 2028 for A and Rs. 1875 for B.