Introduction
This problem is related to compound interest, a crucial concept in business mathematics and banking. Krishna deposits a sum of money which earns compound interest annually. We’re given the amount after 2 years and 3 years. Using this data, we need to calculate two things:
- The principal amount (initial deposit)
- The annual rate of interest
Compound Interest Formula
The general formula for compound interest is:
A = P(1 + r/100)t
Where:
- A = amount at the end of time t
- P = principal or initial deposit
- r = annual rate of interest (%)
- t = number of years
Step 1: Use Data from the Problem
We are given:
- A2 = Rs. 5000 at t = 2 years
- A3 = Rs. 5200 at t = 3 years
We use the compound interest formula to write two equations:
At t = 2 years:
5000 = P(1 + r/100)² — (1)
At t = 3 years:
5200 = P(1 + r/100)³ — (2)
Step 2: Divide Equation (2) by Equation (1)
5200 / 5000 = [(1 + r/100)³] / [(1 + r/100)²]
1.04 = 1 + r/100
⇒ r/100 = 0.04
⇒ r = 4%
So, the rate of interest is 4% per annum.
Step 3: Find the Principal (P)
Now that we know r = 4%, substitute in equation (1):
5000 = P(1 + 0.04)² = P(1.04)²
5000 = P × 1.0816
P = 5000 / 1.0816 ≈ Rs. 4621.14
So, Krishna deposited approximately Rs. 4621.14.
Verification
Let’s verify using the formula:
- Year 2 amount: 4621.14 × (1.04)² = 4621.14 × 1.0816 ≈ Rs. 5000 ✅
- Year 3 amount: 4621.14 × (1.04)³ = 4621.14 × 1.124864 ≈ Rs. 5200 ✅
Conclusion
In compound interest problems, knowing the amount at two different times helps us backtrack to the initial deposit and interest rate. In this case:
- Principal (Initial Deposit) = Rs. 4621.14
- Annual Interest Rate = 4%
These kinds of problems are frequently encountered in finance, investment planning, and banking, making them essential for students to understand in business mathematics and statistics.