Introduction / Context:
This question is similar to a previous multi-deposit compound interest problem, but with a different rate. It tests whether you can correctly handle deposits made at different times while interest is compounded half-yearly. Each deposit experiences a different number of compounding periods within the year, and you must account for this when computing the total interest.
Given Data / Assumptions:
- Rate of compound interest = 5% per half year.
- First deposit = Rs 4800 on 1st January.
- Second deposit = Rs 4800 on 1st July.
- Interest is compounded half-yearly.
- We need the total interest by 31st December of the same year.
Concept / Approach:
The first deposit remains invested for two half-year periods, while the second deposit remains invested for one half-year period. We compute the final amount for each deposit using A = P * (1 + r/100)^n, where r is the half-yearly rate and n is the number of half-yearly periods. Subtracting the original principal from each final amount gives the interest. Then we add all the interest amounts to obtain the total interest.
Step-by-Step Solution:
Step 1: For the 1st deposit of Rs 4800:
Half-yearly rate r = 5%, number of periods n1 = 2.
Step 2: Amount after 2 periods:
A1 = 4800 * (1.05)^2 = 4800 * 1.1025 = Rs 5292.
Step 3: Interest from 1st deposit:
I1 = 5292 - 4800 = Rs 492.
Step 4: For the 2nd deposit of Rs 4800:
Number of periods n2 = 1, same rate 5%.
Step 5: Amount after 1 period:
A2 = 4800 * 1.05 = Rs 5040.
Step 6: Interest from 2nd deposit:
I2 = 5040 - 4800 = Rs 240.
Step 7: Total interest earned:
I_total = I1 + I2 = 492 + 240 = Rs 732.
Verification / Alternative check:
Total final amount = 5292 + 5040 = 10332.
Total principal deposited = 4800 + 4800 = 9600.
Interest = 10332 - 9600 = 732, which confirms the calculation.
Why Other Options Are Wrong:
Rs 1464 is exactly double the correct interest, as if the interest had been miscounted for four half-year periods on both deposits.
Rs 366 and Rs 183 represent partial contributions from just one of the deposits and ignore the full investment period or second deposit.
Common Pitfalls:
Often, students forget that only the first deposit earns interest for two half-yearly periods, while the second earns for just one. Treating both deposits as earning the same number of periods or transforming the half-yearly rate into a rough annual rate without care can lead to wrong answers.
Final Answer:
The total interest earned by the customer at the end of the year is Rs 732.
Discussion & Comments