Anil borrows Rs 2 lakhs at an interest rate of 8% per annum, compounded half-yearly. He repays Rs 10320 at the end of the first year and closes the loan by paying the outstanding amount at the end of the third year. Then, the total interest, in rupees, paid over the three years is nearest to
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Official Correct Answer: C. First, the problem says Anil borrows Rs 2 lakhs at an 8% annual interest rate, compounded half-yearly. He repays Rs 10,320 at the end of the first year and then closes the loan by paying the remaining amount at the end of the third year. We need to find the total interest he paid over the three years, and the options are given.
Okay, so compound interest is calculated using the formula:
$$
A = P \left(1 + \frac{r}{n}\right)^{nt}
$$
where:
- \( A \) is the amount after time \( t \),
- \( P \) is the principal amount,
- \( r \) is the annual interest rate (in decimal),
- \( n \) is the number of times interest is compounded per year,
- \( t \) is the time in years.
In this case, the interest is compounded half-yearly, so \( n = 2 \). The rate is 8%, so \( r = 0.08 \). The principal \( P \) is Rs 2,00,000.
Anil makes a repayment of Rs 10,320 at the end of the first year. Then, he pays the remaining amount at the end of the third year. So, we need to calculate the amount owed after the first year, subtract the repayment, and then calculate the amount owed after the next two years. Finally, we'll find the total interest paid by subtracting the original principal.
Let me break it down step by step.
**Step 1: Calculate the amount after the first year.**
Using the compound interest formula for the first year (\( t = 1 \)):
$$
A_1 = 2,00,000 \left(1 + \frac{0.08}{2}\right)^{2 \times 1}
$$
Simplify the rate per period:
$$
\frac{0.08}{2} = 0.04
$$
So,
$$
A_1 = 2,00,000 (1 + 0.04)^2 = 2,00,000 (1.04)^2
$$
Calculating \( (1.04)^2 \):
$$
1.04 \times 1.04 = 1.0816
$$
Therefore,
$$
A_1 = 2,00,000 \times 1.0816 = 2,16,320
$$
So, after the first year, Anil owes Rs 2,16,320.
**Step 2: Subtract the repayment made at the end of the first year.**
Anil repays Rs 10,320, so the remaining principal is:
$$
A_2 = 2,16,320 - 10,320 = 2,06,000
$$
So, now the principal for the next two years is Rs 2,06,000.
**Step 3: Calculate the amount after the next two years.**
We need to calculate the amount after 2 more years (from year 1 to year 3) on the principal of Rs 2,06,000. Using the compound interest formula again for \( t = 2 \):
$$
A_3 = 2,06,000 \left(1 + \frac{0.08}{2}\right)^{2 \times 2}
$$
Simplify the rate per period again:
$$
\frac{0.08}{2} = 0.04
$$
So,
$$
A_3 = 2,06,000 (1 + 0.04)^4
$$
Calculating \( (1.04)^4 \). Hmm, \( (1.04)^2 = 1.0816 \), so \( (1.0816)^2 \) would be \( (1.04)^4 \).
Calculating \( 1.0816 \times 1.0816 \):
Compute this step by step:
1.0816 × 1.0816:
First, multiply 1.08 × 1.08 = 1.1664.
Then, 0.0016 × 1.0816 = approximately 0.00173056.
Adding these together: 1.1664 + 0.00173056 ≈ 1.16813056.
Wait, that doesn't seem right. Maybe I should compute it more accurately.
Alternatively, use the formula \( (1 + r)^2 = 1 + 2r + r^2 \) for \( r = 0.0816 \).
Wait, no, that's for squaring a binomial. Actually, \( (1.0816)^2 \) is:
1.0816 × 1.0816:
Compute it as:
= (1 + 0.08 + 0.0016) × (1 + 0.08 + 0.0016)
But that might complicate. Alternatively, just multiply 1.0816 × 1.0816:
1.0816 × 1.0816:
First, multiply 1.0816 × 1 = 1.0816
Then, 1.0816 × 0.08 = 0.086528
Then, 1.0816 × 0.0016 = 0.00173056
Adding them together:
1.0816 + 0.086528 = 1.168128
1.168128 + 0.00173056 ≈ 1.16985856
So, approximately 1.16985856.
Therefore,
$$
A_3 = 2,06,000 \times 1.16985856
$$
Calculating this:
First, 2,00,000 × 1.16985856 = 2,33,971.712
Then, 6,000 × 1.16985