Question: 3. कक्षा 12 पढ्दै गरेको विद्यार्थीले रु.10000 वार्षिक चक्रिय ब्याज दरमा 2 वर्षका लागि लगानी गर्छानी गरेको छ । एक वर्षको अन्त्यमा चक्रिय मिश्रधन रु.11200 हुन्छ ।
A student studying in class 12 invests Rs. 10000 for 2 years at an annual compound interest rate. If the compound interest at the end of one year is Rs. 11200.
(क) हरेक वर्षको लागि ब्याजका दर अलग भएमा दुई वर्षको निम्ति चक्रिय ब्याजदर निकाल्ने सूत्र लेख्नुहोस् । 1
Write a formula to calculate the compound interest rate for three years if the interest rate is different every year. (1)
(ख) वार्षिक चक्रिय ब्याज दर पत्ता लगाउनुहोस् । (2)
Find the annual compound interest rate. (2)
(ग) दोस्रो वर्षको अन्त्यमा चक्रिय मिश्रधन कति हुन्छ, पत्ता लगाउनुहोस् । 1
Find the compound interest at the end of the second year. (2)
1. **Problem Statement:** A student invests Rs. 10000 for 2 years at an annual compound interest rate. The compound amount at the end of 1 year is Rs. 11200. We need to find:
(a) The formula for compound interest rate if rates differ each year.
(b) The annual compound interest rate.
(c) The compound amount at the end of the second year.
2. **Formula for compound interest with different rates each year:**
If the interest rates for the first, second, and third years are $r_1$, $r_2$, and $r_3$ respectively (expressed as decimals), then the compound amount after 3 years is:
$$ A = P(1 + r_1)(1 + r_2)(1 + r_3) $$
where $P$ is the principal amount.
3. **Finding the annual compound interest rate:**
Given:
$$ P = 10000, \quad A_1 = 11200 $$
The amount after 1 year is:
$$ A_1 = P(1 + r) $$
Substitute values:
$$ 11200 = 10000(1 + r) $$
Divide both sides by 10000:
$$ \frac{11200}{10000} = \cancel{\frac{10000}{10000}}(1 + r) $$
$$ 1.12 = 1 + r $$
Solve for $r$:
$$ r = 1.12 - 1 = 0.12 $$
So, the annual compound interest rate is 12%.
4. **Finding the compound amount at the end of the second year:**
Using the formula:
$$ A_2 = P(1 + r)^2 $$
Substitute $P = 10000$ and $r = 0.12$:
$$ A_2 = 10000(1.12)^2 $$
Calculate:
$$ A_2 = 10000 \times 1.2544 = 12544 $$
So, the compound amount at the end of the second year is Rs. 12544.