A certain sum has been invested at r% interest compounded annually. The interest earned in the 3rd and the 4th years are Rs. 47952 and Rs. 53946 respectively. Find the principal amount (in Rs.) invested at the beginning of the first year.
Question
A certain sum has been invested at r% interest compounded annually. The interest earned in the 3rd and the 4th years are Rs. 47952 and Rs. 53946 respectively. Find the principal amount (in Rs.) invested at the beginning of the first year.
Solution 1
The problem involves the concept of compound interest. The interest for each year is calculated on the initial principal and the accumulated interest of previous years.
Given that the interest of the 3rd year is Rs. 47952 and the interest of the 4th year is Rs. 53946, we can say that the difference in interest between the 4th and 3rd year is due to the interest rate applied to the 3rd year's interest.
So, the interest rate (r) can be calculated as follows:
r = (Interest of 4th year - Interest of 3rd year) / Interest of 3rd year r = (53946 - 47952) / 47952 r = 5994 / 47952 = 0.125 or 12.5%
Now, we know that the interest of the 3rd year (Rs. 47952) is 12.5% of the principal amount. So, we can calculate the principal amount (P) as follows:
P = Interest of 3rd year / r P = 47952 / 0.125 P = Rs. 383616
So, the principal amount invested at the beginning of the first year is Rs. 383616.
Solution 2
The problem involves the concept of compound interest. The interest for the 3rd year is Rs. 47952 and for the 4th year is Rs. 53946. The difference in the interest of the 3rd and 4th year will give us the interest on the interest of the 2nd year.
Step 1: Find the difference in the interest of the 3rd and 4th year. Interest of 4th year - Interest of 3rd year = Rs. 53946 - Rs. 47952 = Rs. 5994
Step 2: This difference is the interest on the interest of the 2nd year, which is the same as the interest of the 3rd year. So, we can say that Rs. 5994 is r% of Rs. 47952.
Step 3: Find the rate of interest (r). 5994 = (r/100) * 47952 Solving this equation, we get r = (5994 * 100) / 47952 = 12.5%
Step 4: Now, we know that the interest of the 3rd year (Rs. 47952) is 12.5% of the principal amount. So, we can find the principal amount (P) using the formula:
P = (Interest of 3rd year * 100) / rate of interest P = (47952 * 100) / 12.5 P = Rs. 383616
So, the principal amount invested at the beginning of the first year is Rs. 383616.
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