A sum put out at 4% compound interest payable half yearly amounts to Rs 13265.10 in 11/2 years. Find the sum.

To find the original sum of money that, when invested at a 4% compound interest rate payable half-yearly, amounts to Rs 13265.10 after 1.5 years, we can break down the problem step by step.

Understanding Compound Interest

Compound interest means that the interest earned on an investment is added back to the principal amount, so in the next period, interest is earned on the new total. In this case, the interest is compounded half-yearly, which means it is calculated every six months. Given that the interest rate is 4% per annum, we need to adjust this rate for half-yearly compounding:

• Annual interest rate = 4%
• Half-yearly interest rate = 4% / 2 = 2%

Finding the Number of Compounding Periods

Next, we determine how many compounding periods there are in 1.5 years:

• 1.5 years = 1.5 * 2 = 3 compounding periods (since interest is compounded twice a year)

Using the Compound Interest Formula

The formula for compound interest is:

A = P(1 + r/n)^(nt)

Where:

• A = the amount of money accumulated after n years, including interest.
• P = the principal amount (the initial sum of money).
• r = the annual interest rate (decimal).
• n = the number of times that interest is compounded per year.
• t = the time the money is invested for in years.

In this case:

• A = Rs 13265.10
• r = 0.04
• n = 2
• t = 1.5

Plugging in the Values

Now, substituting these values into the formula, we get:

13265.10 = P(1 + 0.04/2)^(2 * 1.5)

This simplifies to:

13265.10 = P(1 + 0.02)^3

Calculating (1 + 0.02)^3:

(1.02)^3 = 1.061208

Now, substituting this back into the equation:

13265.10 = P * 1.061208

Solving for P

To find the principal amount P, we rearrange the equation:

P = 13265.10 / 1.061208

Calculating this gives:

P ≈ 12499.99

Therefore, we can conclude that the original sum of money invested is approximately Rs 12500.

Summary

In summary, by breaking down the calculations step by step and applying the compound interest formula, we determined that the initial investment was roughly Rs 12500. Understanding how to manipulate the formula and account for the compounding periods is crucial for solving these types of problems effectively.

The formula for annual compound interest, including principal sum, is:
A = P (1 + r/n) ^(nt)

Where:

A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = the annual interest rate (decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for

Now according to your question

A = 13265.10 Rs.

t = 11/2 year

r = 4%

n = 2 (as, half yearly)

Therefore, 13265.10=P(1+4/(2×100))^2×11/2

Or, 13265.1= P(51/50)¹¹

Or, P=13265.1(50/51)¹¹

Or, P = 10668.62 (approx.)

Therefore the sum of the amount was Rs. 10668.62 (approx.)