To find the principal sum on which the compound interest at a rate of [tex]\(7 \frac{1}{2} \%\)[/tex] per annum for three years, compounded annually, will be ₹3,101.40, we need to follow these steps:
1. Understand the Problem:
- We know the compound interest (CI) is ₹3,101.40.
- The rate of interest (R) is [tex]\(7.5\%\)[/tex] per annum.
- The time period (T) is 3 years.
- We need to find the principal (P).
2. Recall the Formula for Compound Interest:
The formula for the amount [tex]\(A\)[/tex] with compound interest is given by:
[tex]\[
A = P \left(1 + \frac{R}{100}\right)^T
\][/tex]
where:
- [tex]\(A\)[/tex] is the amount after time [tex]\(T\)[/tex]
- [tex]\(P\)[/tex] is the principal amount
- [tex]\(R\)[/tex] is the annual rate of interest
- [tex]\(T\)[/tex] is the time period in years
3. Relate Compound Interest to Amount:
We know the compound interest [tex]\(CI\)[/tex] can be defined as:
[tex]\[
CI = A - P
\][/tex]
Therefore, the amount [tex]\(A\)[/tex] can be written as:
[tex]\[
A = P + CI
\][/tex]
4. Substitute the Values:
Given [tex]\(CI = ₹3,101.40\)[/tex] and [tex]\(R = 7.5\%\)[/tex], we need to solve for [tex]\(P\)[/tex].
Using the formula for [tex]\(A\)[/tex]:
[tex]\[
A = P \left(1 + \frac{7.5}{100}\right)^3
\][/tex]
We know that:
[tex]\[
A = P + CI
\][/tex]
So:
[tex]\[
P + 3101.40 = P \left(1 + \frac{7.5}{100}\right)^3
\][/tex]
5. Solve for Principal P:
Rearrange the equation to solve for [tex]\(P\)[/tex]:
[tex]\[
P + 3101.40 = P \left(1.075\right)^3
\][/tex]
[tex]\[
P + 3101.40 = P \times 1.242
\][/tex]
Divide both sides by the compounded factor:
[tex]\[
P = \frac{3101.40}{1.242 - 1}
\][/tex]
Performing the division, we get:
[tex]\[
P = \frac{3101.40}{0.242}
\][/tex]
Simplify:
[tex]\[
P = ₹12,800.00
\][/tex]
6. Calculate the Total Amount (A):
Using [tex]\(P = ₹12,800\)[/tex], calculate [tex]\(A\)[/tex]:
[tex]\[
A = P \left(1.075\right)^3
\][/tex]
Similarly, from the equations:
[tex]\[
A = P + CI
\][/tex]
[tex]\[
A = 12,800 + 3101.40 = ₹15,901.40
\][/tex]
Therefore, the principal sum on which the compound interest at [tex]\(7 \frac{1}{2} \%\)[/tex] per annum for three years compounded annually is ₹3,101.40 is ₹12,800. The amount after 3 years will be ₹15,901.40.
