To determine the rate of compound interest per year that results in a compound interest of Rs. 42 on a principal amount of Rs. 200 over a period of 2 years, follow these steps:
1. Identify the given values:
- Principal amount ([tex]\(P\)[/tex]) = Rs. 200
- Compound Interest ([tex]\(CI\)[/tex]) = Rs. 42
- Time period ([tex]\(t\)[/tex]) = 2 years
2. Calculate the total amount [tex]\(A\)[/tex] after 2 years:
The total amount [tex]\(A\)[/tex] is the sum of the principal amount and the compound interest.
[tex]\[
A = P + CI = 200 + 42 = 242 \text{ Rs}
\][/tex]
3. Use the compound interest formula to relate the principal amount and the total amount:
The general formula for compound interest is:
[tex]\[
A = P(1 + r/n)^{nt}
\][/tex]
where:
- [tex]\(A\)[/tex] is the total amount after interest
- [tex]\(P\)[/tex] is the principal amount
- [tex]\(r\)[/tex] is the annual interest rate (as a decimal)
- [tex]\(n\)[/tex] is the number of times interest is compounded per year
- [tex]\(t\)[/tex] is the time period in years
In this problem, the interest is compounded once a year ([tex]\(n = 1\)[/tex]), so the formula simplifies to:
[tex]\[
A = P(1 + r)^t
\][/tex]
4. Insert the known values into the simplified formula:
[tex]\[
242 = 200(1 + r)^2
\][/tex]
5. Solve for the annual interest rate [tex]\(r\)[/tex]:
First, divide both sides by 200 to isolate the term involving [tex]\(r\)[/tex]:
[tex]\[
\frac{242}{200} = (1 + r)^2
\][/tex]
[tex]\[
1.21 = (1 + r)^2
\][/tex]
Next, take the square root of both sides:
[tex]\[
\sqrt{1.21} = 1 + r
\][/tex]
[tex]\[
1.1 = 1 + r
\][/tex]
Finally, solve for [tex]\(r\)[/tex]:
[tex]\[
r = 1.1 - 1
\][/tex]
[tex]\[
r = 0.1
\][/tex]
6. Convert the decimal interest rate to a percentage:
To express the interest rate [tex]\(r\)[/tex] as a percentage, multiply by 100:
[tex]\[
\text{Rate (\%)} = r \times 100 = 0.1 \times 100 = 10\%
\][/tex]
Therefore, the annual rate of compound interest that results in a compound interest of Rs. 42 on a principal amount of Rs. 200 over 2 years is 10%.
