To determine the principal amount that will earn an interest of ₹812 at an interest rate of [tex]\(3 \frac{1}{2} \%\)[/tex] per annum over a period of [tex]\(7 \frac{1}{4}\)[/tex] years, we use the simple interest formula:
[tex]\[ I = P \times R \times T \][/tex]
where
- [tex]\(I\)[/tex] is the interest earned (₹812),
- [tex]\(P\)[/tex] is the principal amount,
- [tex]\(R\)[/tex] is the rate of interest per annum,
- [tex]\(T\)[/tex] is the time in years.
First, let's break down the given values:
- Interest ([tex]\(I\)[/tex]) = ₹812,
- Rate of interest ([tex]\(R\)[/tex]) = [tex]\(3 \frac{1}{2} \%\)[/tex],
- Time ([tex]\(T\)[/tex]) = [tex]\(7 \frac{1}{4}\)[/tex] years.
Converting the rate of interest to a decimal:
[tex]\[ R = 3 \frac{1}{2} \% = 3.5\% = \frac{3.5}{100} = 0.035 \][/tex]
Converting the time period into a decimal:
[tex]\[ 7 \frac{1}{4} = 7 + \frac{1}{4} = 7 + 0.25 = 7.25 \text{ years} \][/tex]
We rearrange the simple interest formula to solve for the principal ([tex]\(P\)[/tex]):
[tex]\[ P = \frac{I}{R \times T} \][/tex]
Substituting in the given values:
[tex]\[ P = \frac{812}{0.035 \times 7.25} \][/tex]
Calculating the denominator:
[tex]\[ 0.035 \times 7.25 = 0.25375 \][/tex]
Now, we divide the interest by this product:
[tex]\[ P = \frac{812}{0.25375} \][/tex]
Performing the division:
[tex]\[ P \approx 3199.9999999999995 \][/tex]
So, the principal amount that will earn an interest of ₹812 at a rate of [tex]\(3 \frac{1}{2} \%\)[/tex] per annum over [tex]\(7 \frac{1}{4}\)[/tex] years is approximately ₹3200.
