A loan of Rs 1000 is fully repaid by two installments of Rs 530 and Rs 594, paid at the end of first and second year, respectively. If the interest is compounded annually, then the rate of interest, in percentage, is

The correct answer is 8.

Let the rate of interest compounded annually be $r$%.

We are given that a loan of Rs. 1000 is repaid in two installments. The first installment paid at the end of the first year is Rs. 530, and the second installment paid at the end of the second year is Rs. 594.

Let's track the outstanding loan amount year by year.

The total amount owed at the end of the first year (before paying the installment) is the principal plus the interest for one year:

$\text{Amount after 1st year}=1000\left(1+\frac{r}{100}\right)$

After paying the first installment of Rs. 530, the remaining loan balance is:

$\text{Remaining balance}=1000\left(1+\frac{r}{100}\right)-530$

This remaining balance acts as the principal for the second year. The total amount owed at the end of the second year is:

$\text{Amount after 2nd year}=\left[1000\left(1+\frac{r}{100}\right)-530\right]\left(1+\frac{r}{100}\right)$

Since the loan is fully repaid with the second installment of Rs. 594, we can set up the following equation:

$\left[1000\left(1+\frac{r}{100}\right)-530\right]\left(1+\frac{r}{100}\right)=594$

To simplify the calculation, let's substitute $x=1+\frac{r}{100}$:

$\left(1000x-530\right)x=594$

$1000x^2-530x-594=0$

Dividing the entire equation by 2, we get:

$500x^2-265x-297=0$

Now, we can solve this quadratic equation using the quadratic formula, $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$:

$x=\frac{265\pm \sqrt{{\left(-265\right)}^2-4\left(500\right)\left(-297\right)}}{2\left(500\right)}$

$x=\frac{265\pm \sqrt{70225+594000}}{1000}$

$x=\frac{265\pm \sqrt{664225}}{1000}$

Calculating the square root of 664225 gives 815:

$x=\frac{265\pm 815}{1000}$

Since the interest rate must be positive, we take the positive root:

$x=\frac{265+815}{1000}=\frac{1080}{1000}=1.08$

Now, substituting back $x=1+\frac{r}{100}$:

$1+\frac{r}{100}=1.08$

$\frac{r}{100}=0.08$

$r=8$

Therefore, the annual rate of interest is 8%.