Solution:

Given that Rajiv lends out 9 rupees to Anni on condition that the amount is payable in 10 months by equal installments of 1 rupee each payable at the start of every month.

To find: The rate of interest per annum.

Let us assume that the rate of interest per annum is 'r'.

Formula:

The present value of an annuity due is given by,
PV = (C/r) x [1 - (1/(1+r)^n)] x (1+r)

Where PV = Present Value of Annuity
C = Cash flow per period
n = Number of periods

Calculation:

Here, C = 1 rupee per month and n = 10 months.

The first installment has to be paid one month from the date the loan is availed. Therefore, the present value of the annuity is calculated for 9 months.

PV = (1/r) x [1 - (1/(1+r)^9)] x (1+r)

Multiplying both sides by 'r':

PV x r = [1 - (1/(1+r)^9)] x (1+r)

PV x r = (1+r) - (1/(1+r)^8)

PV x r^2 + PV = (1+r)^2

Substituting PV = 9 and solving the quadratic equation, we get:

r = 0.122 or -1.122

Since the rate of interest cannot be negative, the rate of interest per annum is 12.2%.

Therefore, the rate of interest per annum is 12.2%.

Explanation:

- The question is about calculating the rate of interest per annum when a loan is given and the repayment is made in equal installments.
- The present value of an annuity due formula is used to calculate the rate of interest per annum.
- The present value of an annuity due formula calculates the present value of a series of equal payments made at the beginning of each period.
- The formula takes into account the cash flow per period, the number of periods, and the rate of interest per period.
- In the given question, the cash flow per period is 1 rupee per month, the number of periods is 10 months, and the rate of interest per period is 'r'.
- The first installment has to be paid one month from the date the loan is availed. Therefore, the present value of the annuity is calculated for 9 months.
- By substituting the values in the formula and solving the equation, we get the rate of interest per annum as 12.2%.