( i )  P = Rs. 5000,   T = 1 year,   A = Rs. 5325

   I = A - P

$$\begin{gathered} \Rightarrow \mathrm{I} = 5325 - 5000 \\ \Rightarrow \mathrm{I} = 325 \end{gathered}$$

So, the interest at the end of first year is Rs. 325

$$\begin{gathered} \mathrm{I} = \frac{\mathrm{PRT}}{100} \\ \Rightarrow \mathrm{R} = \frac{\mathrm{I} \mathrm{X} 100}{\mathrm{P} \mathrm{X} \mathrm{T}} \\ \Rightarrow \mathrm{R} = \frac{325 \mathrm{X} 100}{5000 \mathrm{X} 1} \\ \Rightarrow \mathrm{R} = \frac{32500 }{5000 } = 6.5 \% \end{gathered}$$

So, the rate of interest at the end of the first year is  6.5 %.

( ii ) The amount at the end of the first year will be the principal for the second year.

P = Rs. 5325,  T = 1 year,   R = 6.5 %

$$\begin{gathered} \mathrm{I} = \frac{\mathrm{PRT}}{100} \\ \Rightarrow \mathrm{I} = \frac{5325 \mathrm{X} 6.5 \mathrm{X} 1}{100} \\ \Rightarrow \mathrm{I} = 346.125 \\ \mathrm{A} = \mathrm{P} + \mathrm{I} \\ \Rightarrow \mathrm{A} = 5325 + 346.125 \\ \Rightarrow \mathrm{A} = 5671.125 \\ \Rightarrow \mathrm{A} \approx \mathrm{RS}. 5671 \end{gathered}$$

So, the amount at the end of the second year is  Rs. 5671.

Since the interest is earned on the minimum balance between  10^th  day and the last day of the month as per entries, we have

Minimum balance for April           = Rs. 8300

Minimum balance for May            = Rs. 7600

Minimum balance for June           = Rs. 10300

Minimum balance for July            = Rs. 10300

Minimum balance for August        = Rs. 3900

Minimum balance for September  = Rs. 0

Total balance = Rs. 40400

$\Rightarrow$ Total amount qualifying for interest = Rs. 40400

$\Rightarrow \mathrm{Principle} \mathrm{for} 1 \mathrm{month} \left(\frac{1}{12} \mathrm{year} \right) = \mathrm{Rs}. 40400,$

Rate of interest = 4.5 %  per annum

$\Rightarrow \mathrm{Interest} = \mathrm{Rs}. \frac{40400 \mathrm{x} 4.5 \mathrm{x} {\displaystyle \frac{1}{12}}}{100} = \mathrm{Rs}. 151.50$

Amount = Balance in the account in last month  +  Interest

            = 5900  +  151.50

            = Rs. 6051.50

Thus,  Mrs.  Ravi receives  Rs. 6051.50  on closing the account.

Given    P = Rs. 1500,     I = 496.50,     R = 10 %

A = P + I

$\Rightarrow$ A = Rs. 1500 + Rs. 496.50 = Rs. 1996.50

$$\begin{gathered} \mathrm{A} = \mathrm{P} {\left( 1 + \frac{\mathrm{R}}{100} \right)}^{\mathrm{n}} \\ \Rightarrow 1996.50 = 1500 {\left( 1 + \frac{10}{100} \right)}^{\mathrm{n}} \\ \Rightarrow \frac{1996.50}{1500} = {\left( 1 + \frac{1}{10} \right)}^{\mathrm{n}} \\ \Rightarrow \frac{1996.50}{1500} = {\left( \frac{1 \mathrm{x} 10}{1 \mathrm{x} 10} + \frac{1}{10} \right)}^{\mathrm{n}} \\ \Rightarrow \frac{1996.50}{1500} = {\left( \frac{ 10}{ 10} + \frac{1}{10} \right)}^{\mathrm{n}} \\ \Rightarrow \frac{1996.50}{1500} = {\left( \frac{11}{10} \right)}^{\mathrm{n}} \\ \Rightarrow \frac{1996.50}{1500} = {\left( 1.1 \right)}^{\mathrm{n}} \\ \Rightarrow 1.331 = {\left( 1.1 \right)}^{\mathrm{n}} \\ \Rightarrow ( 1.1 {)}^3 = {\left( 1.1 \right)}^{\mathrm{n}} \\ \Rightarrow \mathrm{n} = 3 \end{gathered}$$

(i)  I = Rs. 1200,   n = 2 x 12 = 24 months,     r =  6%

$$\begin{gathered} \mathrm{I} = \mathrm{P} \mathrm{x} \frac{\mathrm{n} \left( \mathrm{n} + 1 \right)}{2 \mathrm{x} 12} \mathrm{x} \frac{\mathrm{r}}{100} \\ \Rightarrow 1200 = \mathrm{P} \mathrm{x} \frac{24 \mathrm{x} 25}{24} \mathrm{x} \frac{6}{100} \\ \Rightarrow 1200 = \mathrm{P} \mathrm{x} \frac{3}{2} \\ \Rightarrow \mathrm{p} = \frac{1200 \mathrm{x} 2}{3} \\ \Rightarrow \mathrm{P} = \mathrm{Rs}. 800 \end{gathered}$$

So the monthly instalment is  Rs. 800.

( ii ) Total sum deposited = P x n  =  Rs. 800 x 24  =  Rs. 19200

$\therefore$ Amount of maturity =  Total sum deposited + Interest on it

                                 = Rs. 19200 + Rs. 1200

                                 = Rs. 20400