Verified answer

Answer:

The Answer is:- 51263.79 rs and the rate of interest is 6%

Step-by-step explanation:

Given:

A certain sum of money amounts to Rs 57600 in 2 years and Rs 65536 in 4 years at compound interest compounded annually

To find:

The sum and the rate of interest

Solution:

Let "P" represent the sum of money and "R" represent the rate of interest.

The formula to find the amount in compound interest is given as,

\boxed{\bold{A = P\: [1+ \frac{R}{100} ]^n}}

A=P[1+

100

R

]

n

where

A = amount

P = principal

R = rate of interest

n = time period

According to the question and using the formula, we can form two equations such as:

57600 = P\: [1+ \frac{R}{100} ]^2}} ..... (i)

and

65536 = P\: [1+ \frac{R}{100} ]^4}} ...... (ii)

On dividing eq. (ii) by (i), we get

\frac{65536}{57600} = \frac{P[1+\frac{R}{100}^4 ]}{P[1+\frac{R}{100} ]^2}

57600

65536

=

P[1+

100

R

]

2

P[1+

100

R

4

]

⇒ \frac{65536}{57600} = [1+\frac{R}{100}]^2

57600

65536

=[1+

100

R

]

2

⇒ 1.1377 = [1+\frac{R}{100}]^21.1377=[1+

100

R

]

2

taking square root on both sides

⇒ \sqrt{1.1377} = \sqrt{[1+\frac{R}{100} ]^2}

1.1377

=

[1+

100

R

]

2

⇒ 1.06 = 1+\frac{R}{100}1.06=1+

100

R

⇒ \frac{R}{100} = 1.06 - 1

100

R

=1.06−1

⇒ R = 0.06 \times 100R=0.06×100

⇒ \bold{R = 6\%}R=6%

Now, substituting the value of R in eq. (i), we get

57600 = P\: [1+ \frac{6}{100} ]^2}}

⇒ 57600 = P\: [ \frac{106}{100} ]^2}}

⇒ P\:=\: \frac{57600 \times 100\times 100}{106\times 106}}}

⇒ \bold{P = Rs. \: 51263.79}P=Rs.51263.79

Thus, the sum of money is Rs. 51263.79 and the rate of interest is 6%.

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