Question

The difference between the compound interest and the simple interest on a certain sum of money at 15% per annum for three years is rupees 283.50, find the sum​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

The difference between the compound interest and the simple interest on a certain sum of money at 15% per annum for three years is Rs 283.50.

Let assume that sum invested be Rs P.

Case :- 1 Compound interest

Principal = P

Rate of interest, r = 15 % per annum compounded annually.

Time, n = 3 years.

We know,

Compound interest (CI) received on a certain sum of money of Rs P invested at the rate of r % per annum compounded annually for n years is given by

$\boxed{\sf{  \:\rm \: CI = P {\bigg[1 + \dfrac{r}{100} \bigg]}^{n} - P \: \: }}$

Now, on substituting the given values, we get

$\:\rm \: CI = P {\bigg[1 + \dfrac{15}{100} \bigg]}^{3} - P \: \:$

$\:\rm \: CI = P {\bigg[1 + \dfrac{3}{20} \bigg]}^{3} - P \: \:$

$\:\rm \: CI = P {\bigg[\dfrac{20 + 3}{20} \bigg]}^{3} - P \: \:$

$\:\rm \: CI = P {\bigg[\dfrac{23}{20} \bigg]}^{3} - P \: \:$

$\rm \: CI = \dfrac{12167P}{8000} - P$

$\rm \: CI = \dfrac{12167P - 8000P}{8000}$

$\rm\implies \:\rm \: CI = \dfrac{4167 \: P }{8000} - - - - (1)$

Case - 2 Simple interest

Principal = P

Rate of interest, r = 15 % per annum

Time, n = 3 years.

We know,

Simple interest (SI) received on a certain sum of money of Rs P invested at the rate of r % per annum for n years is given by

$\boxed{\sf{  \:\rm \: SI \: = \: \frac{P \times r \times n}{100} \: \: }}$

So, on substituting the values, we get

$\rm \: SI = \dfrac{P \times 15 \times 3}{100}$

$\rm \: SI = \dfrac{P \times 3 \times 3}{20}$

$\rm\implies \:\rm \: SI = \dfrac{9P}{20} - - - (2)$

Now, According to statement

$\rm \: CI \: - \: SI \: = \: 283.50$

On substituting the values from equation (1) and (2), we get

$\rm \: \dfrac{4167P}{8000} - \dfrac{9P}{20} = 283.50$

$\rm \: \dfrac{4167P - 3600P}{8000} = 283.50$

$\rm \: \dfrac{567P}{8000} = 283.50$

$\rm \: P = \dfrac{283.50 \times 8000}{567}$

$\rm\implies \:P \: = \: 4000$

Hence,

Sum invested is Rs 4000.

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Additional Information :-

1. Amount received on a certain sum of money of Rs P invested at the rate of r % per annum compounded annually for n years is given by

$\boxed{\sf{  \:\rm \: Amount = P {\bigg[1 + \dfrac{r}{100} \bigg]}^{n} \: \: }}$

2. Amount received on a certain sum of money of Rs P invested at the rate of r % per annum compounded semi - annually for n years is given by

$\boxed{\sf{  \:\rm \: Amount = P {\bigg[1 + \dfrac{r}{200} \bigg]}^{2n} \: \: }}$

3. Amount received on a certain sum of money of Rs P invested at the rate of r % per annum compounded quarterly for n years is given by

$\boxed{\sf{  \:\rm \: Amount = P {\bigg[1 + \dfrac{r}{400} \bigg]}^{4n} \: \: }}$

Answer 2

CORRECT QUESTION :

• The difference between the compound interest and the simple interest on a certain sum of money at 15% per annum for three years is rupees 283.50, find the sum

information provided :

• If P be the principal amount that is compounded n times in a year at an annual interest rate of r then the future value of the investment after t years is given by

A = P ( 1 + r / n 100

• Simple interest on the principal amount P with interest rate of r after t years is given by ,

SI = P × r / 100 × t

step 1 to do it :

please note the given information :

• Interest rate, r = 15%, Time period, t = 3 and CI- SI = 283.50

first we have Cl is the compound interest and SI is the simple interest.

step 2 :

• Consider x to be the required sum

• then, we will calculate the simple interest

then, we will applying the simple interest. formula :

• SI = P × r /100 × t

Substitute P = x, r = 15 and t = 3

• SI = x × 15 / 100 × 3

• SI = 0.45 x

step 3 :

• first we will calculate the compound interest.

then, we will again applying the simple interest formula :

• CI= A - P

• CI = x ( 1 + 15 /100) - x

• CI = x ( 1. 15 ) - x

step 4 :

Substitute Cl and Sl in equation (1) and calculate x

• CI - SI = 283.50

• x (1.15) ^ 3 - x - 0.45x = 283.5

• 0.070875x = 283.5

• x = 4000

hence, the required sum is Rs 4000

FINAL ANSWER :

• The required sum is Rs 4000

all formula :

• A = P ( 1 + r / n 100

• SI = P × r / 100 × t

• P = 100 × SI / R × T

• R = 100 × S.I / P × T

• T = 100 × S.I / P × R