The difference between compound interest and simple interest for 2 years at 10

Last updated date: 29th Dec 2022

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Hint: Let the sum be \[x\] rupees. We know the compound interest \[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}\] , we need to find P. we know simple interest formula \[S.I = P \times \dfrac{r}{{100}} \times T\] . We know compound interest is the difference between amount and principal amount. Since the difference between compound and simple interest is given we can find the value of \[x\] .

Complete step-by-step answer:
We know,
 \[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}\] , where A is amount, R is rate of interest, n is number of times the interest is compounded per year.
 \[P = x\] , \[n = 2\] \[r = 10\] , substituting we get,
 \[ \Rightarrow A = x{\left( {1 + \dfrac{{10}}{{100}}} \right)^2}\]
 \[ \Rightarrow A = x{\left( {1 + \dfrac{1}{{10}}} \right)^2}\]
Taking L.C.M and simplifying we get,
 \[ \Rightarrow A = x{\left( {\dfrac{{10 + 1}}{{10}}} \right)^2}\]
 \[ \Rightarrow A = x{\left( {\dfrac{{11}}{{10}}} \right)^2}\]
We know that compound interest is the difference between the amount of money accumulated after n years and the principal amount.
 \[C.I = A - P\]
 \[ \Rightarrow C.I = x{\left( {\dfrac{{11}}{{10}}} \right)^2} - x\] .
Now to find the simple interest we have, \[S.I = P \times \dfrac{r}{{100}} \times T\]
Substituting the known values,
 \[ \Rightarrow S.I = x \times \dfrac{{10}}{{100}} \times 2\]
 \[ \Rightarrow S.I = x \times \dfrac{1}{{10}} \times 2\]
 \[ \Rightarrow S.I = \dfrac{x}{5}\]
Given the difference between compound and simple interest is 500
 \[ \Rightarrow C.I - S,I = 500\]
Substituting C.I and S.I we get
 \[ \Rightarrow x{\left( {\dfrac{{11}}{{10}}} \right)^2} - x - \dfrac{x}{5} = 500\]
Simple division \[\dfrac{{11}}{{10}} = 1.1\] and \[\dfrac{1}{5} = 0.2\] we get,
 \[ \Rightarrow x{(1.1)^2} - x - 0.2x = 500\]
 \[ \Rightarrow 1.21x - 1x - 0.20x = 500\]
 \[ \Rightarrow 0.21x - 0.20x = 500\]
Taking x as common,
 \[ \Rightarrow (0.21 - 0.20)x = 500\]
 \[ \Rightarrow 0.01x = 500\]
 \[ \Rightarrow x = \dfrac{{500}}{{0.01}}\]
Multiply numerator and denominator by 100.
 \[ \Rightarrow x = 50,000\]
That is \[P = 50,000\] .
 \[50,000\] Rupees is the sum when the interest is compounded annually.
So, the correct answer is “\[P = 50,000\]”.

Note: Here we used three formulas. Remember the formula for simple interest, compound interest and amount formula. We can also take P as P as it is, and solve for P. Above all we did is substituting the given data in the formula and simplifying. Principal amount is the initial amount you borrow or deposit.

The formula given below can be used to find the difference between compound interest and simple interest for two years.

The above formula is applicable only in the following conditions.

1. The principal in simple interest and compound interest must be same.

2. Rate of interest must be same in simple interest and compound interest.

3. In compound interest, interest has to be compounded annually.

Example 1 :

The difference between the compound interest and simple interest on a certain investment at 10% per year for 2 years is $631. Find the value of the investment.

Solution :

The difference between compound interest and simple interest for 2 years is 631.

Then we have,

P(R/100)2 = 631

Substitute R = 10.

P(10/100)2 = 631

P(1/10)2 = 631

P(1/100) = 631

Multiply both sides by 100.

P = 631 x 100

P = 63100

So, the value of the investment is $63100.

Example 2 :

The compound interest and simple interest on a certain sum for 2 years is $ 1230 and $ 1200 respectively. The rate of interest is same for both compound interest and simple interest and it is compounded annually. What is the principal ?

Solution :

To find the principal, we need rate of interest. So, let us find the rate of interest first.

Step 1 :

Simple interest for two years is $1200. So interest per year in simple interest is $600.

So, C.I for 1st year is $600 and for 2nd year is $630.

(Since it is compounded annually, S.I and C.I for 1st year would be same)

Step 2 :

When we compare the C.I for 1st year and 2nd year, it is clear that the interest earned in 2nd year is 30 more than the first year.

Because, in C.I, interest $600 earned in 1st year earned this $30 in 2nd year.

It can be considered as simple interest for one year.

That is, principle = 600, interest = 30

I = PRT/100

30 = (600 x R x 1)/100

30 = 6R

Divide both sides by 6.

5 = R

So, R = 5%.

Step 3 :

The difference between compound interest and simple interest for two years is

= 1230 - 1200

= 30

Then we have,

P(R/100)2 = 30

Substitute R = 5.

P(5/100)2 = 30

P(1/20)2 = 30

P(1/400) = 30

Multiply both sides by 400.

P = 30 x 400

P = 12000

So, the principal is $12000.

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