Find the difference between C.I and S.I. on Rs 1000 at 10% per annum for 3 years

The difference between compound interest and simple interest is Rs. 5 in 2 years and the rate of interest is 10%. Find the principal amount.

1. Rs. 50
2. Rs. 500
3. Rs. 100
4. Rs. 1000

Given:

Difference between C.I and S.I = Rs. 5

Time = 2 years

Rate = 10%

Where C.I = Compound Interest, S.I = Simple Interest

Formula used:

The difference in C.I and S.I = P (R)2/(100)2

Where P = Principal, R = Rate of Interest

Calculation:

5 = P (10)2/(100)2

⇒ P = (5 × 10000)/100

⇒ P = 500

∴ The Principal amount is Rs. 500

5 is interest on interest

⇒ 5 = 10% of interest

⇒ interest = 50

⇒ 50 = 10% of principal

∴ Principal is Rs 500

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The formula given below can be used to find the difference between compound interest and simple interest for three 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 :

$800 is invested in both simple interest and compound interest at the same rate of interest for three years. If the rate of interest is 20%, find the difference between compound interest and simple interest.

Solution :

The formula for difference between compound interest and simple interest for three years is

C.I - S.I = P(R/100)2(R/100 + 3)

In the above formula, substitute R = 20, P = 800.

C.I - S.I = 800(20/100)2(20/100 + 3)

Simplify

C.I - S.I = 800(1/5)2(1/5 + 3)

= 800(1/25)(16//5)

= 800 x 16/125

= 800 x 16/125

= 102.40

So, the difference between compound interest and simple interest is $102.40.

Example 2 :

The difference between the compound interest and simple interest on a certain principal is at 10% per year for 3 years is $31. Find the principal.

Solution :

The difference between compound interest and simple interest for three years is 31.

Then we have,

P(R/100)2(R/100 + 3) = 31

Substitute R = 10.

P(10/100)2(10/100 + 3) = 31

P(1/10)2(1/10 + 3) = 31 

P(1/10)2(31/10) = 31

P(1/100)(31/10) = 31

P(31/1000) = 31

Multiply both sides by 1000/31.

P = 31 x (1000/31)

P = 1000

So, the principal is $1000.

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Answer

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Hint: To find the difference between the simple interest and compound interest, first calculate the simple interest and then compound interest then subtract simple interest in compound interest.

Complete step-by-step answer:
In the calculation of compound interest if we take small compounding time then the compound interest will be high as the compounding time will increase and the amount of compound interest will decrease.
Given the value of rate of interest = 10%
Principal = Rs 1000
Time = 4 years
We know that simple interest $ = \dfrac{{PRT}}{{100}}$
On putting the given values we get,
$\Rightarrow$ S.I. $ = \dfrac{{1000 \times 10 \times 4}}{{100}}$
S.I. $ = 400$ Rs
Similarly we will find the compound interest
$\Rightarrow$ We know compound interest = Amount−Principal
and amount is given by
$ = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$
On putting the given values we get
A = \[1000{\left( {1 + \dfrac{{10}}{{100}}} \right)^4}\]

\[ = 1000 \times \dfrac{{110}}{{100}} \times \dfrac{{110}}{{100}} \times
\Rightarrow \dfrac{{110}}{{100}} \times \dfrac{{110}}{{100}}\]
\[ = 1464.10\] Rs.
\[C.I. = 1464.10 - 1000 = 464.10\] Rs.
Now we will find the difference between C.I. and S.I.
Difference between C.I and S.I $464.10 - 400 = 64.10$ Rs.

Note: Compound interest is always higher than the simple interest for the same time period and same rate of interest only except the first year. In first year CI and SI are the same.

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