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**Chapter - 19**

**SIMPLE INTEREST AND COMPOUND INTEREST**

Principal is the amount on which interest is paid / earned.

If principal is denoted by P

The rate of interest is denoted by R

The time for which the principal is given on interest = T

Then

**SIMPLE INTEREST **

Simple interest earned / paid is devoted by S.I.

Then (1) S.I. =

Amount (A) = P + SI (Amount is the sum of P and the SI earned)

Amount x 100 = Principal x (100 + RT)

Principal =

**COMPOUND INTEREST**

Amount (A) =

Or CI = A â€“ P

Difference of CI and SI for a period of one year in zero

For 2 years the difference of CI and SI is

For 3 years the difference of CI and SI is

=

**SOLVED EXAMPLES**

**Example 1: Ramesh deposited Rs. 29400 for 6 years at simple interest. After 6 years he received Rs. 4200 as interest. Find the annual rate of interest. **

- P = Rs. 29400 T = 6 years SI = Rs. 4200

For finding out the rate of interest R

**Example 2: I invested an amount of Rs. 17500 at the rate of 8% per annum. After how many years will I get Rs. 16800 as simple interest? **

- P = Rs. 17500 R = 8% SI = Rs.16800

So T =

So time is 12 years

**Example 3: Pritam invested Rs. 16840 at 6% for 5 years. What amount of SI, he will get after 5 years. **

- P = 16840

R = 6%

T = 5 years

= RS. 5052

**Example 4: At what rate percent per annum simple interest a sum of money doubles itself in 15 years. **

- let the principal be Rs. 100

T = 15 years

SI = Rs.100

R = ?

**Example 5: A sum of Rs. 3100 was lent partly at 5% and partly at 8% simple interest. The total interest received after 3 years was Rs. 600. Amount lent on 5% per annum was how much. **

- let Rs. X be lent at 5%, then the remaining amount of Rs. (3100-x) was lent at 8%

So

Or 15x + 24 (3100 â€“ x) = 60000

Or 9x = 74400 â€“ 60000

**SHORT CUT METHOD **

Let us suppose that whole amount was given at 5%

Interest =

But the interest earned in 600. It means some amount which earns Rs. 135 at 3% interest (8% - 5%) was given. This amount is

So 3100 â€“ 1500 = Rs. 1600 was given an interest of 5%.

**Example 6: The simple interest on a sum of money is 4/9 of the principal and the number of years is equal to the rate percent per annum. The rate per annum is what? **

- Let the amount (Principal) be Rs. 1

SI = Rs. 4/9

Let rate be = x%

Then T = x years

As per the formula SI = PRT / 100

So rate per annum =

**Example 7: In 4 years, the simple interest on a sum of money is 9/25 of the principal. Find out the annual rate of interest. **

- Let the principal be Rs. 1

Interest =

T = 4 years

R =

**Example 8: A sum of money at simple interest amounts to Rs. 1012 in 2Â½ years and to Rs. 1067.20 in 4 years. Find out the rate of interest, per year. **

**Solution: **From the question it is clear that

Rs. 1067.20 â€“ Rs. 1012.00 = Rs. 55.20 is the simple interest earned in a period of 1Â½ years. So now we can find the interest earned in 2Â½ year which will be

= 18.40 x 5 = 92.00

So the principal in the beginning was Rs. 1012-92 = Rs. 920

So now P = Rs. 920

T = 5/2 year

Simple Interest = Rs. 92

**COMPOUND INTEREST **

To understand the method of calculating compound interest, let us consider an example.

Example: I borrowed Rs. 40,000 from a bank. The bank charges interest at the rate of 10% per annum. At the end of the year I will have to pay Rs. i.e. Rs. 4000 as interest.

Thus total amount of Rs. 40000 + Rs. 4000 = Rs. 44000 will have to be paid back to the bank.

If due to some reason I am not able to pay, the bank will charge 10% interest for the next year on the principal of Rs. 44000. At the end of the 2^{nd} year the interest will be Rs.

= Rs. 4400.

Thus total interest payable to the bank will be Rs. (4400 + 4000) = Rs. 8400. This interest is Rs. 400 more than the simple interest on 40000 for 2 years at 10%. This Rs. 400 is the interest on the interest of 4000 for the first year. So the interest calculated in this way is called compound interest.

Remarks: For the first unit of time the simple and the compound interest are equal. From the second unit of time onwards the compound interest is more than the simple interest.

**Example 1: find the compound interest on Rs. 3000 for 3 years at 8% per annum. **

- Principal for the first year = Rs. 3000

Rate = 8%

Time = one year

Interest = = Rs. 240

Amount = Principal + interest = Rs. 3000 + 240 = Rs. 3240

Principal for 2^{nd} year = Rs. 3240

Rate = 8%

Time = 1 year

So interest for 2^{nd} year =

So amount = Rs. 3240 + Rs. 259.20 = Rs. 3499.20

Principal for 3^{rd} year = Rs. 3499.20

Rate = 8%

Time = 1 year

Interest for 3^{rd} year =

So compound interest for 3 years

= Rs. 240 + Rs. 259.20 + Rs. 279.94 = Rs. 779.14

**When interest is compounded half yearly.**

**Example 2: Find the compound interest on Rs. 1000 for 1Â½ year at 12% per annum when the interest is compounded half yearly.**

- Rate per annum is 12%

So the rate per half year will be 12/2 % or 6%. Also there will be 3 half years in 1Â½ year. By using the formula

In this case P = 1000

r = 6% per half year

t = 3 half years

So A =

So compound interest = Rs. 1191.02 â€“ Rs. 1000 = Rs. 191.02

**When the interest is compounded quarterly**

**Example 3: Find the compound interest on Rs. 6000 for one year at 16% per annum when the interest is compounded quarterly. **

- Rate per annum is 16%, so rate per quarter is 16/4 = 4%

Time is one year or 4 quarters

By the formula of compound interest

So compound interest for 4 quarters at the rate of 16% on 6000 is 7019.15 â€“ 6000 = Rs. 1019.15

**Example 4: At what rate per cent interest per annum, will Rs. 10000 amount to Rs. 14641 in 2 years. If the interest is compounded half yearly. **

- Here P = Rs. 10000

A = Rs. 14641

R = ?

T = 2 years or 4 half years

Formula isA =

or 14641 =

or

so 10% is half yearly rate

annual rate will be 10 x 2 = 20%

**Example 5: The difference between the compound interest and simple interest on a certain sum at 14% per annum for 2 years is Rs. 147. Find the sum. **

- Let the sum be Rs. 100

SI on 100 for 2 years at 14% is =

CI on 100 for 2 years at 14% = A â€“ P

Difference in CI and SI =

If difference is = 100

If difference 1=

If difference is Rs. 147 =

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