Simple Interest and Compound Interest

Money is not free to borrow. People can always find a use for money, so it costs to borrow money. Different places charge different amounts at different times! It is called Interest. This lesson explains the concept of Simple Interest and Compound Interest. We will develop a basic understanding of these two different types of interests, their uses and properties.

Simple Interest (SI)

It is an easy and quick method of calculating an interest charge on a loan. Simple interest (S.I.) is determined by multiplying the principal (P) with rate of interest (R) and time period (T).

$S.I.= \dfrac{P \times R \times T}{100}$

Example: Henry borrowed Rs. 5000 for 4 years at an interest rate of 5% from a bank. How much of interest is that?

We know,

$S.I.=\dfrac{P \times R \times T}{100}$

Here P= Rs. 5000, R= 5%, T= 4 years

So, $I=\dfrac{5000 \times 5 \times 4}{100}=Rs. 1000$

Ans: Henry has to pay Rs. 1000 as interest.

Clearly, in S.I. the principal remains constant throughout. But the above method is not generally used in day to day financial system like banks, insurance companies, post offices. They use a different method of computing interest. In this method the lender and the borrower agree to fix up a certain time interval, say a year or half a year or a quarter of a year for the computation of the interest and the amount. At the end of the first interval, the interest is computed and is added to the original principal. The amount obtained is added to the second interval of time. The amount of this principal at the end of the second interval of time is taken as the principal of the third interval of time and so on. At the end of the certain specified period, the difference between the amount and money borrowed, that is, the original principal is computed and it is called the compound interest. Let us simplify it.

Compound Interest (CI)

If the borrower and lender agree to fix up a certain interval of time, so that the amount (Principal + Interest) at the end of the interval becomes the principal of the next interval, then the total interest over all the interests, calculated in this way is called the Compound Interest or C.I..

Evidently, C.I. at the end of a certain specified period is equal to the difference between the amount at the end of the period and the original principal.

C.I. = Amount – Principal

Conversion Period: The fixed interval of time at the end of which the interest is calculated and added to the principal at the beginning of the interval is called the conversion period. In other words, the period at the end of which the interest is compounded is called the conversion period. For instance, when the interest is calculated and added to the principal every six months, the conversion period is six months. Likewise, the conversion period is three months when the interest is calculated and added quarterly.

NOTE: If no conversion period is specified, the conversion period is taken to be one year.

Compound Interest Calculation from simple Interest where Interest is compounded annually.

Q1. Find the Compound interest on Rs. 10000 for two years at 5% per annum.

Solution: Principal for the first year = Rs. 10000

Interest for the first year = $Rs \dfrac{10000 \times 5 \times 1}{100} = Rs. 500$

[We are using the formula $S.I.=\dfrac{P \times R \times T}{100}$ ]

$\therefore$ amount at the end of first year = Rs 10000 + Rs. 500 = Rs. 10500

Interest for the second year = $\dfrac{10500 \times 5 \times 1}{100} = Rs. 525$

Principal of the second year was Rs. 10500 and so amount at the end of the second year = Rs. 10500 + Rs. 525 = Rs. 11025

So, Compound interest= Rs. (11025 – 10000) = Rs. 1025

Note: The C.I. can also be found by adding the interest for each year.

Compound Interest Calculation from simple Interest where Interest is compounded half yearly.

If the rate of interest is R% per annum and the interest is compounded half-yearly, then the rate of interest will be R/2% per half year.

Q: Find the compound interest on Rs. 10000 for 1½ years at 20% per annum, interest being payable half-yearly.

Solution: We know, R= 20% per annum
or, 10% per half year.

T= 1½ years = 3 half years

Original Principal (P) = Rs. 10000

I for the first half-year = $Rs. \dfrac{10000 \times 10 \times 1}{100} = Rs. 1000$

P for the second half-year = Rs. 10000+1000= Rs. 11000

I for the second half-year = $Rs.\dfrac{11000 \times 10 \times 1}{100} = Rs. 1100$

Amount at the end of the second half-year = Rs. 11000 + Rs. 1100 = Rs. 12100

P for the third half year= Rs. 12100

I for the third half-year = $Rs. \dfrac{12100 \times 10 \times 1}{100} = Rs. 1210$

Amount at the end of third half-year = Rs. 12100 + Rs. 1210 = Rs. 13310

$\therefore$ C.I. = Rs. 13310 – Rs. 10000 = Rs. 3310.

Computation of C.I. when Interest is compounded quarterly

If the rate of interest is R% per annum and the interest is compounded quarterly, then the rate of interest will be R/4% per quarter.

Q: Find the compound interest on Rs. 10000 for 1 year at 20% per annum, compounded quarterly.

Solution: We have, R= 20% per annum = 20/4 % = 5% per quarter

T= 1 year= 4 quarters

P for the first quarter= Rs. 10000

Interest for the first quarter = $Rs. \dfrac{10000 \times 5 \times 1}{100} = Rs. 500$

Amount at the end of first quarter= Rs. 10000 + Rs. 500 = Rs. 10500

P for the second quarter = Rs. 10500

Interest for the second quarter = $Rs. \dfrac{10500 \times 5 \times 1}{100} = Rs. 525$

Amount at the end of second quarter= Rs. 10500 + Rs. 525 = Rs. 11025

P for the third quarter = Rs. 11025

Interest for the third quarter = $Rs. \dfrac {11025 \times 5 \times 1}{100} = Rs. 551.25$

Amount at the end of third quarter = Rs. 11025 + Rs. 551.25 = Rs. 11576.25

P for the fourth quarter = Rs. 11576.25

Interest for the fourth quarter = $Rs. \dfrac {11576.25 \times 5 \times 1}{100} = Rs. 578.8125$

Amount at the end of fourth quarter = Rs. 11576.25 + Rs. 578.8125= Rs. 12155.0625

$\therefore$ C.I. = Rs 12155.0625 – Rs. 10000 = Rs. 2155.0625 or Rs. 2155.06

Compound Interest formula

Let P be the principal and the rate of interest be R% per annum. If the interest is compounded annually, the amount A and the compound interest, C.I., at the end of n years is given by

$A = P \times (1 + \dfrac{R}{100})^n$

and, $C.I.= A-P = P \times [( 1 + \dfrac{R}{100})^n - 1]$ respectively.

Proof: We have,

P = Principal and rate of interest is R% per annum. Since the interest is given annually.

$\therefore$ the interest after one year $=\dfrac{P \times R}{100}$

$\Rightarrow$ Amount at the end of one year = $P + \dfrac{P \times R}{100} = P [1 + \dfrac {R}{100}]$

Now, this amount is taken as the principal for the second year.

$\therefore$ the interest after the second year $= P [1 + \dfrac{R}{100}] \times \dfrac{R}{100}$

$\Rightarrow$ Amount at the end of second year $= P [1 + \dfrac{R}{100}] + P [1+ \dfrac{R}{100}] \times \dfrac{R}{100}$

$= P [1 + \dfrac{R}{100}] \times [1+ \dfrac{R}{100}]$

$= P [1 + \dfrac{R}{100}]^2$

Considering this amount as the principal of the third year, we have

The interest for the third year $= P [1 + \dfrac{R}{100}]^2 \times \dfrac{R}{100}$

$\Rightarrow$ Amount for the third year $= P [1 + \dfrac{R}{100}]^2 + P [1 + \dfrac{R}{100}]^2 \times \dfrac{R}{100}$

$= P [1 + \dfrac{R}{100}]^2 \times [1 + \dfrac{R}{100}]^2$

$= P [1 + \dfrac{R}{100}]^3$

Hence, if we go on like this further, we have

Amount at the end of n years $= P [ 1 + \dfrac{R}{100}]^n$

Type 1: When the interest is compounded annually

Q: Find the amount of Rs.8000 for 3 years, compounded annually at 10% per annum. Also find the C.I.

Here, P= Rs.8000, R= 10% per annum and n= 3 years.

Using the formula $A = P [1 + \dfrac{R}{100}]^n$ , we get

Amount for 3 years $= Rs. [8000 \times (1 + \dfrac{10}{100})^3]$

$= Rs. [8000 \times \dfrac{11}{10} \times \dfrac{11}{10} \times \dfrac{11}{10}]$

$= Rs. 10648$

Thus the amount after 3 years is Rs.10,648

And the C.I. = Rs. ( 10648 – 8000 ) = Rs. 2648.

Type 2: When the interest is compounded annually but rates are different for different years.

Let Principal = Rs. P, Time= 2 years, and let the rates of interest be p% per annum, during the first year and q% per annum during the second year.

Then the amount after 2 years $= Rs.[P \times (1 + \dfrac{p}{100}) \times (1 + \dfrac{q}{100})]$

This formula can be similarly extended for any number of years.

Q: Find the amount of R.s. 50000 after 2 years, compounded annually; the rate of interest being 8% p.a. during the first year and 9% p.a. during the second year. Also, find the compound interest.

Here, P = R.s. 50000, p= 8% p.a. and q= 9% p.a.
Using the formula $A= [P \times (1 + \dfrac{p}{100}) \times (1 + \dfrac{q}{100})]$ , we have,

amount after 2 years $= Rs. [50000 \times (1 +\dfrac{8}{100}) \times (1 +\dfrac{9}{100})]$

$= Rs. [50000 \times \dfrac{27}{25} \times \dfrac{109}{100}] = Rs. 58860.$

$\therefore$ amount after 2 years = Rs. 58860

And, the C.I. is Rs. [ 58860 – 50000 ] = Rs. 8860.

Type 3: When interest is compounded annually but time is a fraction

Suppose time is $\dfrac{3}{5}$ years.

Then, amount $= P \times [1 + \dfrac{R}{100}]^2n \times [1 + \dfrac{3}{5} \times R ]$

Type 4: Interest Compounded Half-Yearly

Let the Principal be Rs. P, The rate of interest be R% and time be n years.

Suppose the interest is compounded half-yearly. Then,

rate = (R/2) % per half year, time= 2n half-years and amount $= Rs. [1 + \dfrac{R}{2 \times 100}]^{2n}$

$\therefore$ Compound Interest = amount – principal

Type 5: Interest Compounded Quarterly

Let the Principal be Rs. P, The rate of interest be R% and time be n years.

Suppose the interest is compounded quarterly. Then,

rate= (R/4) % per half year, time= 4n quarters and amount $= Rs. [1 + \dfrac{R}{4 \times 100}]^{4n}$

$\therefore$ Compound Interest = amount – principal

Comments

Satyaveer says

January 1, 2018 at 2:23 pm

Principle =30000
Rate =2%
Time = 3yr 10 months 25 days
Find compound interest

- ramkumar gawde says
  
  January 24, 2020 at 8:10 am
  
  calculate the amount of 1000 @ 4% p.a. compound interest for 3 years.
  
- safiq says
  
  August 21, 2020 at 9:50 am
  
  What principle will produce taka 55.50 interest in two years at 5% per year simple interest?
  
PRIYA says

January 28, 2018 at 2:58 pm

Nice and thx

jacqlyne says

April 24, 2018 at 7:34 pm

well done

SARU MAHARJAN says

August 11, 2019 at 1:52 am

can you please help me solve this question :

by what percent more in the yearly compound interest on rs.4000 for 3 years at 10% p.a. than simple interest on Rs. 5000 for 3 years at 8% p.a. find it

Mercy Williams says

November 8, 2019 at 10:37 pm

Please help me to solve this
P = 500,000
R = 10%
T = 5years 9months
Find the interest and amount

Zerat says

April 6, 2020 at 9:14 am

Calculate compound interest for Rs 5000 for 4 years at 3 1/2% p.a….
Find the solve this Question

Zerat says

April 6, 2020 at 9:17 am

Calculate c.i on Rs 7500 for 21/2 years at 8% p.a..

- Mahima Singh says
  
  November 21, 2021 at 3:11 am
  
  P = Rs. 7500, R = 8% p. a., N = 2 years
  
  Compound interest = P(1 + R)n – P
  = P[(1 + R)n – 1]
  = 7500 (1.082 – 1)
  = Rs. 1248
  
  Simple interest = PNR/100
  = 7500 × 8 × 2/100
  = Rs. 1200
  Difference = 1248 – 1200 = Rs. 48
  
Godgiftoweshi says

July 8, 2020 at 9:48 am

A trader needs # 300000 to improve he business, she then deposited #110000 in her savings account at5 percent per annum compound interest. She then adds #50000 to her savings account st the end of each year.
1.find total savings for 3 yrs.
2.how much is her savings greater or less than #300000 at the end of the third yr.
Pls solve it for me pls.

- Mahima Singh says
  
  November 21, 2021 at 3:14 am
  
  she deposits 110,000.
  at end of first year, she has 110,000 * 1.05 = 115,500.
  she adds 50,000 to get 165,500.
  at end of second year, she has 165,500 * 1.05 = 173,775.
  she adds 50,000 to get 223,775.
  at end of third year, she has 223,775 * 1.05 – 234963.75.
  she adds 50,000 to get 284,963.75.
  she needs 300,000
  she is short by 300,000 – 284,963.75 = 15,036.25
  
Vandana parmar says

August 11, 2020 at 7:57 am

calculate C.I on Rs.500 for 30year at 2% per annum

SARAH says

February 15, 2021 at 4:54 pm

P=10000
T=6months
R=10%
Interest is compounded annually…
Can u please help me solve this question!!!

rajesh kumar neupane says

May 31, 2022 at 12:46 pm

what sum must be deposited today at 10% p.a. compounded quarterly if the goal is to have a compounded amount of Rs 50000, 6 years from today? how much interest will be earned during this period.