Simple Interest

 Interest is the extra money paid by institutions like banks or post offices on money deposited (kept) with them. Interest is also paid by people when they borrow money. 

 

With Simple interest, the interest is calculated on the same amount of money in each time period, and, therefore, the interest earned in each time period is the same. i.e., If the interest on a sum borrowed for certain period is reckoned uniformly, then it is called simple interest.

Let the principal = P, Rate = R% per annum (p.a) and Time = T years. Then ,

 

Example - 1

A sum of Rs 10,000 is borrowed at a rate of interest 15% per annum for 2 years. Find the simple interest on this sum and the amount to be paid at the end of 2 years.

 

 Solution :

 On Rs 100, interest charged for 1 year is Rs 15.

 So, on Rs 10,000, interest charged =  

                         Interest for 2 years =   

 Amount to be paid at the end of 2 years = Principal + Interest

Compound Interest

Compound interest is calculated on the principal plus the interest for the previous period. The principal amount increases with every time period, as the interest payable is added to the principal. This means interest is not only earned on the principal, but also on the interest of the previous time periods. 

Therefore, the compound interest calculated is more than the simple interest on the same amount of money deposited.
  

Let us take an example and find the interest year by year. Each year our sum or principal changes.

 

Calculating Compound Interest

 

Example - 2

A sum of Rs 20,000 is borrowed by Heena for 2 years at an interest of 8% compounded annually. Find the Compound Interest (C.I.) and the amount she has to pay at the end of 2 years. 

Aslam asked the teacher whether this means that they should find the interest year by year. The teacher said ‘yes’, and asked him to use the following steps :

 

1.  Find the Simple Interest (S.I.) for one year.

 Let the principal for the first year be P₁. Here, P₁ = Rs 20,000

 SI₁ = SI at 8% p.a. for 1st year = Rs

 

 2.  Then find the amount which will be paid or received. This becomes principal for the next year.

 Amount at the end of 1st year = P₁ + SI₁ = Rs 20000 + Rs 1600

 = Rs 21600 = P₂ (Principal for 2nd year)

 

 3.  Again find the interest on this sum for another year.

 SI₂ = SI at 8% p.a.for 2nd year = Rs

 =  Rs 1728

 

 4. Find the amount which has to be paid or received at the end of second year.

 Amount at the end of 2nd year = P₂ + SI₂

 = Rs 21600 + Rs 1728

 = Rs 23328

 Total interest given = Rs 1600 + Rs 1728

 = Rs 3328

 

Reeta asked whether the amount would be different for simple interest. The teacher told her to find the interest for two years and see for herself.

SI for 2 years 

Reeta said that when compound interest was used Heena would pay Rs 128 more. Let us look at the difference between simple interest and compound interest. We start with Rs 100. Try completing the chart.

 

		Under Simple Interest	Under Compound Interest
First year	Principal	Rs 100.00	Rs 100.00
	Interest at 10%	Rs 10.00	Rs 10.00
	Year-end amount	Rs 110.00	Rs 110.00
Second year	Principal	Rs 100.00	Rs 110.00
	Interest at 10%	Rs 10.00	Rs 11.00
	Year-end amount	Rs(110 + 10) = Rs 120	Rs 121.00
Third year	Principal	Rs 100.00	Rs 121.00
	Interest at 10%	Rs 10.00	Rs 12.10
	Year-end amount	Rs(120 + 10) = Rs 130	Rs 133.10

 

 Note that in 3 years, 

Interest earned by Simple Interest = Rs (130 – 100) = Rs 30, whereas,

Interest earned by Compound Interest = Rs (133.10 – 100) = Rs 33.10

Note also that the Principal remains the same under Simple interest, while it changes year after year under compound interest.

 

Deducing a Formula for Compound Interest

Example-3

Zubeda asked her teacher, ‘Is there an easier way to find compound interest?’ The teacher said ‘There is a shorter way of finding compound interest. Let us try to find it.’

Suppose P₁ is the sum on which interest is compounded annually at a rate of R % per annum.

Let P₁ = Rs 5000 and R = 5 % per annum. Then by the steps mentioned above

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Proceeding in this way the amount at the end of years will be

 

Or, We can say  

So, Zubeda said, but using this we get only the formula for the amount to be paid at the end of years, and not the formula for compound interest.

Aruna at once said that we know CI = A – P, so we can easily find the compound interest too.

 

Example - 4

 

Find CI on Rs 12600 for 2 years at 10 %  per annum compounded annually.

 

Solution:

 We have, where Principal (P) = Rs 12600, Rate (R) = 10,

 Number of years (n) = 2

 

 

 

 Rate Compounded Annually or Half Yearly (Semi Annually)

 You may want to know why ‘compounded annually’ was mentioned after ‘rate’. Does it mean anything?

 It does, because we can also have interest rates compounded half yearly or quarterly. Let us see what happens to Rs 100 over a period of one year if an interest is compounded annually or half yearly.

The time period after which the interest is added each time to form a new principal is called the conversion period. When the interest is compounded half yearly, there are two conversion periods in a year each after 6 months. In such situations, the half yearly rate will be half of the annual rate. What will happen if interest is compounded quarterly? In this case, there are 4 conversion periods in a year and the quarterly rate will be one-fourth of the annual rate.

 

P = Rs 100 at 10% per annum compounded annually	P = Rs 100 at 10% per annum compounded half yearly
The time period taken is 1 year	The time period is 6 months or year

 

 Example - 5

 Find CI paid when a sum of Rs 10,000 is invested for 1 year and 3 months at     per annum compounded annually.

 Solution :

 Mayuri first converted the time in years.

 

 Mayuri tried putting the values in the known formula and came up with:

 

 Now she was stuck. She asked her teacher how would she find a power which is fractional?

 The teacher then gave her a hint:

 Find the amount for the whole part, i.e., 1 year in this case. Then use this as principal to get simple interest for   year more.Thus,   

  

                       

Now this would act as principal for the next year. We find the SI on Rs 10,850  for  year

 

       

 

         

 

Interest for first year = Rs 10850 – Rs 10000 = Rs 850

And, interest for the next

Therefore, total compound Interest = 850 + 230.56 = Rs 1080.56.

 

 Applications of Compound Interest Formula

 

There are some situations where we could use the formula for calculation of amount in CI. Here are a few.

 

    (i)   Increase (or decrease) in population.

 

   (ii)   The growth of a bacteria if the rate of growth is known.

 

   (iii)   The value of an item, if its price increases or decreases in the intermediate years.

 

Example - 6

The population of a city was 20,000 in the year 1997. It increased at the rate of 5 % p.a. Find the population at the end of the year 2000.

 

Solution :

 There is 5% increase in population every year, so every new year has new population. Thus, we can say it is increasing in    compounded form.

 Population in the beginning of 1998 = 20000 (we treat this as the principal for the 1st year)

      Increase at

 Population in

        increase at

 

Population in

 

 

Increase at 

 

 

At the end of the population

 

Or,  Population at the end of

 

 

 

So, the estimated population

Aruna asked what is to be done if there is a decrease. The teacher then considered the following example.

 

Example - 7

 

A TV was bought at a price of Rs 21,000. After one year the value of the TV was depreciated by 5 %  (Depreciation means reduction of value due to use and age of the item). Find the value of the TV after one year

 

Solution:

 

Principal = Rs 21,000

 

Reduction = 5 %  of Rs 21000 per year

 

 

value at the end of 1 year = Rs 21000 – Rs 1050 = Rs 19,950

 

Alternately, We may directly get this as follows

 

value at the end of 1 year =