Simple and Compound Interest

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 ,

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»S«/mi»«mo».«/mo»«mi mathvariant=¨normal¨»I«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mfrac»«mrow»«mi mathvariant=¨normal¨»P«/mi»«mo»*«/mo»«mi mathvariant=¨normal¨»R«/mi»«mo»*«/mo»«mi mathvariant=¨normal¨»T«/mi»«/mrow»«mn»100«/mn»«/mfrac»«/math»

 

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 =  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»15«/mn»«mn»100«/mn»«/mfrac»«mo»§#215;«/mo»«mn»10000«/mn»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»1500«/mn»«/math»

                         Interest for 2 years =   «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»1500«/mn»«mo»§#215;«/mo»«mn»2«/mn»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»3000«/mn»«/math»

 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 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mn»20000«/mn»«mo»§#215;«/mo»«mn»8«/mn»«/mrow»«mn»100«/mn»«/mfrac»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»1600«/mn»«/math»

 

 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 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mn»21600«/mn»«mo»§#215;«/mo»«mn»8«/mn»«/mrow»«mn»100«/mn»«/mfrac»«/math»

 =  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  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mfrac»«mrow»«mn»20000«/mn»«mo»§#215;«/mo»«mn»8«/mn»«mo»§#215;«/mo»«mn»2«/mn»«/mrow»«mn»100«/mn»«/mfrac»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»3200«/mn»«/math»

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

 

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»1«/mn»«mo».«/mo»«mo»§nbsp;«/mo»«mi»S«/mi»«msub»«mi»I«/mi»«mn»1«/mn»«/msub»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mfrac»«mrow»«mn»5000«/mn»«mo»§#215;«/mo»«mn»5«/mn»«mo»§#215;«/mo»«mn»1«/mn»«/mrow»«mn»100«/mn»«/mfrac»«mo»§nbsp;«/mo»«mi»o«/mi»«mi»r«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»S«/mi»«msub»«mi»I«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mfrac»«mrow»«msub»«mi»P«/mi»«mn»1«/mn»«/msub»«mo»§#215;«/mo»«mi»R«/mi»«mo»§#215;«/mo»«mn»1«/mn»«/mrow»«mn»100«/mn»«/mfrac»«/math»

 

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«mi»o«/mi»«mo»,«/mo»«mo»§nbsp;«/mo»«msub»«mi»A«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»5000«/mn»«mo»+«/mo»«mfrac»«mrow»«mn»500«/mn»«mo»§#215;«/mo»«mn»5«/mn»«mo»§#215;«/mo»«mn»1«/mn»«/mrow»«mn»100«/mn»«/mfrac»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»o«/mi»«mi»r«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msub»«mi»A«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«msub»«mi»P«/mi»«mn»1«/mn»«/msub»«mo»+«/mo»«mi»S«/mi»«msub»«mi»I«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«msub»«mi»P«/mi»«mn»1«/mn»«/msub»«mo»+«/mo»«mfrac»«mrow»«msub»«mi»P«/mi»«mn»1«/mn»«/msub»«mi»R«/mi»«/mrow»«mn»100«/mn»«/mfrac»«/math»

 


 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»5000«/mn»«mo»§nbsp;«/mo»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mn»5«/mn»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mo»=«/mo»«mo»§nbsp;«/mo»«msub»«mi»p«/mi»«mn»2«/mn»«/msub»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«msub»«mi»p«/mi»«mn»1«/mn»«/msub»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mi»R«/mi»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mo»=«/mo»«msub»«mi»p«/mi»«mrow»«mn»2«/mn»«mo»§nbsp;«/mo»«/mrow»«/msub»«/math»

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo».«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»S«/mi»«msub»«mi»I«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»5000«/mn»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mn»5«/mn»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mo»§#215;«/mo»«mfrac»«mrow»«mn»5«/mn»«mo»§#215;«/mo»«mn»1«/mn»«/mrow»«mn»100«/mn»«/mfrac»«mo»§nbsp;«/mo»«mi»o«/mi»«mi»r«/mi»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»S«/mi»«msub»«mi»I«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mfrac»«mrow»«msub»«mi»P«/mi»«mn»2«/mn»«/msub»«mo»§#215;«/mo»«mi»R«/mi»«mo»§#215;«/mo»«mn»1«/mn»«/mrow»«mn»100«/mn»«/mfrac»«/math»

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mfrac»«mrow»«mn»5000«/mn»«mo»§#215;«/mo»«mn»5«/mn»«/mrow»«mn»100«/mn»«/mfrac»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mn»5«/mn»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msub»«mi»P«/mi»«mn»1«/mn»«/msub»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mi»R«/mi»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mo»§#215;«/mo»«mfrac»«mi»R«/mi»«mn»100«/mn»«/mfrac»«/math»

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mrow»«msub»«mi»P«/mi»«mn»1«/mn»«/msub»«mi»R«/mi»«/mrow»«mn»100«/mn»«/mfrac»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mi»R«/mi»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«/math»

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»A«/mi»«mn»2«/mn»«/msub»«mo»§nbsp;«/mo»«mo»=«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»5000«/mn»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mn»5«/mn»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mo»+«/mo»«mi»R«/mi»«mi»S«/mi»«mfrac»«mrow»«mn»5000«/mn»«mo»§#215;«/mo»«mn»5«/mn»«/mrow»«mn»100«/mn»«/mfrac»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mn»5«/mn»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mo»§nbsp;«/mo»«msub»«mi»A«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«msub»«mi»P«/mi»«mn»2«/mn»«/msub»«mo»+«/mo»«mi»S«/mi»«msub»«mi»I«/mi»«mn»2«/mn»«/msub»«/math»

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»5000«/mn»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mn»5«/mn»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mn»5«/mn»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«msub»«mi»P«/mi»«mn»1«/mn»«/msub»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mi»R«/mi»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mo»+«/mo»«msub»«mi»P«/mi»«mn»1«/mn»«/msub»«mfrac»«mi»R«/mi»«mn»100«/mn»«/mfrac»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mi»R«/mi»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«/math»

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»5000«/mn»«msup»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mn»5«/mn»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»=«/mo»«msub»«mi»P«/mi»«mn»3«/mn»«/msub»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«msub»«mi»P«/mi»«mn»1«/mn»«/msub»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mi»R«/mi»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mi»R«/mi»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«/math»

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mo»=«/mo»«mo»§nbsp;«/mo»«msub»«mi»P«/mi»«mn»1«/mn»«/msub»«mo»§nbsp;«/mo»«msup»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mi»R«/mi»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»=«/mo»«msub»«mi»P«/mi»«mn»3«/mn»«/msub»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«/mtable»«/math»

 

Proceeding in this way the amount at the end of«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§nbsp;«/mo»«mi»n«/mi»«mo»§nbsp;«/mo»«/math» years will be

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»A«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«msub»«mi»P«/mi»«mn»1«/mn»«/msub»«msup»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mi»R«/mi»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mi»n«/mi»«/msup»«/math»

 

Or, We can say   «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»A«/mi»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»P«/mi»«msup»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mi»R«/mi»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mi»n«/mi»«/msup»«/math»

So, Zubeda said, but using this we get only the formula for the amount to be paid at the end of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§nbsp;«/mo»«mi»n«/mi»«mo»§nbsp;«/mo»«/math» 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, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»A«/mi»«mo»=«/mo»«mi»P«/mi»«msup»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mi»R«/mi»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mi»n«/mi»«/msup»«mo»,«/mo»«/math» where Principal (P) = Rs 12600, Rate (R) = 10,

 Number of years (n) = 2

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»12600«/mn»«msup»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mn»10«/mn»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»12600«/mn»«msup»«mfenced»«mfrac»«mn»11«/mn»«mn»10«/mn»«/mfrac»«/mfenced»«mn»2«/mn»«/msup»«/math»

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»12600«/mn»«mo»§#215;«/mo»«mfrac»«mn»11«/mn»«mn»10«/mn»«/mfrac»«mo»§#215;«/mo»«mfrac»«mn»11«/mn»«mn»10«/mn»«/mfrac»«mo»§nbsp;«/mo»«mo»=«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»15246«/mn»«/math»

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«mi»I«/mi»«mo»=«/mo»«mi»A«/mi»«mo»-«/mo»«mi»P«/mi»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»15246«/mn»«mo»-«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»12600«/mn»«mo»=«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»2646«/mn»«/math»

 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 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/math» year
 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»1«/mn»«mo»=«/mo»«mi»R«/mi»«mi»S«/mi»«mfrac»«mrow»«mn»100«/mn»«mo»§#215;«/mo»«mn»10«/mn»«mo»§#215;«/mo»«mn»1«/mn»«/mrow»«mn»100«/mn»«/mfrac»«mo»=«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»10«/mn»«/math»  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»I«/mi»«mo»=«/mo»«mi»R«/mi»«mi»S«/mi»«mfrac»«mrow»«mn»100«/mn»«mo»§#215;«/mo»«mn»10«/mn»«mo»§#215;«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mrow»«mn»100«/mn»«/mfrac»«mo»=«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»5«/mn»«/math»
 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi»A«/mi»«mo»=«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»100«/mn»«mo»+«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»10«/mn»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»110«/mn»«/mtd»«/mtr»«/mtable»«/math»  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi»A«/mi»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»100«/mn»«mo»+«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»5«/mn»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»105«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»N«/mi»«mi»o«/mi»«mi»w«/mi»«mo»§nbsp;«/mo»«mi»f«/mi»«mi»o«/mi»«mi»r«/mi»«mo»§nbsp;«/mo»«mi»n«/mi»«mi»e«/mi»«mi»x«/mi»«mi»t«/mi»«mo»§nbsp;«/mo»«mn»6«/mn»«mo»§nbsp;«/mo»«mi»m«/mi»«mi»o«/mi»«mi»n«/mi»«mi»t«/mi»«mi»h«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mi»t«/mi»«mi»h«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»P«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»105«/mn»«/mtd»«/mtr»«/mtable»«/math»
   «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi»S«/mi»«mi»o«/mi»«mo»,«/mo»«mo»§nbsp;«/mo»«mi»I«/mi»«mo»=«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mfrac»«mrow»«mn»105«/mn»«mo»§#215;«/mo»«mn»10«/mn»«mo»§#215;«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mrow»«mn»100«/mn»«/mfrac»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»5«/mn»«mo».«/mo»«mn»25«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»a«/mi»«mi»n«/mi»«mi»d«/mi»«mo»§nbsp;«/mo»«mi»A«/mi»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»105«/mn»«mo»§nbsp;«/mo»«mo»+«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»5«/mn»«mo».«/mo»«mn»25«/mn»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»110«/mn»«mo».«/mo»«mn»25«/mn»«/mtd»«/mtr»«/mtable»«/math»

 

 Example - 5

 Find CI paid when a sum of Rs 10,000 is invested for 1 year and 3 months at  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»8«/mn»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»%«/mo»«/math»   per annum compounded annually.

 Solution :

 Mayuri first converted the time in years.

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»1«/mn»«mo»§nbsp;«/mo»«mi»y«/mi»«mi»e«/mi»«mi»a«/mi»«mi»r«/mi»«mo»§nbsp;«/mo»«mn»3«/mn»«mo»§nbsp;«/mo»«mi»m«/mi»«mi»o«/mi»«mi»n«/mi»«mi»t«/mi»«mi»h«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mn»1«/mn»«mfrac»«mn»3«/mn»«mn»12«/mn»«/mfrac»«mi»y«/mi»«mi»e«/mi»«mi»a«/mi»«mi»r«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mn»1«/mn»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mi»y«/mi»«mi»e«/mi»«mi»a«/mi»«mi»r«/mi»«mi»s«/mi»«/math»

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

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»A«/mi»«mo»=«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»10000«/mn»«msup»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mn»17«/mn»«mn»200«/mn»«/mfrac»«/mrow»«/mfenced»«mrow»«mn»1«/mn»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/mrow»«/msup»«/math»

 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 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/math»  year more.Thus,   «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»A«/mi»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»10000«/mn»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mn»17«/mn»«mn»200«/mn»«/mfrac»«/mrow»«/mfenced»«/math»

  

                       «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»10000«/mn»«mo»§#215;«/mo»«mfrac»«mn»217«/mn»«mn»200«/mn»«/mfrac»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»10«/mn»«mo»,«/mo»«mn»850«/mn»«/math»

Now this would act as principal for the next «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/math» year. We find the SI on Rs 10,850  for  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mo»§nbsp;«/mo»«/math» year

 

       «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»S«/mi»«mi»I«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mfrac»«mrow»«mn»10850«/mn»«mo»§#215;«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mo»§#215;«/mo»«mn»17«/mn»«/mrow»«mrow»«mn»100«/mn»«mo»§#215;«/mo»«mn»2«/mn»«/mrow»«/mfrac»«/math»

 

          «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mfrac»«mrow»«mn»10850«/mn»«mo»§#215;«/mo»«mn»1«/mn»«mo»§#215;«/mo»«mn»17«/mn»«/mrow»«mn»800«/mn»«/mfrac»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»230«/mn»«mo».«/mo»«mn»56«/mn»«/math»

 

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

And, interest for the next «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mi»y«/mi»«mi»e«/mi»«mi»a«/mi»«mi»r«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»230«/mn»«mo».«/mo»«mn»56«/mn»«/math»

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 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»5«/mn»«mo»§nbsp;«/mo»«mo»%«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mfrac»«mn»5«/mn»«mn»100«/mn»«/mfrac»«mo»§#215;«/mo»«mn»20000«/mn»«mo»§nbsp;«/mo»«mo»=«/mo»«mn»1000«/mn»«/math»

 Population in «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»1999«/mn»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mn»20000«/mn»«mo»+«/mo»«mn»1000«/mn»«mo»=«/mo»«mn»21000«/mn»«/math»

        increase at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»5«/mn»«mo»%«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mfrac»«mn»5«/mn»«mn»100«/mn»«/mfrac»«mo»§#215;«/mo»«mn»21000«/mn»«mo»=«/mo»«mn»1050«/mn»«/math»

 

Population in «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2000«/mn»«mo»=«/mo»«mn»21000«/mn»«mo»+«/mo»«mn»1050«/mn»«/math»

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mn»22050«/mn»«/math»

 

Increase at  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»5«/mn»«mo»§nbsp;«/mo»«mo»%«/mo»«mo»§nbsp;«/mo»«mfrac»«mn»5«/mn»«mn»100«/mn»«/mfrac»«mo»§#215;«/mo»«mn»22050«/mn»«/math»

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mn»1102«/mn»«mo».«/mo»«mn»5«/mn»«/math»

 

At the end of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2000«/mn»«/math» the population «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mn»22050«/mn»«mo»+«/mo»«mn»1102«/mn»«mo».«/mo»«mn»5«/mn»«mo»=«/mo»«mn»23152«/mn»«mo».«/mo»«mn»5«/mn»«/math»

 

Or,  Population at the end of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2000«/mn»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mn»20000«/mn»«msup»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mn»5«/mn»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«mn»3«/mn»«/msup»«/math»

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mn»20000«/mn»«mo»§#215;«/mo»«mfrac»«mn»21«/mn»«mn»20«/mn»«/mfrac»«mo»§#215;«/mo»«mfrac»«mn»21«/mn»«mn»20«/mn»«/mfrac»«mo»§#215;«/mo»«mfrac»«mn»21«/mn»«mn»20«/mn»«/mfrac»«/math»

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mn»23152«/mn»«mo».«/mo»«mn»5«/mn»«/math»

 

So, the estimated population «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mn»23153«/mn»«mo».«/mo»«/math»

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

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mfrac»«mrow»«mn»21000«/mn»«mo»§#215;«/mo»«mn»5«/mn»«mo»§#215;«/mo»«mn»1«/mn»«/mrow»«mn»100«/mn»«/mfrac»«mo»=«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»1050«/mn»«/math»

 

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 = «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»21000«/mn»«mo»§nbsp;«/mo»«mfenced»«mrow»«mn»1«/mn»«mo»-«/mo»«mfrac»«mn»5«/mn»«mn»100«/mn»«/mfrac»«/mrow»«/mfenced»«/math»

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»21000«/mn»«mo»§nbsp;«/mo»«mo»§#215;«/mo»«mo»§nbsp;«/mo»«mfrac»«mn»19«/mn»«mn»20«/mn»«/mfrac»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»S«/mi»«mo»§nbsp;«/mo»«mn»19«/mn»«mo»,«/mo»«mn»950«/mn»«/math»

 

 

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