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Compound Interest

When interest at the end of each conversion period is added to the principal and the amount thus obtained is taken as the principal for the next period, the interest obtained is called Compound Interest.
 

By conversion period, we mean the fixed interval of time at the end of which interest is calculated.
 

For example, if the interest is compounded semi-annually, then every six months make 1 conversion period. If the interest is compounded monthly, then every month forms a conversion period and so on.
 

Suppose ₹1000 is borrowed for 2 years at 10% compound interest.
 

Then, interest for the first year will be Description: 56324.png 
 

Now, this interest will be added to the original principal giving us ₹(1000 + 100) = ₹1100.
 

This will become the principal for the second year. Hence, interest for the second year will be Description: 56349.png Thus, the total interest for the two years becomes ₹210.
 

If we had used simple interest for the whole period, interest would have been ₹200.

 

Consider the following terms and notations:
 

Description: 56371.png 

 

Then,

  • Description: 56383.png
  • Description: 56389.png
  • Description: 56395.png

Note: In general, if the interest is compounded m times a year, then the amount after n years will be

 

Description: 56803.png 

  • When interesst is compounded annually but time is in fraction, say Description: 62428.png years, then
     
    Description: 56819.png
  • ƒ When rates are different for different years, say R1%, R2%, R3% for 1st, 2nd and 3rd year respectively, then
    Description: 56831.png

Example
Find the compound interest on ₹6,000 at 3% p.a. for 3 years, compounded annually.
Solution
Given, P = ₹6000 R = 3% p.a. n = 3 years
Description: 56838.png 
Amount = ₹6,556.362
C.I. = ₹(6556.362 - 6000) (Since C.I. = A - P)
C.I. = ₹556.362

 

Example
In what time will ₹1000 become ₹1331 at 10% p.a. compounded annually?
Solution
Given, P = ₹1000 A = ₹1331 R = 10% p.a.
Time is unknown. Taking time as ‘n’ and substituting values in the formula for amount,
Description: 56880.png 
Therefore, the time period comes out to be 3 years.

 

Example
If ₹500 amounts to ₹583.20 in 2 years compounded annually, find the rate of interest p.a.
Solution
Given, P = ₹500 A = ₹583.20 n = 2 years Rate is unknown.
Description: 56906.png 

 

Example
Find the C.I. on ₹2,000 at 5% p.a. for 2 years, the interest being compounded half yearly.
Solution
Given, P = ₹2,000 R = 5% p.a. n = 2 years
Interest is being compounded half-yearly which means there are two conversion periods per year. Rate will become R/2 and time will become 2n.
Description: 56949.png 
C.I. = A - P = 2207.62 - 2000 = ₹207.62

 

Example
Find the amount on ₹2,000 at 20% p.a. in 2 years being compounded quarterly.
Solution
Given, P = ₹2000, R = 20% p.a. n = 2 years
Interest is compounded quarterly, which means 4 times a year. Hence, rate will become R/4 and conversion periods will be 4n.
Description: 56971.png 

 

Example
Find the C.I. on ₹8,000 at 15% p.a. for 2 years 4 months, compounded annually.
Solution
Given,Description: 56990.png years; R = 15%
Description: 56997.png 
Description: 57032.png 

 

Example
What will be the amount if a sum of ₹3,000 is placed at C.I. for 3 years while rate of interest for the first, second and third year is 4, 5 and 6 per cent, respectively?
Solution
Given, P = ₹3,000 T = 3 years R1 = 4% R2 = 5% R3 = 6%
Description: 57039.png 





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