Quantitative Notes on Compound Interest
Now, Let’s discuss the basic difference between Simple Interest and Compound Interest.
Principal = 1000, rate of interest (r) = 10%, time = 3yrs
Simple Interest
SI for 1^{st} yr = (1000×10×1)/100 = 100,
SI for 2^{nd} yr = 100 (In SI it will be the same as 1^{st} yr)
SI for 3^{rd} yr = 100
Compound Interest:
CI for 1^{st} yr = 100
CI for 2^{nd} yr will not be same as 1^{st} yr because principal for 2^{nd} yr is the amount of 1^{st} yr.
So, CI (2^{nd} yr) = (1100×10×1)/100 = 110
CI for 3^{rd} yr will also not be the same as 1^{st} yr and 2^{nd} yr because principal for 3^{rd} yr is the amount of 2^{nd} yr.
principal (3^{rd} yr) = Amount (2^{nd} yr) = Principal(2^{nd} yr)+Interest(2^{nd} yr) = 1100+110 = 1210
CI (3^{rd} yr) = (1210×10×1)/100 = 121
Hence total CI for 3yrs = 100+110+121 = 331
Amount after 3 yrs = 1331
Interest is always calculated on the Principal. But in case of CI, Principal is get changed every year.
If we calculate it by net rate concept then the Principal will remain same.
Concept1: How to calculate net CI rate for 2 years?
Let rate is r% per annum for 2 years
Net CI rate for 2yrs can be calculated by = 2r+(r^{2}/100)
If rate is 1%, net CI rate for 2yrs = 2×1+(1^{2}/100) = 2.01%
If rate is 3%, net CI rate for 2yrs = 2×3+(3^{2}/100) = 6.09%
If rate is 14%, net CI rate for 2yrs = 2×14+(14^{2}/100) = 29.96%
We suggest you to learn the table given below:
% Rate per annum | Net CI rate for 2 yrs | % Rate per annum | Net CI rate for 2 yrs |
2% | 4.04% | 9% | 18.81% |
3% | 6.09% | 10% | 21% |
4% | 8.16% | 11% | 23.21% |
5% | 10.25% | 12% | 25.44% |
6% | 12.36% | 13% | 27.69% |
7% | 14.49% | 14% | 29.96% |
8% | 16.64% | 15% | 32.25% |
Concept2: How to calculate net CI rate for 3 years?
Let rate is r% per annum for 3 years
Net CI rate for 3yrs can be calculated = 3r+3(r^{2}/100)+1(r^{3}/10000)
If rate is 3% p.a., net CI rate for 3 yrs
= 3×3+3(9/100)+1(27/10000)
= 9+.27+.0027 = 9.2727
If rate is 12% p.a., net CI rate for 3 yrs
= 3×12+3(144/100)+1(1728/10000)
= 36+4.32+.1728
= 40.4928
Representation while calculating net rate %.
Let’s calculate it for the rate 3% p.a.
write, r/r^{2}/r^{3} = 3/9/27
then,3r/3r^{2}/1r^{3} = 9/27/27
= 9.2727
We suggest you learn the table given below:
% Rate per annum | Net CI rate for 3 yrs | % Rate per annum | Net CI rate for 3 yrs |
1% | 3.31% | 6% | 19.1016% |
2% | 6.1208% | 7% | 22.5043% |
3% | 9.2727% | 8% | 25.9712% |
4% | 12.4864% | 9% | 29.5029% |
5% | 15.7625% | 10% | 33.10% |
Concept3: If the r% p.a. is in fraction:
Example: if the rate is 16(2/3) % and the principal is 216, then calculate CI for 2yrs and 3yrs.
Solution: We can write 16(2/3)% = 1/6 (Discussed in percentage study notes)
For 2 years
216×(1/6)= 36, Now multiply 36 by 2 = 72
36× (1/6) = 6 , multiply 6 by 1 = 6
Add both the above value = 72+6 = 78
CI for 2yrs = 78
For 3 years
216×(1/6) = 36, Now multiply 36 by 3 = 108
36× (1/6) = 6, multiply 6 by 3 = 18
6× (1/6) = 1, multiply 1 by 1 = 1
Add all the above values = (108+18+1)= 127
CI for 3yrs = 127
Concept4: When r% is given p.a. and CI has to be calculated half-yearly or quarterly basis.
Yearly | factor | r% (per annum) | Time (n yrs) |
Half yearly | 6months = (6/12) =1/2 | Factor× r% = (r/2) % | 2n |
Quarterly | 3months= (3/12) =1/4 | (1/4) × r% = (r/4) % | 4n |
9 months | 9months= (9/12) = 3/4 | (3/4) × r% = (3r/4) % | 4n/3 |
8 months | 8months= (8/12) = 2/3 | (2/3) × r% = (2r/3) % | 3n/2 |
Example: If r% = 10% per annum. Find the CI on 5000 for 2 years if it is compounded half-yearly.
Solution: Rate is calculated half yearly so new r% = (10/2)% = 5%
Given time is 2 yrs, acc.to half yearly, it will be 2×2 = 4
Now we have to calculate CI for 4yrs @ 5%
We know 5% = (1/20)
So, 5000×(1/20) = 250, multiply 250 by 4 = 1000
250× (1/20) = 12.5, multiply 12.5 by 6 = 75
12.5× (1/20) = 0.625, multiply 0.625 by 4 = 2.5
0.625× (1/20)= .03125 multiply .03125 by 1 = .03125
Add all the above values
(1000+75+2.5+0.03125)
= 1077.53125
Concept 5: When different rates are given for 2 years.
If a% is given for 1^{st} year and b% is given for 2^{nd} year.
Net rate of CI for 2 yrs = (a+b+ab/100) % (discussed in percentage study notes)
Note: The net CI rate will be the same if b% is given for 1^{st} year and a% is given for 2^{nd} year.
Example: If principal is 1000 Rs and r(1^{st} yr) = 4% and r(2^{nd} yr) = 6%. Calculate the CI after 2yrs.
Solution:
Net CI rate = 4+6+(4×6)/100
= 10.24%
Now CI = 1000×10.24% = 102.4 Rs
Concept6: When difference between CI and SI is given.
We know, net CI for 2yrs = 2r+(r^{2}/100) %,
net SI for 2 yrs = 2r%
So, difference = (r^{2}/100)%
Example: If the difference between CI and SI is Rs.10 and the principal is Rs.1000.Calculate the rate % per annum.
Solution: difference = 10 Rs.
So difference% = (10/1000)×100 = 1%
We know that, if rate of interest is 10%
then, net CI rate (2yrs) = 21%
net SI rate (2yrs) = 20%
difference = 1%
Definitely we can say r% per annum is 10%.
Example: Calculate the difference between CI and SI for 3 yrs if Principal = 8000 and r = 6% p.a.
Solution: Net rate CI(3yrs) = 19.1016%
Net rate SI (3yrs) = 18%
Difference = 1.1016%
So, difference = 1.1016% of 8000 = 88.128
Example: If difference between CI and SI is Rs.64 and r = 8% p.a.. Calculate the Principal and Amount?
Solution: If r = 8% p.a.
then, net rate CI (2yrs) – net rate SI (2yrs)
= 16.64% -16% = 0.64%
Given, difference is Rs. 64
So, 0.64% = 64
100% = 10000
Hence, Principal is 10000 Rs.
Amount = principal× (116.64%)= 10000× 116.64% = Rs.11664
Concept7: Calculation of Instalment
Example: A man borrowed Rs.8,400 at 10% p.a. CI. He pays equal annual repayment of X rs and clear off his debts in 2 yrs. What is the value of X?
For 3 yrs: If r% p.a. is given, convert it into fraction(a/b)
Example: A man borrowed Rs.1820 at 20% p.a. CI. He pays equal annual repayment of X rs and clear off his debts in 3 yrs. What is the value of X?
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