Compound Interest with solved examples using shortcut method for competitive exams

Compound interest:

Basic formula:

A = P (1 + R/100)ⁿ

C.I = P [ (1 +r/100)ⁿ -1]

A= Amount

P = Principal

r = Rate

n = number of years

Rate of interest:

Type                                                                   Rate                                                       Time

yearly                                                                  R                                                             n

Half Yearly                                                         R/2                                                         2n

Quarterly                                                           R/4                                                         4n

Ex.  Find C.I on R-s. 16000 at the rate of 5% for 2 years, compounded annually?

Sol.Method 1:

C.I = A – P

A = 16000 (1+ 5/100)²

    = 16000 × 4441/400

A = 17640

C.I = 17640 -16000 = 1640

Method 2:

1-year interest 800       

2-year interest 800+   5% of 800  =   40

C.I is 800+800+40 = 1640

Few shortcut Formulae;

1.Difference between CI and SI for 2 years, CI – SI = P(R/100)² = P R²/ 10000

2.Difference between CI and SI for 3 years, CI – SI = P(R/100)² ×(R+300)/100

3.For finding CI percent of 2 years CI = 2R.R²

Ex. If R = 5% Find CI for 2 years.

Sol. Then the rate of interest is 2 × 5. 5² = 10.25%

4.For finding CI percent of 3 years, CI % = 3R.3R²/R³

5.CI for 4 years C.I = 2R.R²

   Then again use this formula again

Ex.R = 20% 

Sol. C.I = 2*20.(20)2 = 44%

C.I = 2*44.(44)2 = 88.1936% 

Move 19 to the left side and add it to 88 it becomes 107.36%

Ex. If R=5% Find CI for 3 years.

Sol. 3*5. 3 * 5² /5³ = 15.75/125 = 15.7625%.

Another method for finding CI is.

Ex.  R = 10%.

Sol.10% = 1/10, 10³ or 1000 is principal

100                     

100      20                   

100      10     1 

Then,

100+ 100 + 100 is                  SI

20 +10+ 1 is                           CI - SI

100+100+100+20 +10 + 1 is CI 

Few more solved Examples:

Ex.1 Find compound interest if Rate=10%,Time = 2 years, Simple interest is 1200 Rs.

Sol. Compound Interest = 2R.R² = 10.100 = 11%

SI = 20% = 1200 Rs

1% = 600 Rs.

C.I = 11% = 600 * 11 = 6600 Rs. Ans.

Ex.2 Any principle becomes 500 in 4 years and 800 in 8 years then find the principle.

Sol. 800/500 = [P(1+R/100)8]/[ P(1+R/100)4]

800/500 = (1+R/100)⁸/(1+R/100)⁴
8/5 = (1+R/100)⁴
now, 500 = (1+R/100)⁴
500 = p *5/8
P = 2500/8 

Method 2.

P = A²/A = (500*500)/800 = 2500/8
The amount of less year is the numerator
The amount of more years is the denominator

Ex.3 Any principle becomes 21780 Rs. in 2 years and 23958 in 3 years then find the rate.

Sol. 21780 = P(1+R/100)2

23958 = P(1+R/100)3

Divide both equations

23958/21780 = (1+R/100)

(23958/21780) - 1 = R/100

R = 10%

Method 2.

A3 = 23958

A2 = 21780

A3-A2 = 2178 is 10%  21780

R= 10% Ans.

Ex.4 Find the CI for 1 year 73 days with a rate of 20% and P=40000 Rs.

Sol.1 year 73 is equal to 1 year  and 1/5 year

Rate of percentage of 1 year 20% and 1/5 year is 4%

net effect = 20+4+0 (20*4)/100 = 24.8%

100% = 40000

24.8% = (40000/100)*24.8  Ans

Ex.5 Find the quarterly compound interest for 9 months with a 20% rate and 50000 Rs. principle.

Sol.Interest is quarterly so we divide 20% by 4  

R=5% = 1/20

P = (20)3 = 8000

400

400    20

400    40    1

1261 = CI

8000 = 50000

1=50000/8000

1261 = (50000*1261)/8000

CI =7881.25 Ans.

Method 2

R=5%,    R=5%,    R=5%  For 9 months 5% rate is per 3 month

suppose it is 5% per year for 3 years

CI for 3 years = 3*5. 3 * 5² /5³ = 15.75/125 = 15.7625%.

100% = 50000

15.7625 = (50000 * 15.7625)/100 = 7881..25 Ans

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