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Amount when Rates of Interest are Different for Different Years

When rates of interest for the successive fixed period are R %, R %, R % ..., then the final amount
2
1
3
 R   R   R 
2
1
A is given by A = P 1 + 100  1 + 100  1 + 100 .....
3


 
 

When interest is compounded half-yearly
Let principal = P, time (T) = n years, rate of interest = R% p.a. If the interest is compounded half-
yearly. Then, Rate of interest =  R % per half-year and Time = (2n) half-years. And the amount
 

2 
(A) and the compound interest (CI) is calculated by
R 

A = P 1 + 200 2n ,




R 

R 
C.I. = A – P = P 1 + 200 2n − P = P   1 + 200 2n − 1   






 
When interest is compounded quarterly
Let principal = P, time = n years, rate of interest = R% p.a. if the interest is compounded quarterly.
R
Then, Rate of interest =    % per quarter and Time = (4n) quarters. And the amount (A) and the

4 
compound interest (CI) is calculated by
R 

A = P 1 + 400 4n



 R  4n   R  4n 
C.I. = A – P = P 1 + 400 − P = P   1 + 400 − 1  





 
Example 23: Find the amount on `12,000 for 2 years, if interest is compounded annually at 8% p.a.
Solution: Here, P = `12,000, R = 8% p.a., T = 2 years
 R  T
Using the formula, A = P 1 + 100  , we get amount after 2 years



8 

2 
A = 12000 1 + 100  2 = 12000 1 + 25 2 = 12000 × 27 × 27 = `13996.80





25
25
So, Amount after 2 years = `13,996.80.
1
Example 24: Find the compound interest on `5,12,000 for 3 years, compounded annually at 12 % p.a.
1 2
Solution: Here, P = `5,12,000, R = 12 % p.a. and T = 3 years
2
 R  T
Using the formula, A = P 1 + 100  , we get


25


Amount after 3 years = 512000 1 + 2 100   3 = 512000 1 + 1 3



8


×
9 9 9 729
= 512000 × × × = 512000 × = `7,29,000
8 8 8 512
So, Compound Interest = `(7,29,000 – 5,12,000) = `2,17,000.
Mathematics-8 182