Page 128 - ICSE Math 8

Computation of compound interest when the interest is compounded quarterly

R
If, P = Principal R% = % per quarter Maths Info
4
 R  4n When the interest is compounded
n = 4n A = P 1+ 4 100×   quarterly, the rate of interest becomes



R 
4
 
 R  4n    R  4n  quarter   % and the time becomes

Then, C.I. = A – P = P 1+ 4 100×   – P = P    1+ 400 − 1  four times (4n).


 
 

Example 12: Manav took a loan of ` 3,90,625 from a private financer. The financer charges interest at the
rate of 16% per annum, compounded quarterly. What amount will discharge Manav’s debt after
1 year? Also find the compound interest on the principal.
Solution: P = ` 3,90,625, R = 16% per annum, n = 1 year
 R  4n  16  41×
\ Amount after 1 year = P 1+ 400 = ` 3,90,625 1+ 400 





 1  4  26 4
= ` 3,90,625 1+ 25  = ` 3,90,625 ×   25 


 26 26 26 26
= ` 390 625 × 25 × 25 × 25 × 25  = ` 4,56,976
,
,


An amount of ` 4,56,976 will discharge Manav from his debt after 1 year.
\ C.I. = A – P = ` (4,56,976 – 3,90,625) = ` 66,351
Computation of compound interest when the interest is compounded annually but the rates are
different for different years
If, P = Principal
R %, R %, R %, ......, R % = rates of interest for 1st, 2nd, 3rd, ......, nth year respectively
1
2
3
n
Ê R ˆ Ê R ˆ Ê R ˆ Ê R ˆ
n
A = P 1+ 100¯ Ë 1+ 100¯ Ë 1+ 100¯ ˜ ...... 1+ 100¯ ˜ then C.I. = A – P
1
2
3
˜ Á
Á
Á
˜ Á
Ë
Ë
Example 13: Find the amount and compound interest on ` 6,000 in 3 years if the rate of interest is 5% for
the first year, 4% for the second year and 3% for the third year.
Solution: P = ` 6,000, R = 5% per annum, R = 4% per annum, R = 3% per annum
2
3
1
Ê R ˆ Ê R ˆ Ê R ˆ
2
1
Amount after 3 years = P 1+ 100¯ Ë 1+ 100¯ Ë 1+ 100¯
3
Á
˜
˜ Á
˜ Á
Ë
Ê 5 ˆ Ê 4 ˆ Ê 3 ˆ
= ` 6,000 1+ 100¯ Ë 1+ 100¯ Ë 1+ 100¯
˜ Á
˜
Á
˜ Á
Ë
Ê 1 ˆ Ê 1 ˆ Ê 3 ˆ 21 26 103
= ` 6,000 1+ 20¯ Ë 1+ 25¯ Ë 1+ 100¯ = ` 6,000 × 20 × 25 × 100
˜
˜ Á
Á
˜ Á
Ë
= ` 6,748.56
\ C.I. = A – P = ` (6,748.56 – 6,000) = ` 748.56
Computation of compound interest when the interest is compounded annually but the time is
in fraction Ê lR ˆ
n
l Ê R ˆ Á ˜
If, P = Principal R% = rate of interest per annum Time = n years A = P 1+ 100¯ 1+ m ˜
Á
˜ Á
Ë
then C.I. = A – P m Á Ë 100 ˜
¯
116