Compound Interest Formula
Simple interest is calculated on the principal or on the original amount of the loan. If principle = p, rate of interest = r, time = t, Then SI = (p * t * r)/100. But Compound interest is calculated on the principal amount and also on the accumulated interest of previous periods. It is also known as “interest on interest”.
- When interest compounded annually, Amount A = P (1 + R/100)n
- When interest compounded Half-yearly, Amount A = P {1 + (R/2)/100}2n
[Half-yearly: Calculating twice a year so rate(R) is divided by 2 and number of years (n) multiplied by 2]
Now Compound interest (C.I) = Amount – Principle
C.I = P[(1 + r/100)n – 1]
Some Key Points and Useful Formulas
Final Amount
The Interest at end of a certain period is added to the original sum(P) to get the amount. Now This amount becomes the principle for the next period. This process will be repeated until the amount for the last period is found which is the Final Amount (A).
Basic rules for the compound interest :
Rule 1:
If rate of interest is R1% For first year , R2% for second year and R3% for the third year , then
A=P*(1+R1/100)(1+R2/100)(1+R3/100)
Rule 2 :
IF principle =P, rate=R% , AND time=T years then
a) If the interest is compounded annually:
A=P*(1+R/100)^T
b) If the interest is compounded half yearly (two times in year)
A=P*(1+R/100)^2T
c) If the interest is compounded quarterly (four times in year):
A=P*(1+R/100)^4T
Rule 3: IF difference between simple interest and compound interest is given :
- If the difference between compound interest and simple interest on a certain sum of money for 2 years at R% rate is given then
sum = difference *(100/R)^2
- If the difference between compound interest and simple interest on a certain sum of money for 2 years at R% rate is asked then
difference = p (r/100)^2
- If the difference between simple interest and compounded interest on a certain sum of money for 3 years at R% rate is given then
p(r/100)^2(300+r/100)
RULE 4: if sum A becomes B in T1 years at compounded interest , then after T2 years
sum = (b^t2/t1)/ A^t2/t1-1
TRICK: If a certain sum becomes ‘m’ in times in ‘t’ years the rate of compound interest r is equal to
=100[(m^1/t -1]
Compound Interest Of Consecutive Years
If we have the same sum and at the same rate of interest. The C.I of a particular year is always more than C.I of Previous Year. (C.I of 3rd year is greater than C.I of 2nd year). The difference between C.I for any two consecutive years is the interest of one year on C.I of the preceding year.
C.I of 3rd year – C.I of 2nd year = C.I of 2nd year * r * 1/100
[r = rate; t = 1 year]
The difference between the amounts of any two consecutive years is the interest of one year on the amount of the preceding year.
Amount of 3rd year – Amount of 2nd year = Amount of 2nd year * r * 1/100
[r = rate; t = 1 year]
Key Results
When we have the same sum and same rate,
C.I for nth year = C.I for (n – 1)th year + Interest for one year on C.I for (n – 1)th year.
C.I for 6th year = C.I for 5th year + Interest for one year on C.I for 5th year
For Amount,
The amount for 6th year = Amount for 5th year + Interest for one year on Amount for 5th year.
Some Other Applications of Amount
Growth: This is mainly used for growth if industries related.
Production after n years = initial production * (1 + r/100)n
Depreciation: When the cost of a product depreciates by r% every year, then its value after n years is
Present value * (1 + r/100)n
Population Problems: When the population of a town, city, village increases at a certain rate per year.
Population after n years = present population * (1 + r/100)n
Examples
Example 1: Find the Compound Interest when principal = Rs 6000, rate = 10% per annum and time = 2 years?
Solution:
Interest for first year = (6000 * 10 * 1)/100 = 600
Amount at the end of first year = 6000 + 600 = 6600
Principal interest for second year = (6600 * 10 * 1) / 100 = 660
Amount at the end of second year = 6600 + 660 = 7260
Compound Interest = 7260 – 6000 = 1260
Example 2: What will be the compound interest on Rs 8000 in two years when the rate of interest is 2% per annum?
Solution:
Given principal P = 8000
rate r = 2%
time = 2years
by formula ,
A = P (1 + R/100)n
= 8000 (1 + 2/100)2
= 8000 (102/100)2
= 8323
Compound interest = A – P
= 8323 – 8000
= Rs 323
Example 3: Hari deposited Rs. 4000 with a finance company for 2 years at an interest of 5% per annum. What is the compound interest that Rohit gets after 2 years?
Solution:
Given
principal P = 4000
rate r = 5%
time = 2years
By formula ,
A = P (1 + R/100)n
= 4000 (1 + 5/100)2
= 4000 (105/100)2
= 4410
Compound Interest = A – P
= 4410 – 4000
= 410
Example 4: Find the compound interest on Rs. 2000 at the rate of 4 % per annum for 1.5 years. When interest is compounded half-yearly?
Solution:
Given,
principal p = 2000
rate r = 4%
time = 1.5 ( i.e 3 half years )
by formula ,
A = P (1 + R/200)2n
= 2000 (1 + 4/200)3
= 2000 (204/200)3
= 2122
Compound Interest = A – P
= 2122 – 2000
= 122
Example 5: What is the compound interest on 10000 for one year at the rate of 20% per annum, if the interest compounded quarterly?
Solution:
Given,
Principal P = Rs 10000
rate R = 12% (12/4 = 3 % per quarter year)
Time = 1 year (1 * 4 = 4 quarters)
by formula,
A = P (1 + R/100)n
= 10000 (1 + 3/100)4
= 10000 (103/100)4
= 11255
Compound Interest = A – P
= 11255 – 10000
= 1255
Example 6: Find the Compound interest at the rate of 5% per annum for 2 years on that principal which in 2 years at the rate of 5% per annum given Rs. 400 as simple interest?
Solution:
Given
Simple interest SI = 400
rate R = 5%
time T = 2years
by formula,
Simple interest = (P * T * R)/100
P = (SI * 100)/T * R
= (400 * 100)/2 * 5
= 40000/10
= Rs 4000
Rate of Compound Interest = 5%
time = 2 years
by formula ,
A = P (1 + R/100)
= 4000 (1 + 5/100)
= 4410
Compound Interest = A – P
= 4410 – 4000
= 410
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