Compound Interest Formula: Compound interest is defined as the interest on a certain sum or amount, where the interest gets accrued successively for every year from the previous periods. People may have noticed that when a certain sum of money in a bank is kept on a savings account basis, the money gets increased every year due to the addition of the annual interest amount. It is because the bank’s interest is calculated on the previous year’s amount. It is known as interest compounded or Compound Interest (C.I.).

In this article, we have provided the compound interest formula along with some examples to help students become confident on this topic.

Compound Interest Formula: Overview

Compound interest is the interest calculated on the principal and the interest earned previously. Compounding is when the interest is calculated not on the principal amount, but also the interest earned in the previous periods. So, the total interest for the successive period includes the interest on principal plus interest in the prior period. It is called “interest on interest”.

It is different from Simple Interest (SI), in which previously accumulated interest is not added to the principal amount of the current period, so there is no compounding.
Compound interest is standard in finance and economics.

Let us understand what compound interest with an example is:
Suppose you borrow ₹5000 from a moneylender on a 10% per annum interest rate. You promise to return the money after two years.

Case 1: Simple Interest Formula:
We know that 10% of 5000 = 500. So at the end of the 1st year, the lender will get ₹500 extra as interest. Similarly, at the end of the 2nd year, the lender will get ₹500 as interest. So, the total interest at the end of 2 years = 500 + 500 = ₹1000. The total amount to return back is 5000 + 1000 = ₹6000.

Case 2: Compound Interest Formula:
At a 10% interest rate, the lender will get ₹500 extra as interest at the end of 1st year. For the 2nd year, the principal amount becomes ₹5000 + ₹500 = ₹5500.
So, for the 2nd year, the lender will get 10% of ₹5500 = ₹550 as the interest. The total interest at the end of 2 years = 500 + 550 = ₹1050. The total amount to return back is 5000 + 1050 = ₹6050.

Now you can see that compound interest gives more return on the same principal amount for an extended period of time.

Some of the real-life applications of compound interest are:
(i) To calculate the growth of bacteria.
(ii) To calculate the increase or decrease in population.
(iii) To determine the rise or depreciation in the value of an item.

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Compounding Interest Calculator

Here we have provided the formula of CI. With the formula provided below, you can quickly know how to calculate compound interest for any principal amount for years.

Compound Interest = Amount – Principal

The amount is calculated with the help of the following formula:

The general formula of compound interest in maths is:

\(C.I.\;=\;(A\;-\;P)\) \(=P(1+\frac rn)^{nt}-P\) \(=P\lbrack(1+\frac rn)^{nt}-1\rbrack\)

Here,
A = amount
P = principal
r = rate of interest
n = number of times interest is compounded per year
t = time (in years)

If the principal amount is annually compounded, the CI formula is:

\(C.I.\;=\;P\lbrack(1\;+\;\frac r{100})^t\;-\;1\rbrack\)

We also have the CI formula for half-yearly and quarterly, which we will discuss in the subsequent sections.

Compounding Interest Calculator Formula Derivation

Here we have derived the compound interest formula when compounded annually.

Let, Principal amount = P, Time = n years, Rate = r
Simple Interest (S.I.) for the first year:

\(SI_1=(\frac{P\times r\times t}{100})\)

Amount after first year = P + SI₁ = P + (P × r × t)/100 
= P(1+r/100) 
= P₂

\(SI_2=(\frac{P_2\times r\times t}{100})\)

Amount after second year = P₂ + SI₂
= P₂ + (P₂ × r × t)/100
= P₂(1 + r/100)
= P(1 + r/100)(1 + r/100)
= P(1+r/100)² 
Similarly, if we proceed further to n years, we can deduce:
A = P(1 + r/100)ⁿ
C.I. = (A – P)
= P[(1 + r/100)ⁿ  – 1]

Compounding Interest Calculator Half Yearly Formula

When Compound Interest is calculated for a time duration of half-year, we divide the rate by two and multiply the time by 2 in the general formula. So, the compound interest formula half-yearly becomes:

\(A\;=\;P\lbrack1\;+\;\frac{r/2}{100}\rbrack^{2t}\)

\(CI=\;P\lbrack1\;+\;\frac{r/2}{100}\rbrack^{2t}\;-\;P\)

Derivation:

Here we calculate the compound interest half-yearly on a principal, P kept for one year at an interest rate r % compounded half-yearly.

Since interest is compounded half-yearly, the principal amount will change at the end of the first 6 months. The interest for the next six months will be calculated on the amount remaining after the first six months. Simple interest at the end of the first six months,

SI₁ = (P × r × 1)/(100 × 2)
Amount at the end of the first six months,
A₁ = P + SI₁
= P + (P × r × 1)/(2 × 100)
= P[1 + r/(2 × 100)]
= P₂
Simple interest for the next six months, now the principal amount has changed to P₂
SI₂ = (P₂ × r × 1)/(100 × 2)
Amount at the end of 1 year,
A₂ = P₂ + SI₂
= P₂ + (P₂ × r × 1)/(2 × 100)
= P₂[1 + r/(2 × 100)]
= P(1 + r/2×100)(1 + r/2×100)
= P[1 + r/(2 × 100)]²
Now we have the final amount at the end of 1 year:
A = P[1 + r/(2 × 100)]²
Rearranging the above equation,
A = P[1 + (r/2)/100)^2×1
Let r/2 = r′; 2t = t′, the above equation can be written as, [for the above case t = 1 year]
A = P(1 + r′/100)^t′

Compound Interest Formula When Amount Is Compounded Quarterly

Here we have provided the Compound Interest Formula when the amount is compounded quarterly. When the rate is compounded quarterly, we divide the rate by four and multiply the time by 4 in the general formula.

Compound Interest Quarterly Formula:

\(A\;=\;P\lbrack1\;+\;\frac{r/4}{100}\rbrack^{4t}\)

\(CI=\;P\lbrack1\;+\;\frac{r/4}{100}\rbrack^{4t}\;-\;P\)

Here,
A = Amount
P = Principal
C.I. = Compound interest
r = Rate of interest per year
t = Number of years

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Compounding Interest Calculator Formula Continuous

Continuously compounded interest is the mathematical limit of the general compound interest formula, with interest compounded many times each year.

In other words, you are paid every possible time increment. Mathematically, the formula is:

\(CCI=\lim_{n\rightarrow\infty}P\lbrack(1+\frac rn)^{nt}-1\rbrack\)

Here,
CCI = Continuous Compound Interest
P = Principal
r = Interest Rate
t = Time (in years)
n = Number of times interest is compounded per year

Now that we have provided the compound interest formulas, let’s have a summary of the formulas in the table below:

Time	Compound Interest Formula	Amount
1 year [Compounded annually]	P(1 + r)t – P	P(1 + r)t
6 months [Compounded half yearly]	P[1 + (r/2)2t] – P	P[1 + (r/2)2t]
3 months [Compounded quarterly]	P[1 + (r/4)4t] – P	P[1 + (r/4)4t]
1 month [Monthly compound interest formula]	P[1 + (r/12)12t] – P	P[1 + (r/12)12t]
365 days [Daily compound interest formula]	P[1 + (r/365)365t] – P	P[1 + (r/365)365t]

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How To Find Compound Interest?

Compound interest can be found when we have the principal amount, rate of interest, time, and the number of times the interest is compounded.

Using the formula for compound interest, we can substitute all the values in the formula and get the result. Sometimes, the value of compound interest is given, and we have to deduce other values such as the final amount, principal amount, or rate of interest.

Solved Compound Interest Formula Examples

We have provided compound interest formula examples with solutions to help you understand the concepts in a better manner:

Q.1: Rohit deposited Rs. 8000 with a finance company for 3 years at an interest of 15% per annum. What is the compound interest that Rohit gets after 3 years?
Ans: Principal, P = Rs 8000
Rate, r = 15%
Time, t = 3 years
By using the formula,
A = P(1 + r/100)ⁿ
= 8000 (1 + 15/100)³
= 8000 (115/100)³
= Rs 12167
∴ Compound Interest = (A – P)
= Rs 12167 – Rs 8000
= Rs 4167

Q.2: Find the compound interest on Rs. 160000 for one year at the rate of 20% per annum, if the interest is compounded quarterly?
Ans: Principal (p) = Rs 160000
Rate, r = 20%
= 20/4
= 5% (for quarter year)
Time = 1year
= 1 × 4
= 4 quarters
By using the formula,
A = P (1 + R/100) ⁿ
= 160000 (1 + 5/100)⁴
= 160000 (105/100)⁴
= Rs 194481
∴ Compound Interest = (A – P)
= Rs 194481 – Rs 160000
= Rs 34481

Q.3: The count of a certain breed of bacteria was found to increase at the rate of 2% per hour. Find the bacteria at the end of 2 hours if the count was initially 600000?
Ans: Since the population of bacteria increases at the rate of 2% per hour, we use the formula,
A = P(1 + r/100)ⁿ
Thus, the population at the end of 2 hours = 600000(1 + 2/100)²
= 600000(1 + 0.02)² 
= 600000(1.02)² 
= 624240

Q.4: Roma borrowed Rs. 64000 from a bank for 1½ years at the rate of 10% per annum. Compare the total compound interest payable by Roma after 1½ years, if the interest is compounded half-yearly?
Ans: Principal, P = Rs 64000
Rate, r = 10 %
= 10/2 % (for half a year)
Time = 1 ½ years = 3/2 × 2 = 3 (half year)
By using the formula,
A = P (1 + r/100) ⁿ
= 64000 (1 + 10/2×100)³
= 64000 (210/200)³
= Rs 74088
∴ Compound Interest = (A – P)
= Rs 74088 – Rs 64000
= Rs 10088

Q5: The price of a radio is Rs 1400 and it depreciates by 8% per month. Find its value after 3 months?
Ans: For the depreciation, we have the formula A = P(1 – R/100)ⁿ
Thus, the price of the radio after 3 months = 1400(1 – 8/100)³
= 1400(1 – 0.08)³ 
= 1400(0.92)³ 
= Rs 1090 (Approx.)

Q6: Find the compound interest at the rate of 10% per annum for two years on that principal which in two years at the rate of 10% per annum given Rs. 200 as simple interest?
Ans: Simple interest, S.I. = Rs 200
Rate, r = 10 %
Time, t = 2 years
So, by using the formula,
Simple interest = (P×r×t)/100
P = (S.I. × 100)/t×r
= (200 × 100)/2 × 10
= 20000/20
= Rs 1000
Now,
Rate of compound interest = 10%
Time = 2 years
By using the formula,
A = P (1 + r/100) ⁿ
= 1000 (1 + 10/100)²
= 1000 (110/100)²
= Rs 1210
∴ Compound Interest = (A – P)
= Rs 1210 – Rs 1000
= Rs 210

Q7: Ramesh deposited Rs. 7500 in a bank which pays him 12% interest per annum compounded quarterly. What is the amount which he receives after 9 months?
Ans: Principal, P = Rs 7500
Rate, r = 12 %
= 12/4
= 3 % (for quarterly)
Time = 9 months
= 9/12 years
= 9/12 × 4
= 3 (for quarter in a year)
By using the formula,
A = P (1 + r/100) ⁿ
= 7500 (1 + 3/100)³
= 7500 (103/100)³
= Rs 8195.45
∴ Required amount is Rs 8195.45

Q8: A town had 10,000 residents in 2000. Its population declines at a rate of 10% per annum. What will be its total population in 2005?
Ans: The population of the town decreases by 10% every year. Thus, it has a new population every year. So the population for the next year is calculated on the current year population. For the decrease, we have the formula A = P(1 – r/100)ⁿ
Therefore, the population at the end of 5 years = 10000(1 – 10/100)⁵
= 10000(1 – 0.1)⁵ 
= 10000 x 0.9⁵ 
= 5904 (Approx.)

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Practice Questions On Compound Interest

Here we have provided some practice questions on compounding interest calculator Class 8 for you to practice:

Q1: The population of a town was 160000 three years ago. If it has increased by 3%, 2.5%, and 5% in the last 3 years. Find the present population of the town. Q2: The difference between SI and CI of a certain sum of money is Rs.48 at 20% per annum for 2 years. Find the principal. Q3: At what interest rate compounded annually, will Rs.5000 amount to Rs.6050 in 2 years? Q4: Compute the amount and the compound interest in each of the following by using the formulae when : (i) Principal = Rs 3000, Rate = 5%, Time = 2 years (ii) Principal = Rs 3000, Rate = 18%, Time = 2 years (iii) Principal = Rs 5000, Rate = 10 paise per rupee per annum, Time = 2 years Q5: Amit borrowed Rs. 16000 at 17 ½ % per annum simple interest. On the same day, he lent it to Ashu at the same rate but compounded annually. What does he gain at the end of 2 years? Q6: Kamal borrowed Rs. 57600 from LIC against her policy at 12 ½ % per annum to build a house. Find the amount that she pays to the LIC after 1 ½ year if the interest is calculated half-yearly. Q7: What is the difference between the compound interests on Rs. 5000 for 1½ years at 4% per annum compounded yearly and half-yearly? Q8: What is the least number of complete years in which a sum of money put out at 20% compound interest will be more than doubled?

FAQs On Compound Interest Formula

Q.1: How do you calculate compound interest?
Ans: Compound interest is calculated on the sum total of the principal amount plus the interest earned over the previous periods.

Q.2: Is compound interest good or bad?
Ans: If you have deposited money in a bank, then compound interest will benefit you. However, if you have taken a loan or mortgage, the compound interest will get you to pay more.

Q.3: Why is compound interest so powerful?
Ans: Compound interest makes a principal amount grow at a faster rate than simple interest. This is because, in addition to earning returns on the money you invest, you also earn returns on those returns at the end of every compounding period.

Q.4: What is the formula for compounding interest calculator?
Ans: The formula to calculate compound interest is: CI=P[(1+r100)t−1]

Q.5: What is the compound interest formula used for?
Ans: The compound interest formula is the way to determine the total amount at the end of a period after the interest is added. Compound interest accrues over the period of a loan or mortgage.