In this platform, you will learn about how to find Compound Interest when Interest is Compounded Quarterly. Few of you might feel the process of calculating the Compound Interest with Growing Principal is a bit difficult if the time duration is long. Refer to Solved Examples on finding Quarterly Compounded Interest and learn how to solve the related problems. You can find the formula and its importance in the next sections.
10th Grade Math Compound Interest concepts are explained in detail on our website for free. On this page, we will discuss what is compound interest, the quarterly compound interest formula, and how to find compound interest when interest is compounded quarterly its derivation with the solved examples.
What does it mean if Interest is Compounded Quarterly?
The compound interest quarterly is defined as the process of calculating and adding the interest amount to the loan or investment which is earned quarterly where the interest earned will also be reinvested. It is useful in calculating the fixed deposit income as most of the banks offer interest income on the deposits which compound quarterly.
The compounding effect can be seen as the interest is paid. The compounded interest monthly will gives a higher profit as it is calculated on a higher balance each month. Consider an example, the value after 2 years is t=2, Earns 3% compounded quarterly is r=0.015 and m=4 since the compound interest quarterly means 4 times a year, the principal P=3500.
Compound Interest Quarterly Formula | How to Calculate Compound Interest Quarterly?
The compound interest is calculated on the interest accumulated and the principal amount over the previous period. In the case, that interest is compounded Quarterly, the formula used to calculate compound interest is,
Compound Interest = Amount – Principal
C.I. = A – P
Amount A = P{(1 + r/n)nT}
Here,
P = Principal Amount
R = Rate of Interest
T = Number of years
A = Amount in t years.
n = number of times the amount is compounding.
Compound Interest Quarterly Formula: If the Rate of Interest is Annual and Interest is Compounded Quarterly then the number of years is multiplied by 4 which means 4n and the annual interest rate is cut down by one-fourth of the year. In such cases, the Formula for Quarterly Compound Interest is given as under:
Let us assume the Principal = P, Rate of Interest = r/4 %, and time = 4n, Amount = A, Compound Interest = CI then,
A = P(1+(r/4)/100)4n
In the above formula, the rate of interest is divided by 4 whereas the time is multiplied by 4.
We know the Compound Interest CI = A – P
CI = P(1+(r/4)/100)4n – P
CI = P{1+(r/4)/100)4n – 1}
If you are aware or know any of the three values then you can automatically find the other one.
See More:
- Compound Interest by Using Formula
- Compound Interest when Interest is Compounded Yearly
- Compound Interest when Interest is Compounded Half-Yearly
Compound Interest When Interest is Compounded Quarterly Examples
Example 1:Â Find the compound interest when $1,00, 000 is invested for 6 months at 5 % per annum, compounded quarterly?
Solution:
Given that,
The Principal Amount is $1,00, 000.
The Rate of Interest is 5% per annum.
n = 6 months = 1/2 year
The interest rate is compounded Quarterly by dividing the interest rate by 4 which means r/4 and it is multiplied the time by 4 times i.e. 4n
So, the Amount (A) = P(1+(r/4)/100)4n
Now, substitute the inputs in the above formula to find the amount.
So, the Amount A = 1,00,000(1+(5/4)/100)4*1/2
= 1,00,000(1+5/400)2
= $ 1,02,515
Thus, the compound interest is, CI = A – P
i.e., $ 1,02,515 – $ 1,00,000 =$2515.
Hence, the compound interest quarterly is $2515 and the amount is $1,02,515.
Example 2: Find the amount and the compound interest on Rs.12,000 compounded quarterly for 9 months at the rate of 10% per annum?
Solution:
As given in the question,
The Principal Amount is Rs.12, 000.
The Rate of Interest is 10% per annum.
The number of times n is 9 months i.e., 9/12 = 3/4 year.
Let the Interest Rate is Compounded Quarterly by dividing the interest rate by 4 i.e. r/4 and multiply the time by 4 which means 4n.
So, the Amount A = P(1+(r/4)/100)4n
After, substitute the input values in the above formula. We get,
A= 12,000(1+(10/4)/100)4*3/4
A = 12,000(1+10/400)3Â = 12,000(1+0.025)3
= 12,000(1.025)3
A = Rs. 12922.
So, the Amount A is Rs. 12922.
Now, find the compound interest.
CI = A – P
Place A and P values in the formula. We get,
C.I = 12922 – 12000 = Rs. 922
Therefore, after quarterly, the compound interest is Rs.922 and the amount is Rs.12922.
Example 3:Â How long does it take for $15000 to double if the amount is compounded quarterly at 10% annual interest?
Solution:
In the given question,
The principal amount is P = $15000.
The rate of interest is, r = 10% =10/100 = 0.1.
The amount is, A = 15000 x 2 = $30000
We will find the time taken for $15000 to double.
Consider that, the required time in years is t.
By using the quarterly compound interest formula, A = P (1 + r / 4)4t
Now, substitute the given inputs in the above formula. We get,
30000 =15000(1+0.1/4)4t
Now, dividing both sides by 15000, then it will be
2 =(1.025)4t
Taking ln on both sides
ln 2 = 4t ln 1.025
t = ln 2 / 4ln 1.025
t = 7.
So, it takes 7 years for $15000 to become double.
Therefore, the amount of $15000 will becomes double after 7years.
Example 4: Sam deposited $7500 in a bank that pays him 12% interest P.A. compounded quarterly. Find the amount which he receives after 9 months?
Solution:
In the given question,
The principal amount is P = $ 7500
Rate of interest compounded for 9 months is 12% p.a. and
n = 9 months = 9/12 year = 3/4 year
The formula used to calculate amount is,
Amount after 9 months = P ( 1 + R/100 )4t
Substitute the given inputs in the formula.
A = 7500 x ( 1 + 0.12/4 )4n
= 7500 x ( 1 + 0.03)4 x 3/4
= 7500 x ( 1.03)3
= 7500 x 1.092727
A= 8195.45
Hence, the amount is 8195.45
Therefore, the amount after 9months is 8195.45.
FAQ’s on Compound Interest when Interest is Compounded Quarterly
1. Who benefits from compound interest?
The investors benefit from the compound interest since the interesting pair here on the principle plus on the interest which they already earned.
2. What is the interest compounded quarterly formula?
The formula for interest compounded quarterly is given by:
A = P(1 + (R/4)/100)4T
Where, A is the Amount, P = principal amount, T= time, and R= rate of interest.
3. How do you find the compound interest rate?
The compound interest rate can be found using the formula,
A = P(1 + r/n)^{nt}
A = Total amount
P = Principal
r = Annual nominal interest rate as a decimal
n = Number of compounding periods
t = Time (in years)
Thus, the compound interest (CI) is A – P.
4. What is the formula used to calculate compound interest when interest is compounded Quarterly?
The formula used to calculate the quarterly compound interest is CI = P{(1 + (r/4)/100)4T} – P
where CI is the compound interest
P is the initial principal amount.
T is the time period.
R is the rate of interest per annum.
A is the Total Amount.
Summary
Compound Interest when interest is compounded Quarterly along the formula is given in this article. It becomes easy if you read this article. Every part of this article is given with an explanation. Know what is the difference to find compound interest when it is compounded yearly, Half Yearly, and Quarterly. Know every difference and how to solve compound interest problems by referring to our articles. So, don’t miss the chance to learn the concepts of compound interest.