Compound interest is interest that builds up over a set length of time on both principal and interest. The principal is also used to account for the interest that has accrued on a principal over time. It is the idea that we employ the most frequently on a daily basis. It is the new way of calculating interest now utilized in all international financial and commercial operations. Some of its major applications are:
Population Growth or Decline.
Growth of Bacteria
Increase or Decrease in an Item's Value.
Terminology of Compound Interest
Principal (P): The amount of money lent for a specific amount of time at a specific interest rate.
Time (t): It is the length of time the principal is lent, commonly expressed in years.
Interest (I): It is the return on an investment made by lending a principal for a specific amount of time.
Rate(r): It is the percentage of interest received for a loan of a certain amount.
Amount (A): The total sum of money remaining at the end is the amount. It is the total of the initial principal and all compound interest that has been earned.
Compound Interest Equation
Following the computation of the total amount over a period of time using the initial principal and the interest rate, the compound interest is determined. The formula for calculating the amount is given below for an initial principal of P, an annual interest rate of r, a time period of t in years, and a frequency of n times the interest is compounded annually.
Compound Interest: \[CI\, = \,P\,{\left( {1 + \frac{R}{{100}}} \right)^t} - P\]
Amount: \[A = P{(1 + \dfrac{r}{n})^{nt}}\]
What does Compounded Quarterly Mean?
Different formulas can be used to calculate compound interest for a given principal over various time periods. If the interest calculation period is quarterly, the sum is compounded four times a year and the interest is calculated every three months. The money left over after the first three months will be used to compute interest for the subsequent three months (second quarter). Additionally, interest will be computed for the third quarter on the amount left over after the first six months and for the final quarter on the amount left over after the first nine months. Thus, the following is the quarterly compound interest formula:
Compounded quarterly equation: \[C.I\, = \,P\,{\left( {1 + \frac{{\frac{r}{4}}}{{100}}} \right)^{4t}} - \,P\]
How to Calculate Quarterly Compound Interest?
When the amount compounds every three months, it indicates that it does so four times per year. i.e., n = 4.We will learn how to calculate compound interest quarterly by solving the following examples:
1.Find the compound interest when Rs 100000 is invested for 9 months at 6% per annum, compounded quarterly.
Explanation: Here principal (P) = Rs 100000
Rate of interest (r) = 6%
Time (n)=\[\frac{9}{{12}} = \frac{3}{4}\] year.
Therefore, the amount of money accumulated for n years=
\[A\, = \,P\,{\left( {1 + \frac{{\frac{r}{4}}}{{100}}} \right)^{4n}}\]
\[ = \,100000\,{\left( {1 + \frac{{\frac{8}{4}}}{{100}}} \right)^{4\, \times \frac{3}{4}}}\]
\[ = 100000{\left( {1 + \frac{2}{{100}}} \right)^3}\]
\[ = 100000{\left( {\frac{{51}}{{50}}} \right)^3}\]
\[ = 100000 \times \frac{{51}}{{50}} \times \frac{{51}}{{50}} \times \frac{{51}}{{50}}\]
\[ = 106120.8\]
Therefore, Compound Interest = Total Amount - Principal
= 106120.8 – 100000
= Rs 6120.8
Summary
The profit made from lending money is known as interest. It is always calculated using a specific interest rate for a specific amount of time. In compound interest, the principal (amount on which interest is calculated) is renewed each year, and compound interest is calculated in the same way as simple interest. Adding interest to the current principal sum is referred to as "compounding." We can calculate compound interest weekly, monthly, quarterly, half-yearly or yearly.
Solved Questions
1. Find the amount and the compound interest on Rs 1, 20,000 compounded quarterly for 9 months at the rate of 4% per annum.
Ans: Here Principal (p) = Rs 1, 20,000
Rate of interest = 4%
Time = 9 months which will be \[\frac{9}{{12}} = \frac{3}{4}\] years
Hence, Amount will be \[A\, = \,P\,{\left( {1 + \frac{{\frac{r}{4}}}{{100}}} \right)^{4n}}\]
\[ = \,120000\,{\left( {1 + \frac{{\frac{4}{4}}}{{100}}} \right)^{4\, \times \frac{3}{4}}}\]
\[ = 120000{\left( {1 + \frac{1}{{100}}} \right)^3}\]
\[ = 120000{\left( {\frac{{101}}{{100}}} \right)^3}\]
\[ = 123636.12\]
Therefore, Compound Interest = Total Amount – Principal
= 1, 23,636.12 – 1, 20,000
= Rs 3636.12
2. If Rs 1200 is invested at a compound interest rate 8% per annum compounded quarterly for 12 months, find the compound interest.
Ans: Here, Principal (p) = Rs 1200
Rate of interest = 8 %
Time = \[\dfrac{{12}}{{12}} = 1\] years
Hence, amount on the accumulated sum will be:
\[ = \,1200\,{\left( {1 + \dfrac{{\frac{8}{4}}}{{100}}} \right)^{4\, \times 1}}\]
\[ = 1200{\left( {1 + \frac{2}{{100}}} \right)^4}\]
\[ = 1200{\left( {\frac{{51}}{{50}}} \right)^4}\]
\[ = 1298.91\]
Therefore, Compound Interest = Total Amount – Principal
= 1298.91-1200
=Rs 98.91
3. What is the compound interest (CI) onRs.6000for1years at12%per annum compounded annually?
Ans: Here Principal (P) = 6000
Rate of interest: 12 %
Time: 1 year
Hence, amount on the given sum will be:
\[ = \,6000\,{\left( {1 + \frac{{\frac{12}{4}}}{{100}}} \right)^{4\, \times \frac{1}{1}}}\]
\[ = 6000{\left( {1 + \frac{3}{{100}}} \right)^4}\]
\[ = 6000{\left( {\frac{{103}}{{100}}} \right)^4}\]
\[ = 6753.05\]
Hence, C.I = Amount – Principal
= 6753.05 – 6000
= Rs 735.05
