Compound Interest Without Using Formula: Definition, Method, Examples

Compound Interest Without Using Formula: The principal plus the interest from the previous period is used to compute compound interest. As the interest payable is added to the principal, the principal amount increases with each time period, implying that interest is earned not just on the principal but also on previous periods’ interest.

As a result, we may conclude that compound interest is more than simple interest on the same amount of money invested. In this article, we will learn how to calculate the compound interest without using the formula.

Introduction

Sometimes, in need, we borrow money from a bank or some other agency doing financial business. In general, the money is borrowed for a specified period and returned at the end. At the end of the period, we pay the money borrowed plus some extra money for utilising the lender’s money.

The money borrowed is called the principal, the extra money paid for using the lender’s money is called interest, and the total money, paid to the lender at the end of the specified period is called the amount.
\({\rm{Amount}} = {\rm{Principal}} + {\rm{Interest}}\)
Or \({\rm{A}} = {\rm{P}} + {\rm{I}}\)

Interest (Simple Interest or SI)

Suppose interest is computed on the initial principal throughout the loan period, regardless of the duration of the period for which it is borrowed. In that case, it is considered to be a simple interest.

\(S.I. = \frac{{{\rm{Principal}} \times {\rm{Rate}} \times {\rm{Time}}}}{{100}}\)
or \(\frac{{{\rm{P}} \times {\rm{R}} \times {\rm{T}}}}{{100}}\)

When we say interest, it always means simple interest.

Compound Interest (CI)

When money is lent at compound interest, the interest due at the end of a set period (one year, half-year, etc.) is not paid to the moneylender but instead added to the amount lent. The sum earned thus becomes the next period’s principal. This procedure is repeated until the latest period’s amount is determined.
The necessary compound interest is the difference between the final amount and the original principal.
\({\rm{Compound}}\,{\rm{Interest}} = {\rm{Final}}\,{\rm{Amount}} – {\rm{Original}}\,{\rm{Principal}}\)
\(C.I. = A – P\)
The table below clarifies the difference between simple interest (S.I.) and compound interest (C.I.).
In the table, we have taken the sum borrowed (principal) \(=₹ 1000\) at \(10\%\) per annum and for \(3\) years.

Compound Interest Without Using Formula

Compound Interest as a Repeated Simple Interest Computation with a Growing Principal:

As shown in the table above, the principal for the \(1\)st year is \(₹1000,\) and interest (C.I.)on it is \(₹100.\) The principal for the \(2\)nd year is \(₹1100\) and interest (C.I.)on it is \(₹110;\) whereas, the principal for the \(3\)rd year is \(₹1210,\) and the interest (C.I.)on it is \(₹121.\)

It is observed that the compound interest is growing every year, which increases the principal for next year.

As shown in the table given above, compound interest in \(3\) years
\(=\)C.I. of \(1\)st year \(+\) C.I. of \(2\)nd year \(+\) C.I. of \(3\)rd year
\(=₹100+₹110+₹121=₹331 \)

Also compound interest in \(3\) years
\(=\) Amount at the end of \(3\) years \(–\) original sum (Principal for \(1\)st year)
\(=₹13310-₹1000=₹331 \)

Compound Interest Using Formula

To make the above said calculation easy and fast, we use certain formulae.
1. When the interest is compounded yearly, the formula for finding the amount is:
\(A = P{\left( {1 + \frac{r}{{100}}} \right)^n}\)
Where \(A=\)amount
\(P=\)principal
\(r=\)rate of interest compounded yearly
and \(n=\)number of years

Example: Calculate the amount on \(₹7500\) in \(2\) years and at \(6\%\) compounded annually.
Solution: Given, \(P=₹7500;\,n=2\) years and \(r=6\%\)
\(A = ₹7500{\left( {1 + \frac{6}{{100}}} \right)^2}\)
\( = ₹7500{\left( {\frac{{106}}{{100}}} \right)^2} =₹ 8427\)
Therefore, required amount \(=₹8427\)
And C.I. \(=₹8427-₹7500=₹927\)

2. When the rates for successive years are different, then
\(A = P\left( {1 + \frac{{{r_1}}}{{100}}} \right)\left( {1 + \frac{{{r_2}}}{{100}}} \right)\left( {1 + \frac{{{r_3}}}{{100}}} \right)…..\) and so on
Where \({r_1}\% ,\,{r_2}\% ,\,{r_3}\% ….\) and so on are the rates for successive years.

Example: Calculate the amount and the compound interest on \(₹12000\) in \(3\) years when the interest rates for successive years are \(8\%,\,10\%\) and \(15\%,\) respectively.
Solution: Required amount \(A = P\left( {1 + \frac{{{r_1}}}{{100}}} \right)\left( {1 + \frac{{{r_2}}}{{100}}} \right)\left( {1 + \frac{{{r_3}}}{{100}}} \right)\)
\(A = ₹12000\left( {1 + \frac{8}{{100}}} \right)\left( {1 + \frac{{10}}{{100}}} \right)\left( {1 + \frac{{15}}{{100}}} \right)\)
\(=₹16394.40 \)
And C.I. \(=₹16394.40-₹12000=₹4394.40\)

Monthly Compound Interest Formula

The formula for the compound interest is derived from the difference between the final amount and the principal, which is:
\(C.I. = {\rm{Amount}} – {\rm{Principal}}\)
The formula of monthly compound interest is:
\(C.I. = P{\left( {1 + \left( {\frac{r}{{12}}} \right)} \right)^{12r}} – P\)
Where,
1. \(P\) is the principal amount,
2. \(r\) is the interest rate in decimal form,
3. \(t\) is the time.

Solved Examples – Compound Interest Without Using Formula

Q.1. \(₹8000\) is lent at \(5\%\) compound interest per year for \(2\) years. Find the amount and the compound interest.
Ans: For the first year:
Principal \((P)=₹8000;\) Rate \(R)=5\%\)
Time \((T)=1\) year
Interest \( = \frac{{P \times R \times T}}{{100}}\)
\( =₹ \frac{{8000 \times 5 \times 1}}{{100}} = ₹400\)
\({\rm{Amount}} = {\rm{Principal}} + {\rm{Interest}}\)
\(=₹8000+₹400=₹8400 \)
According to the compound interest definition, the first year’s amount will work as principal for the next (second) year.
Therefore, for the second year:
Principal \((P) = ₹8400\)
Rate \(R)=5\%\)
Time \((T)=1\) year
Interest \( = ₹\frac{{8400 \times 5 \times 1}}{{100}} =₹ 420\)
Amount at the end of the second year \(=₹8400+₹420=₹8820\)
Compound interest \(=\) Final Amount \(–\) Initial Principal
\(=₹8820-₹8000=₹820 .\)

Q.2. Calculate the amount and the compound interest on \(₹10000\) at \(8\%\) per annum, and in \(1\) year, interest is compounded half-yearly.
Ans: For first \(\frac{1}{2}\) year: Principal \(P=₹10000;\) Rate \((R)=8\%\) and Time \(\left( T \right) = \frac{1}{2}\) year.
Therefore, Interest \(I =₹ \frac{{10000 \times 8 \times 1}}{{100 \times 2}} = ₹400\)
And \(A = P + I\)
\(=₹10000+₹400=₹10400\)
For second \(\frac{1}{2}\) year: Principal \(P=₹10400;\) Rate \((R)=8\%\) and Time \(\left( T \right) = \frac{1}{2}\) year
Therefore, Interest \(I =₹ \frac{{10400 \times 8 \times 1}}{{100 \times 2}} = ₹416\)
And \(A = P + I\)
\(=₹10400+₹416=₹10816\)
Required amount \(=₹10816\)
And compound interest \( = A – P\)
\(=₹10816-₹10000=₹816.\)

Q.3. Calculate the compound interest increased on \(₹16000\) in \(3\) years, when the interest rates for successive years are \(10\% ,\,12\% \) and \(15\% ,\) respectively.
Ans: For the first year: Principal \(P=₹16000;\) Rate \(R=10\%\) and Time \(T=1\) year
Therefore, Interest \( =₹ \frac{{16000 \times 10 \times 1}}{{100}} = ₹1600\)
And amount \(=₹16000+₹1600=₹17600\)
For the second year: Principal P=₹17600; Rate R=12% and Time T=1 year Therefore, Interest =₹17600×12×1100=₹2112 And amount =₹17600+₹2112=₹19712 For the third year: Principal \(P=₹19712;\) Rate \(R=15\%\) and Time \(T=1\) year
Therefore, Interest \( = ₹\frac{{19712 \times 15 \times 1}}{{100}} = ₹2956.80\)
And amount \(=₹19712+₹2956.80=₹22668.80\)
Therefore, C.I. accrued \(=\) Final amount \(–\) Initial principal
\(=₹22668.80-₹16000=₹6668.80.\)

Q.4. Calculate the compound interest due in \(1\frac{1}{2}\) years on \(₹6000\) at \(10\%\) compound annually.
Ans: For the first year: Principal \(P=₹6000;\) Rate \(R=10\%\) and Time \(T=1\) year
Therefore, interest \( = ₹\frac{{6000 \times 10 \times 1}}{{100}} = ₹600\)
And amount \(A = P + I = ₹6000 + ₹600 = ₹6600\)
For the second year: Principal \(P=₹6600;\) Rate \(R=10\%\) and Time \(T=1\) year
Therefore, interest \( = ₹\frac{{6600 \times 10 \times 1}}{{100}} = ₹660\)
And amount \(=₹6600+₹660=₹7260\)
For the \(\frac{1}{2}\) year: Principal \(P=₹7260;\) Rate \(R=10\%\) and Time \(T = \frac{1}{2}\) year
Therefore, interest \( = ₹\frac{{7260 \times 10 \times 1}}{{100 \times 2}} = ₹363\)
And amount \(=₹7260+₹363=₹7623\)
Since the amount in \(1\frac{1}{2}\) years is \(₹7623,\) and the original principal is \(₹6000.\)
Therefore, compound interest \(=₹7623-₹6000=₹1623.\)

Q.5. Calculate the difference between the compound interest and the simple interest on \(₹4000\) at \(8\%\) per annum and in \(2\) years.
Ans: For C.I.: Principal \(P=₹4000;\) Rate \(R=8\%\) and Time \(T=2\) years
Therefore, simple interest \( = ₹\frac{{4000 \times 8 \times 2}}{{100}} = ₹640\)
For C.I.: Principal for first-year \(=₹4000\)
Interest on it \( = \frac{{4000 \times 8 \times 1}}{{100}} = 320\)
Amount \(=₹4000+₹320=₹4320\)
Therefore, principal for the second year \(=₹4320\)
Interest on it \( =₹ \frac{{4320 \times 8 \times 1}}{{100}} =₹ 345.60\)
Therefore, C.I. for \(2\) years \(=₹320+₹345.60\)
\(=₹665.60\)
Required difference between C.I. and \(S.I. = C.I. – S.I.\)
\(=₹665.60-₹640\)
\(=₹25.60.\)

Summary

In this article, we have discussed what simple interest and compound interest is. Also, we discussed how to compute compound interest without using formula as a repeated simple interest computation with a growing principal and how to calculate compound interest using the formula along with some solved examples and frequently asked questions.

Frequently Asked Questions (FAQs)

Q.1. How do you find compound interest without formula?
Ans: Compound interest without using formula can be calculated as a repeated simple interest computation with a growing principal.

Q.2. How do you calculate compound interest directly?
Ans: When the interest is compounded yearly, the formula for finding the amount is:
\(A = P{\left( {1 + \frac{r}{{100}}} \right)^n}\)
Where \(A=\) amount;
\(P=\) principal;
\(r=\) rate of interest compounded yearly;
and \(n=\) number of years
When the rates for successive years are different, then:
\(A = P\left( {1 + \frac{{{r_1}}}{{100}}} \right)\left( {1 + \frac{{{r_2}}}{{100}}} \right)\left( {1 + \frac{{{r_3}}}{{100}}} \right)…\)
Where \({r_1}\% ,\,{r_2}\% ,\,{r_3}\% ,\,….\) and so on are the rates for successive years.

Q.3. What is the easiest way to learn compound interest?
Ans: The easiest way to learn compound interest is by calculating the simple interest and amount for the first year and considering that as the principal for the second year, and proceeding further.

Q.4. What is the formula of compound interest with example?
Ans: When the interest is compounded yearly, the formula for finding the amount is:
\(A = P{\left( {1 + \frac{r}{{100}}} \right)^n}\)
Where \(A=\) amount,
\(P=\) principal,
\(r=\) rate of interest compounded yearly
and \(n=\) number of years
Example: Calculate the amount on \(₹7500\) in \(2\) years and at \(6\%\) compounded annually.
Solution: Given, \(P=₹7500;\,n=2\) years and \(r=6\%\)
\(A =₹ 7500{\left( {1 + \frac{6}{{100}}} \right)^2}\)
\( = ₹7500 \times {\left( {\frac{{106}}{{100}}} \right)^2} = ₹8427\)
Therefore, required amount \(=₹8427\)
And C.I. \(=₹8427-₹7500=₹927.\)

Q.5. What is the formula for monthly compound interest?
Ans: The formula for the compound interest is derived from the difference between the final amount and the principal, which is
\(C.I. = {\rm{Amount}} – {\rm{Principal}}\)
The formula of monthly compound interest is:
\(C.I. = P{\left( {1 + \left( {\frac{r}{{12}}} \right)} \right)^{12r}} – P\)
Where \(P\) is the principal amount, \(r\) is the interest rate, \(t\) is the time.

We hope this detailed article on compound interest without using formulas helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!