MATHS QUERY
Simple, Compound & Rate of Interest
Introduction
Principal, amount, simple interest, compound interest and rate of interests are the terms mostly used by banks or the money lenders who lends money to money borrower. Bank gives interest on the amount deposited by a customer. On the other hand, money lender charges interests on the amount borrowed by money borrower.
How does a bank give interest?
Bank keeps customer’s money in the savings or deposit accounts. Bank keeps on adding more money by themselves on top of what a customer had deposited at the begining.
How much extra money bank adds on to customer’s amount depends upon how much and how long customer keeps money in the bank. More a customer deposits, more the bank adds on into customer’s account and more time a customer keeps money in the bank, more extra money bank adds on.
So, let’s understand it briefly in terms of the basic terms of principal, interest, rate of interest and time period.
- Money that a customer deposits while opening a new bank account is called as the principal.
- Time for which a customer keeps money in bank is called as time period.
- Bank adds extra amount of money into customer’s account in addition to the principal amount, is called as interest.
- Bank gives interest after a specific period of time, which is called as rate of interest. Rate of interest is written in percentage figures only.
How does a money lender charge interest?
Money lender is the one who gives money to money borrowers. Money borrowers are the people who lends money from money lenders, which is the other way around. Money borrower returns back the borrowed money plus the interest to money lender. So borrower returned back more than the borrowed money.
Let’s dive in and understand the above terms in more detail with examples.
Principal
Amount of money deposited by a customer into the bank or amount of money borrowed by a borrower from money lender is called as principal. It is also called as a sum.
Example 1
A customer deposits an amount of 100 $ in the bank.
So, 100 $ is the principal.
Example 2
Money borrower borrows an amount of 200 $ from money lender.
So, 200 $ is the principal.
Interest
Amount of extra money paid by borrower to the lender, in addition to what was borrowed is called as interest.
Example 1
Bank adds 20 $ to customer’s deposited amount of 100 $.
So, 20 $ is the interest given by the bank to the customer.
Example 2
Money borrower gives back to money lender an extra 40 $ in addition to the borrowed 200 $.
So, 40 $ is the interest paid by money borrower to the money lender.
Amount
The total amount of money paid by borrower to the lender at the end of a specific period of time is called
amount.
It is calculated with the following formula:
\(Amount\; = \;Principal \;+\; Interest\)
Example 1
Bank gives interest of 20 $ on customer’s deposited principal amount of 100 $.
\(Amount\; = \;Principal \;+\; Interest\)
\(\therefore\), \(Amount\; = \;100 \;+\; 20 = 120\)
So, amount in customer’s account after receiving the interest = 120 $.
Example 2
Money borrower gives back to money lender an extra 40 $ in addition to the borrowed 200 $.
\(Amount\; = \;Principal \;+\; Interest\)
\(\therefore\), \(Amount\; = \;200 \;+\; 40 = 240\)
So, amount of money returned back by borrower = 240 $.
Simple interest
If interest is calculated uniformly on original amount of money throughout the specific period of
time, this type of interest is called as simple interest.
It is calculated with the following formula:
\(Simple \;interest\; = \frac{Principal \;\times \;Rate\; of\; interest\; \times \;time}{100}\)
or \(SI\; = \frac{\;P \;\times\; R\; \times \;T}{100}\)
Example 1
A sum of $1000 is lent for 2 years at the rate of 5% per annum. Find the simple interest.
Principal = 1000
Rate of interest = 5% per annum
Time = 2 years
\(SI\; = \frac{\;P \;\times\; R\; \times \;T}{100}\)
\(= \frac{1000 \;\times \;5\; \times \;2}{100}\)
\(= 100 $\)
Example 2
Find the amount paid by a farmer if he borrows Rs 24000 for the time of 4 years at the rate of 6% per
annum.
Principal = 24000
Rate of interest = 6%
Time = 4 years
\(SI\; = \frac{\;P \;\times\; R\; \times \;T}{100}\)
\(= \frac{24000 \;\times \;6\; \times \;4}{100}\)
\(= 576\)
\(Amount\; = \;Principal \;+\; Interest\)
\(Amount\; = \;24000 \;+\; 576\)
\(= \;24576 $\)
Compound interest
It is the interest calculated on principal value plus the interest on the new principal amount.
In other words, we can say that interest earned on the amount of previous period of time.
It is the interest which is earned on the principal plus previously accumulated interest.
It is calculated as:
\(Compound \;interest\; = \;Amount \;-\; Principal\)
\(CI\; = \;A \;-\; P\)
Amount can be calculated as:
\(A\; = P(1\;+\;\frac{R}{100})^n\)
where, P = Principal, R = Rate of interest and n = Number of years
Calculate compound interest on Rs 6000 for 3 years at 2% per annum.
Method 1
P = 6000
R = 2% per annum
T = 3 years
As we know, \(A\; = P(1\;+\;\frac{R}{100})^n\)
\(A\; = 6000(1\;+\;\frac{2}{100})^3\)
\(A\; = 6000(\frac{100+2}{100})^3\)
\(A\; = 6000(\frac{102}{100})^3\)
\(A\; = 6367.248\)
\(CI\; = \;6367.248 \;-\; 6000\)
\(CI\; = \;367.248\)
Method 2
\(P_1 = \) Principal for first year = Rs 6000
T = 1 year
R = 2% per annum
Interest at the end of first year = \(\frac{6000 \; \times \;2\; \times \;1}{100} = 120\)
Amount at the end of first year = \(6000 + 120 = 6120\)
\(P_2 = \) Principal for second year = Rs 6120
T = 1 year
R = 2% per annum
Interest at the end of second year = \(\frac{6120 \; \times \;2\; \times \;1}{100}\)
\(=\frac{1224}{10} = 122.4\)
Amount at the end of second year = \(6120 + 122.4 = 6242.4\)
\(P_3 = \) Principal for third year = Rs 6242.4
T = 1 year
R = 2% per annum
Interest at the end of third year = \(\frac{6242.4 \; \times \;2\; \times \;1}{100}\)
\(= 124.48\)
Amount at the end of third year = \(6242.4 + 124.848 = 6367.248\)
Compound interest = Final amount – Original principal
\(=6367.248 – 6000\)
\(=367.248\)
In the above example, we can also find compound interest by adding interest of consecutive year.
CI = interest of Ist year + interest of 2nd year+ interest of 3rd year
= 120 + 122.4 + 124.848
= 367.248
Difference between compound and simple interest
In simple interest, principal value remains same for the whole period. Whereas, in compound interest,
principal value remains changing with time and it remains not same for the whole period.
Compound interest keeps on increasing every year, whereas simple interest remains constant for
every year.
Also, in compound interest, the principal value increases with time period which make the interest
increases accordingly.
If the values of principal, rate of interest and time period are kept same to calculate simple interest and compound interest, the calculated value of compound interest is always found greater than simple interest. Let’s use an example to find out how compound interest is always greater than simple interest for the above case.
Calculate simple interest and compound interest for $3000 at 4% per year in 3 years.
Simple interest calculations
Principal = 3000
Rate of interest = 4%
Time period = 3 years
Simple interest = \(\frac{\;P \;\times\; R\; \times \;T}{100}\)
\(=\frac{3000 \;\times\; 4\; \times \;3}{100}\)
\(=\frac{36000}{100}\)
\(=360\)
Compound interest calculations
\(A\; = P(1\;+\;\frac{R}{100})^n\)
\(= 3000(1\;+\;\frac{4}{100})^3\)
\(= 3000(\frac{100+4}{100})^3\)
\(= 3000(\frac{104}{100})^3\)
\(= 3000 \times \frac{104}{100} \times \frac{104}{100}\times \frac{104}{100}\)
\(= \frac{421824}{125}\)
Compound interest = amount – principal
\(= 3374.592 -3000\)
= 374.592
To understand this concept more briefly, lets have a look at the following tables with year wise
calculations.
Table: Year wise simple interest calculations
| Principal | Rate of interest | Time | Simple interest | Amount | |
|---|---|---|---|---|---|
| First year | 3000 | 4 | 1 year | 120 | 3000 + 120 = 3120 |
| Second year | 3000 | 4 | 1 year | 120 | 3000 + 120 = 3120 |
| Third year | 3000 | 4 | 1 year | 120 | 3000 + 120 = 3120 |
| Total | 360 |
Table: Year wise compound interest calculations
| Principal | Rate of interest | Time | Simple interest | Amount | |
|---|---|---|---|---|---|
| First year | 3000 | 4 | 1 year | 120 | 3000 + 120 = 3120 |
| Second year | 3120 | 4 | 1 year | 124.8 | 3120 + 124.8 = 3244.8 |
| Third year | 3244.8 | 4 | 1 year | 129.792 | 3244.8 + 129.792 = 3374.592 |
| Total | 374.592 |
Difference between simple interest and compound interest
Simple interest for 3 years = 120 + 120 + 120 = 360
Compound interest for 3 years = 120 + 124.8 + 129.792 = 374.592
Difference between them = 374.592 – 360 = 14.592