Simple and Compound Interest

1. Introduction

Interest is the fee charged by a money lender to a borrower for the use of borrowed money, usually expressed as an annual percentage of the principal. The interest rate is defined as the interest per year as a percentage of the principal amount.

Interest can be categorized into two types:

(i) Simple interest

(ii) Compound interest

With simple interest, the amount of the deposit remains the same, and the interest is paid to the depositor at the end of each interval of time. With compound interest, the amount of the deposit increases because the interest is added to the deposit at the end of each interval of time.

For example, let's suppose the deposit is Rs 8000, the annual interest rate is 20%, and the payment intervals are quarterly.

If this is simple interest, the financial institution will pay the depositor Rs 400 at the end of each quarter, totaling Rs 1600 interest earned for the year (20% of Rs 8000). The depositor's total assets after one year will be Rs 9600.

If this is compound interest, the payment will still be Rs 400 at the end of the first quarter, but the interest will be added to the deposit, increasing it to Rs 8400. At the end of the second quarter, the interest will be calculated using this larger amount, resulting in Rs 420, which will be added to the deposit, making the new total Rs 8820. This process continues for subsequent quarters.

SIMPLE INTEREST

When interest is calculated on the original principal for any length of time, it is called simple interest. In other words, simple interest for a fixed time remains constant if the principal money is fixed. 

Simple interest (SI) = Principal × Time × Rate\[SI = \frac{{P \times R \times T}}{{100}}\]Where \(P\) is principal, \(T\) is the time period and \(R\) is the interest rate. 

Amount = Principal + Interest 

Calculation of Simple Interest:

Case 1: when the interest rate is constant for the entire period.

Example: In how many years will the sum of money triple itself, at 25% per annum simple interest. 

Solution: Let the sum of money be P, so A is 3P, 

S.I. = A – P = 3P – P = 2P

Rate = 25%, Time = \(T\) years

\(2P = \frac{{P \times 25 \times T}}{{100}}\), hence \(T = 8\) years

Case 2: when the interest rate is not constant for the entire period.

If rate of simple interest differs from year to year, then to calculate total amount at the end of the time period, we consider different interest rates for the different given time periods.

Suppose the interest rates are R1, R2 and R3 for the time periods \({T_1},\,\,{T_2}\) and \({T_3}\). If the principal money is P, then the total amount after \({T_1} + {T_2} + {T_3}\)time,

=\(P + P\left[ {\frac{{{R_1} \times {T_1}}}{{100}} + \frac{{{R_2} \times {T_2}}}{{100}} + \frac{{{R_3} \times {T_3}}}{{100}}} \right]\)

Example 1:

Find the interest to be paid on a loan of Rs 6000 at 5% per year for the first year and 10% per year for second year for 5 years.

P = Rs 6000,
Simple Interest =\(6000\,\left[ {\frac{{5 \times 1}}{{100}} + \frac{{10 \times 1}}{{100}}} \right]\) = 900
Example 2:

A certain sum is lent out at a certain rate of simple interest, if simple interest at the end of the period is \(\frac{1}{{36}}\) time that of the principal. Find the rate of interest if the magnitude of rate of interest and time period is equal.

Suppose the principal is \(p\) and rate of interest (and time period) = \(r\)
Then \(\frac{p}{{36}} = \frac{{p \times r \times r}}{{100}}\)
\( \Rightarrow r = \sqrt {\frac{{100}}{{36}}} = \frac{{10}}{6} = 1.67\% \)
Example 3:

If simple interest on Rs. 420 increases by Rs. 140, when the time is increased by 5 years. Find the rate of interest per annum.

Here \({t_2} - {t_1} = 5\)
\(\frac{{420 \times R \times {t_2}}}{{100}} - \frac{{420 \times R \times {t_1}}}{{100}} = 140\)
\(\frac{{420({t_2} - {t_1})R}}{{100}} = 140\)
\(\frac{{420(5) \times R}}{{100}} = 140\)
\( \Rightarrow R = 6.67\% \)
Example 4:

A certain sum is lent out for a certain period. Had it been lent out on 10% higher rate of interest, it would have fetched Rs. 960 more in 4 years. Find the principle and rate of interest.

Difference in S.I. = 960
\({R_2} - {R_1} = 10\% \), \(T = 4\)yrs.
Difference in S.I.= \(\frac{{P \times {R_2} \times t}}{{100}} - \frac{{P \times {R_1} \times t}}{{100}}\)
= \(\frac{{P \times t({R_2} - {R_1})}}{{100}}\)
\(960 = \frac{{P \times 4(10)}}{{100}}\)
\(P = 2400\)
We cannot find the original rate of interest; as any two rate of interest whose difference is 10 is acceptable here.
Example 5:

A certain sum is lent out at a certain rate of interest for a certain period and S.I. is Rs. 400. Had the sum is lent out at 40% higher rate of interest for 30% less time period then what would have been S.I. in the second case.

As interest rate becomes 1.4 times and the time period becomes 0.70 times of the initial value, the new interest earned = 1.40×0.70 = 0.98 times of the initial interest.
Interest in the second case = 0.98×400 = 392.