This problem is related to the concepts of simple interest and compound interest. This makes it necessary to shed light on these two concepts. Simple interest could be defined as the rate at which money is borrowed or lent. In the following section, there would be discussion of the significant formulas and terms, which would assist in solving and understanding the problems on compound interest and simple interest. Firstly, the concept of simple interest would be defined and then the definitions and formulas would be used for solving problems that is expected to come from the section. Therefore, it is imperative to begin with the definitions.

When an individual lends money to a borrower, the borrower generally needs to incur an additional monetary amount to the lender. This additional money is considered in the form of interest. This interest could be expressed in terms of the amount taken by the borrower in the initial stage. In case; the interest on a sum borrowed for a certain point of time is reckoned uniformly, it would be termed as flat rate or simple interest. Before starting the formula for simple interest, it is necessary to cite some terms first that would be used in the formula.

Principal: The money lent out or borrowed for a specific period is termed as the sum or the principal.

Interest: Interest could be defined as the additional amount of money, which is paid by the borrower for using the money of the lender.

For instance, it is assumed that the principal amount is equal to P. The rate at which the interest is levied is equivalent to R% per year and the time for which the money is lent is assumed to be T years. Therefore, it could be written as follows:

Simple Interest = (P x R x T)/100

The principal amount could be computed by using the following formula:

P = 100 x {Simple Interest/(R x T)}

In a similar manner, time could be computed with the help of the below-stated formula:

T = 100 x {Simple Interest/(P x R)}

  In case of the simple interest formula, the computation of interest is made on the initial investment amount and there is no interest on interest unlike the formula of compound interest. The equation of simple interest finds its usage in consumer loans and car loan, which is extended by the banks and financial institutions. In addition, the interest incurred on savings bank accounts and term deposits by the banks is based on the equation of simple interest.

 The computation of compound interest is made by obtaining figures for the principal amount, time period, rate of interest and compound frequency along with entering them into the compound interest formula for returning the interest amount to be accrued over that timeframe. The compound interest concept is the interest added back to the principal amount in order to gain interest on the already collected interest during the next compounding period.

The rate at which the accrual of compound interest is made relies on the compounding frequency such that when the number of compounding periods is more, the compound interest is more as well. Hence, the sum of compound interest outstanding on $100 compounded at 10% yearly would be lower than that on $100 compounded at 5% semi-annually over the similar timeframe. Since the effect of interest-on-interest could produce increasingly positive returns depending on the initial amount of principal, it is sometimes denoted by the term “miracle of compound interest”.

The calculation of compound interest is made by multiplying the initial amount of principal by one along with the yearly rate of interest increased to the compounding period number less one. The overall initial loan amount is then deducted from the resulting outcome. The formula to compute compound interest is shown as follows:

Compound Interest = Total amount of Interest and Principal in Future or Future Value - Principal Amount at Present or Present Value

Compound Interest = [P x (1 + i)ⁿ] - P

Compound Interest = P [(1 + i)ⁿ - 1]

In the above formula, P denotes the principal, i is equal to nominal annual interest, r is rate denoted in percentage terms and n is denoted by the number of compounding periods.

For the provided problem, it could be seen that the difference between compound interest and simple interest on a sum of money lent for 2 years at the rate of 10% is Rs 40. Therefore,

CI = Amount - Principal

CI = P x (1 + r/100)ⁿ - P

SI = PRT/100

CI - SI = Rs 40

P x (1 + r/100)ⁿ - P - PRT/100 = Rs 40

P x [(1 + 10/100)² - 1 - (10x2)/100] = Rs 40

P x [(11/10)² - 1 - 1/5] = Rs 40

P x [121/100 - 1 - 1/5] = Rs 40

P x [21/100 - 1/5] = Rs 40

P x [(21-20)/100] = Rs 40

P x 1/100 = Rs 40

P = Rs 40 x 100

P = Rs 4,000

Therefore, the sum is computed is Rs 4,000