Find the Principal: Solving Simple and Compound Interest Problems

Understanding how to calculate simple and compound interests is crucial in finance and can be applied to various real-life situations. Let's explore how to find the principal using given interest amounts over multiple years.

Simple Interest Problem

Consider the following scenario: A sum of money becomes 2016 in 2 years and 2124 in 3 years at simple interest. The objective is to determine the sum of money (principal).

Let P be the principal.

Let r be the rate of interest per year.

Let I be the interest.

After 2 years, the amount is 2016:A2  P   2I  2016After 3 years, the amount is 2124:A3  P   3I  2124

Since ( I P cdot r ), we can express the interest over the periods as:

For 2 years:2I  2P cdot rFor 3 years:3I  3P cdot r

Subtract the equations:

P   3I - (P   2I)  2124 - 2016I  108

Substitute ( I 108 ) back into one of the original equations:

P   2 cdot 108  2016P   216  2016P  2016 - 216P  1800

The principal, or sum of money, is 1800.

Compound Interest Problem

Given:

Simple Interest (S.I.) for 3 years 3000

Simple Interest (S.I.) for 2 years 3000 ÷ 2 × 2 2000

Difference (Compounded Interest - Simple Interest) 2050 - 2000 50

The Simple Interest formula is ( S.I. frac{P cdot R cdot T}{100} )

3000  frac{P cdot R cdot 3}{100}P cdot R  frac{3000 cdot 100}{3}  100000Difference  frac{P cdot R^2}{10000}  50

From this, we find ( P cdot R^2 500000 ).

Substituting ( P cdot R 100000 ) into the difference equation:

frac{500000}{100000}  550  frac{5 cdot 100000}{P}P  frac{5 cdot 100000}{50}  20000

The principal is 20000.

Understanding the Difference Between Simple and Compound Interests

Simple interest is calculated on the principal amount for each period. In contrast, compound interest is calculated on the principal plus any interest accumulated until the last period. In the given scenario:

Simple Interest for 3 years  3000 (P cdot R cdot 3 / 100)3000  Principal cdot 1000 cdot 3 / 100 (since R  1000)Principal  20000

Compounded interest includes interest on the interest, leading to a higher final amount. The difference between simple and compound interest is 50, which indicates the additional interest earned in the third year due to the compounding effect.

Using the formula for compound interest:

Compound Interest  C(1   frac{r}{100})^t - C3000  Principal cdot frac{r cdot 3}{100} (Simple Interest formula)1000  Principal cdot r (rearranged)r  1000 / Principal (Substituting into the compound interest formula)

A verification step confirms that:

2050 - 2000  50 (due to the compounding effect)

In conclusion, the principal in the given scenario is 20000, showcasing the importance of understanding the differences between simple and compound interests in financial calculations.