Calculating Principal Amount with Simple Interest

This problem involves finding the original sum of money (the principal) when we are given the total amounts after different time periods under simple interest.

Let the principal sum be $P$ and the annual simple interest rate be $R$. The formula for the amount ($A$) after $T$ years at simple interest is:

$\text{Amount} = \text{Principal} + \text{Simple Interest}$

$A = P + \frac{P \times R \times T}{100}$

Or, $A = P + \text{Interest per year} \times T$

We are given two pieces of information:

• Amount after 4 years is Rs. 1,285.
• Amount after 5 years is Rs. 1,330.

Let $I$ be the simple interest earned per year. Since it is simple interest, the interest earned each year is constant.

The difference between the amount after 5 years and the amount after 4 years is the simple interest earned in the 5th year (which is equal to the interest earned in any single year).

Step 1: Calculate the simple interest earned per year.

Simple Interest for 1 year = Amount in 5 years - Amount in 4 years

Simple Interest for 1 year = $\text{Rs. } 1330 - \text{Rs. } 1285$

Simple Interest for 1 year = $\text{Rs. } 45$

So, the simple interest earned each year is Rs. 45.

Step 2: Calculate the total simple interest earned in 4 years.

Total Simple Interest in 4 years = Simple Interest per year $\times$ 4

Total Simple Interest in 4 years = $\text{Rs. } 45 \times 4$

Total Simple Interest in 4 years = $\text{Rs. } 180$

The total simple interest earned over 4 years is Rs. 180.

Step 3: Calculate the principal sum.

The amount after 4 years is the sum of the principal and the total simple interest earned in 4 years.

Amount in 4 years = Principal + Total Simple Interest in 4 years

$\text{Rs. } 1285 = \text{Principal} + \text{Rs. } 180$

Principal = $\text{Rs. } 1285 - \text{Rs. } 180$

Principal = $\text{Rs. } 1105$

The principal sum is Rs. 1105.

We can verify this using the amount after 5 years:

Total Simple Interest in 5 years = Simple Interest per year $\times$ 5 = $\text{Rs. } 45 \times 5 = \text{Rs. } 225$

Amount in 5 years = Principal + Total Simple Interest in 5 years = $\text{Rs. } 1105 + \text{Rs. } 225 = \text{Rs. } 1330$. This matches the given information.

The difference in amount over 1 year is the annual simple interest.

Revision Table: Key Simple Interest Concepts

Additional Information on Simple Interest Problems

• Simple interest is easier to calculate than compound interest because the principal remains constant throughout the investment period.
• When solving simple interest problems where amounts at different times are given, the difference between these amounts for a one-year period directly gives the annual simple interest.
• If the time difference is more than one year, say $N$ years, the interest difference represents the total simple interest for $N$ years. The interest for one year can then be found by dividing the total interest by $N$.
• Always ensure the time period ($T$) and the rate ($R$) are in consistent units (usually years for time and per annum for rate).