Understanding the Simple Interest Problem

This problem asks us to find the original principal sum based on the difference in simple interest earned when the rate is increased. We are given the time period and the amount of extra interest earned due to a specific increase in the simple interest rate.

Simple Interest Formula

The formula for calculating simple interest (SI) is:

\(SI = \frac{P \times R \times T}{100}\)

Where:

• \(P\) is the Principal sum (the initial amount of money).
• \(R\) is the Rate of interest per annum.
• \(T\) is the Time period in years.

Analyzing the Given Information

We are given the following details:

• Time (\(T\)) = 3 years
• Let the original Rate be \(R\%\) per annum.
• The new Rate is \(R' = (R + 2)\%\) per annum.
• The increase in simple interest is Rs. 420.

This means the simple interest earned with the new rate (\(SI'\)) is Rs. 420 more than the simple interest earned with the original rate (\(SI\)).

So, \(SI' - SI = 420\).

Setting up the Equations

Using the simple interest formula, we can write the expressions for \(SI\) and \(SI'\):

Original Simple Interest (\(SI\)):

\(SI = \frac{P \times R \times 3}{100}\)

Simple Interest with the higher rate (\(SI'\)):

\(SI' = \frac{P \times (R+2) \times 3}{100}\)

Solving for the Principal Sum

Now, we use the given difference in interest to find the principal sum \(P\):

\(SI' - SI = 420\)

Substitute the expressions for \(SI'\) and \(SI\):

\(\frac{P \times (R+2) \times 3}{100} - \frac{P \times R \times 3}{100} = 420\)

Multiply both sides by 100 to clear the denominators:

\(3P(R+2) - 3PR = 420 \times 100\)

\(3PR + 6P - 3PR = 42000\)

Notice that the \(3PR\) terms cancel out:

\(6P = 42000\)

Now, solve for \(P\):

\(P = \frac{42000}{6}\)

\(P = 7000\)

Thus, the original sum (principal) is Rs. 7,000.

Verification

Let's quickly verify this. Suppose the original rate \(R\) was 5%. The interest would be \(\frac{7000 \times 5 \times 3}{100} = \frac{105000}{100} = 1050\). If the rate was 7% (2% higher), the interest would be \(\frac{7000 \times 7 \times 3}{100} = \frac{147000}{100} = 1470\). The difference is \(1470 - 1050 = 420\), which matches the problem statement. This confirms our calculation for the sum is correct.

Conclusion

The sum put at simple interest was Rs. 7,000.

Revision Table: Simple Interest Concepts

Additional Information on Simple Interest Calculations

Simple interest is the easiest type of interest to calculate. It is based only on the principal amount. Unlike compound interest, the interest earned in previous periods is not added to the principal for calculating interest in subsequent periods.

The key components influencing simple interest are the principal amount, the interest rate, and the time period. Changing any of these factors will change the total simple interest earned or paid.

In problems where the rate changes, as in this question, the difference in interest is solely due to the change in rate applied to the same principal over the same time. The formula for the difference in simple interest for a change in rate \(\Delta R\) over time \(T\) for principal \(P\) is:

\(\text{Difference in SI} = \frac{P \times \Delta R \times T}{100}\)

In this problem, \(\Delta R = 2\%\), \(T = 3\) years, and the Difference in SI is Rs. 420. Using this shortcut formula:

\(420 = \frac{P \times 2 \times 3}{100}\)

\(420 = \frac{6P}{100}\)

\(42000 = 6P\)

\(P = \frac{42000}{6}\)

\(P = 7000\)

This provides an alternative, quicker way to solve such problems directly.