Understanding Simple Interest Calculation and Period

This problem involves comparing simple interest earned on two different principal amounts over the same period and finding that period. We are given the principals, the rate of interest, and the difference between the simple interests earned.

Given Information for Simple Interest Problem

Let's list the details provided in the question:

• Principal 1 (\(P_1\)): ₹1000
• Principal 2 (\(P_2\)): ₹800
• Rate of Interest (\(R\)): 10% per annum (This rate is the same for both principals)
• Difference in Simple Interest (\(SI_1 - SI_2\)): ₹100
• Period (\(T\)): Unknown (Same for both principals)

We want to find the value of the period \(T\) in years.

Simple Interest Formula

The formula used to calculate simple interest (\(SI\)) is:

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

Where:

• \(P\) is the Principal amount
• \(R\) is the Rate of Interest per annum
• \(T\) is the Period in years

Calculating Simple Interest for Each Case

Using the simple interest formula, we can write expressions for the simple interest earned on each principal for the period \(T\).

For Principal 1 (\(P_1 = ₹1000\)):

\( SI_1 = \frac{1000 \times 10 \times T}{100} \)

\( SI_1 = \frac{10000 \times T}{100} \)

\( SI_1 = 100T \)

For Principal 2 (\(P_2 = ₹800\)):

\( SI_2 = \frac{800 \times 10 \times T}{100} \)

\( SI_2 = \frac{8000 \times T}{100} \)

\( SI_2 = 80T \)

Setting Up the Equation Based on Interest Difference

The problem states that the simple interest on ₹1000 is ₹100 more than that on ₹800. This gives us the equation:

\( SI_1 - SI_2 = 100 \)

Substitute the expressions for \(SI_1\) and \(SI_2\) that we found:

\( 100T - 80T = 100 \)

Solving for the Period (T)

Now, we solve the equation for \(T\):

\( (100 - 80)T = 100 \)

\( 20T = 100 \)

To find \(T\), divide both sides by 20:

\( T = \frac{100}{20} \)

\( T = 5 \)

The period is 5 years.

Let's verify this result:

• Simple interest on ₹1000 at 10% for 5 years: \( \frac{1000 \times 10 \times 5}{100} = \frac{50000}{100} = ₹500 \)
• Simple interest on ₹800 at 10% for 5 years: \( \frac{800 \times 10 \times 5}{100} = \frac{40000}{100} = ₹400 \)
• Difference: \( ₹500 - ₹400 = ₹100 \). This matches the condition given in the problem.

So, the period is 5 years.

Given difference = ₹100

Therefore, \(20T = 100\)

\(T = 5\) years.

Revision Table: Simple Interest Concepts

Additional Information: Exploring Simple Interest Further

Simple interest is a basic concept in finance. Unlike compound interest, where interest is calculated on the principal plus accumulated interest, simple interest is always calculated only on the initial principal amount.

• Calculation Simplicity: Simple interest is easier to calculate compared to compound interest.
• Applications: It is often used for short-term loans, simple deposits, and calculations involving fixed principal amounts.
• Graphing Simple Interest: When plotted on a graph with time on the x-axis and simple interest on the y-axis, the relationship is linear, showing a constant increase in interest per unit of time.
• Effect of Variables:
  • Increasing the principal (\(P\)) increases the simple interest earned or paid.
  • Increasing the rate (\(R\)) increases the simple interest earned or paid.
  • Increasing the period (\(T\)) increases the simple interest earned or paid.

Understanding simple interest is fundamental before moving on to more complex concepts like compound interest or annuities.