Understanding the Simple Interest Problem

This question asks us to find the simple interest earned on two parts of a total sum, where the sum is divided such that the simple interest generated by each part at different rates is equal. We are given the total sum, the annual simple interest rates for the two parts, and the time period (one year). The key is that the simple interest on each part is equal.

Setting up the Equations for Simple Interest

Let the total sum be Rs. 6000. This sum is divided into two parts. Let these parts be \(P_1\) and \(P_2\). Therefore, we know:

\(P_1 + P_2 = 6000 \quad (Equation\ 1)\)

The time period for calculating simple interest is 1 year for both parts (\(T = 1\)).

The annual rate of simple interest for the first part is 10 percent (\(R_1 = 10\%\)).

The annual rate of simple interest for the second part is 15 percent (\(R_2 = 15\%\)).

The formula for simple interest (SI) is:

\(SI = \frac{Principal \times Rate \times Time}{100}\)

For the first part, the simple interest (\(SI_1\)) is:

\(SI_1 = \frac{P_1 \times R_1 \times T}{100} = \frac{P_1 \times 10 \times 1}{100} = \frac{10 P_1}{100} = \frac{P_1}{10}\)

For the second part, the simple interest (\(SI_2\)) is:

\(SI_2 = \frac{P_2 \times R_2 \times T}{100} = \frac{P_2 \times 15 \times 1}{100} = \frac{15 P_2}{100} = \frac{3 P_2}{20}\)

Using the Equal Interest Condition

The problem states that the annual simple interest on the first part is equal to the annual simple interest on the second part. So, \(SI_1 = SI_2\).

This gives us the equation:

\(\frac{P_1}{10} = \frac{3 P_2}{20}\)

To simplify this equation, we can cross-multiply or multiply both sides by the least common multiple of 10 and 20, which is 20:

\(20 \times \frac{P_1}{10} = 20 \times \frac{3 P_2}{20}\)

\(2 P_1 = 3 P_2 \quad (Equation\ 2)\)

Solving for the Principal Amounts

Now we have a system of two linear equations with two variables, \(P_1\) and \(P_2\):

1. \(P_1 + P_2 = 6000\)
2. \(2 P_1 = 3 P_2\)

From Equation 1, we can express \(P_1\) in terms of \(P_2\):

\(P_1 = 6000 - P_2\)

Substitute this expression for \(P_1\) into Equation 2:

\(2 (6000 - P_2) = 3 P_2\)

Distribute the 2 on the left side:

\(12000 - 2 P_2 = 3 P_2\)

Add \(2 P_2\) to both sides of the equation:

\(12000 = 3 P_2 + 2 P_2\)

\(12000 = 5 P_2\)

Now, solve for \(P_2\) by dividing both sides by 5:

\(P_2 = \frac{12000}{5}\)

\(P_2 = 2400\)

So, the second part of the sum is Rs. 2400.

Now substitute the value of \(P_2\) back into Equation 1 to find \(P_1\):

\(P_1 + 2400 = 6000\)

\(P_1 = 6000 - 2400\)

\(P_1 = 3600\)

So, the first part of the sum is Rs. 3600.

Calculating the Simple Interest on Each Part

We need to find the simple interest on each part for one year. Since the simple interest is equal for both parts, we only need to calculate it for one part.

Using the first part (\(P_1 = 3600\), \(R_1 = 10\%\), \(T = 1\) year):

\(SI_1 = \frac{P_1}{10} = \frac{3600}{10} = 360\)

Using the second part (\(P_2 = 2400\), \(R_2 = 15\%\), \(T = 1\) year):

\(SI_2 = \frac{3 P_2}{20} = \frac{3 \times 2400}{20} = \frac{7200}{20} = 360\)

As expected, the simple interest on each part is indeed equal, and it is Rs. 360.

Final Answer Summary

The sum of Rs. 6000 was divided into two parts: Rs. 3600 and Rs. 2400. The annual simple interest on each part is Rs. 360.

Revision Table: Simple Interest Calculation

Here's a quick table summarising the key elements and results of the simple interest calculation for this problem.

Additional Information: Simple Interest Concepts

Understanding simple interest is fundamental in many financial calculations. Simple interest is calculated only on the principal amount, and it does not compound (interest is not added to the principal to earn more interest). Here are some related concepts:

• Principal (P): The initial amount of money borrowed or invested.
• Rate (R): The percentage at which simple interest is charged or earned annually. It's usually given as a percentage per annum (p.a.).
• Time (T): The duration for which the principal is borrowed or invested, usually in years.
• Simple Interest (SI): The interest calculated using the formula \(SI = \frac{P \times R \times T}{100}\).
• Amount (A): The total sum of the principal and the interest earned. \(A = P + SI\).

In problems involving the division of a sum with equal simple interest, setting up simultaneous equations based on the total sum and the equality of interests is a common method to find the individual principal amounts before calculating the interest.