Understanding the Simple Interest Problem

This question involves dividing a total sum of money into two parts and calculating the simple interest earned on each part. The key condition is that the simple interest from both parts is equal, even though the principal amounts, rates, and time periods are different.

We are given:

• Total sum of money = Rs. 2600.00
• Part 1: Rate of Interest (\(R_1\)) = 10% per annum, Time (\(T_1\)) = 5 years
• Part 2: Rate of Interest (\(R_2\)) = 9% per annum, Time (\(T_2\)) = 6 years
• Condition: Simple Interest on Part 1 = Simple Interest on Part 2

We need to find the amount lent out as the first part (at 10% per annum).

Applying the Simple Interest Formula

The formula for calculating 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 Time period in years

Setting up Equations for the Two Parts

Let the first part of the sum lent out at 10% be \( P_1 \). Let the second part of the sum lent out at 9% be \( P_2 \).

According to the question, the total sum is Rs. 2600.00. So, we have our first equation:

Equation 1: \( P_1 + P_2 = 2600 \)

Now, let's calculate the simple interest for each part:

• Simple Interest on the first part (\(SI_1\)) = \( \frac{P_1 \times R_1 \times T_1}{100} = \frac{P_1 \times 10 \times 5}{100} = \frac{50 P_1}{100} \)
• Simple Interest on the second part (\(SI_2\)) = \( \frac{P_2 \times R_2 \times T_2}{100} = \frac{P_2 \times 9 \times 6}{100} = \frac{54 P_2}{100} \)

The question states that the simple interest on the first part is the same as the interest on the second part:

Equation 2: \( SI_1 = SI_2 \)

\( \frac{50 P_1}{100} = \frac{54 P_2}{100} \)

Solving the Equations

Now we need to solve the two equations simultaneously to find the values of \( P_1 \) and \( P_2 \).

From Equation 2, we can simplify by multiplying both sides by 100:

\( 50 P_1 = 54 P_2 \)

We can divide both sides by 2 to simplify further:

\( 25 P_1 = 27 P_2 \)

From this relationship, we can express \( P_2 \) in terms of \( P_1 \):

\( P_2 = \frac{25 P_1}{27} \)

Now substitute this expression for \( P_2 \) into Equation 1:

\( P_1 + \frac{25 P_1}{27} = 2600 \)

To solve for \( P_1 \), find a common denominator (which is 27):

\( \frac{27 P_1}{27} + \frac{25 P_1}{27} = 2600 \)

\( \frac{27 P_1 + 25 P_1}{27} = 2600 \)

\( \frac{52 P_1}{27} = 2600 \)

Now, isolate \( P_1 \) by multiplying both sides by \( \frac{27}{52} \):

\( P_1 = 2600 \times \frac{27}{52} \)

We can simplify the calculation. \( 2600 = 52 \times 50 \). So, \( \frac{2600}{52} = 50 \).

\( P_1 = 50 \times 27 \)

Calculating the final value for \( P_1 \):

\( P_1 = 1350 \)

So, the part lent out at 10% per annum is Rs. 1350.

Finding the Second Part (Optional)

Although not asked, we can find the second part (\( P_2 \)) using Equation 1:

\( P_2 = 2600 - P_1 \)

\( P_2 = 2600 - 1350 \)

\( P_2 = 1250 \)

The second part lent out at 9% per annum is Rs. 1250.

Verification

Let's verify if the simple interest is indeed equal for both parts with \( P_1 = 1350 \) and \( P_2 = 1250 \).

• \( SI_1 = \frac{1350 \times 10 \times 5}{100} = \frac{1350 \times 50}{100} = \frac{67500}{100} = 675 \)
• \( SI_2 = \frac{1250 \times 9 \times 6}{100} = \frac{1250 \times 54}{100} = \frac{67500}{100} = 675 \)

Since \( SI_1 = SI_2 = 675 \), our calculation is correct.

Final Answer

The part lent out at 10% per annum is Rs. 1350.

Let's look at the given options:

The calculated amount for the part lent at 10% is Rs. 1350, which matches option 4.

Revision Table: Simple Interest Calculation Steps

Additional Information: Simple Interest Concepts

Simple interest is a method of calculating the interest charge on a loan or investment. It is the easiest interest method to calculate.

• Principal (P): The initial amount of money borrowed or invested.
• Rate (R): The percentage of the principal charged as interest over a specific period, usually per year.
• Time (T): The duration for which the money is borrowed or invested, usually in years.
• Simple Interest (SI): The interest calculated only on the principal amount.
• Amount (A): The total sum of money at the end of the period, which is the principal plus the simple interest (\(A = P + SI\)).

Simple interest differs from compound interest, where interest is calculated on the principal amount and also on the accumulated interest from previous periods.