Calculating Simple Interest and Sum Lent

This problem involves simple interest. Simple interest is calculated only on the principal amount, or the initial sum lent. It does not compound, meaning interest is not earned on previously accumulated interest.

The formula for calculating simple interest (SI) is:

$$\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}$$

Where:

• \(\text{P}\) is the Principal amount (the sum lent).
• \(\text{R}\) is the annual Rate of interest.
• \(\text{T}\) is the Time period in years.

In this specific problem, we are given the following information:

• Annual Rate of interest (\(\text{R}\)) = 8 percent
• Time period (\(\text{T}\)) = 9 years
• The interest received in 9 years is Rs. 560 less than the sum lent.

Let's denote the sum lent (Principal) as \(\text{P}\).

According to the problem, the simple interest (\(\text{SI}\)) received is Rs. 560 less than the sum lent. We can write this relationship as an equation:

$$\text{SI} = \text{P} - 560$$

Now, let's calculate the simple interest (\(\text{SI}\)) using the formula with the given rate and time, in terms of \(\text{P}\):

$$\text{SI} = \frac{\text{P} \times 8 \times 9}{100}$$

$$\text{SI} = \frac{72\text{P}}{100}$$

We can simplify the fraction \(\frac{72}{100}\) by dividing both the numerator and denominator by their greatest common divisor, which is 4:

$$\text{SI} = \frac{18\text{P}}{25}$$

Now we have two expressions for the simple interest (\(\text{SI}\)):

1. \(\text{SI} = \text{P} - 560\) (from the problem statement)
2. \(\text{SI} = \frac{18\text{P}}{25}\) (from the simple interest formula)

Since both expressions represent the same simple interest, we can set them equal to each other:

$$\frac{18\text{P}}{25} = \text{P} - 560$$

To solve for \(\text{P}\), we need to get all the terms involving \(\text{P}\) on one side of the equation. First, let's eliminate the denominator by multiplying both sides of the equation by 25:

$$25 \times \left(\frac{18\text{P}}{25}\right) = 25 \times (\text{P} - 560)$$

$$18\text{P} = 25\text{P} - 25 \times 560$$

Calculate the product of 25 and 560:

$$25 \times 560 = 14000$$

So the equation becomes:

$$18\text{P} = 25\text{P} - 14000$$

Now, move the \(18\text{P}\) term to the right side or move the \(25\text{P}\) term to the left side. Let's move \(18\text{P}\) to the right side by subtracting \(18\text{P}\) from both sides:

$$0 = 25\text{P} - 18\text{P} - 14000$$

$$0 = 7\text{P} - 14000$$

Now, move the constant term to the other side by adding 14000 to both sides:

$$14000 = 7\text{P}$$

Finally, to find \(\text{P}\), divide both sides by 7:

$$\text{P} = \frac{14000}{7}$$

$$\text{P} = 2000$$

So, the sum lent is Rs. 2000.

Let's verify this. If P = 2000, then SI = P - 560 = 2000 - 560 = 1440. Using the formula, SI = \(\frac{2000 \times 8 \times 9}{100} = \frac{20 \times 8 \times 9}{1} = 160 \times 9 = 1440\). The simple interest calculated by the formula matches the condition that it is 560 less than the principal (2000 - 560 = 1440). This confirms our calculation is correct.

The sum lent is Rs. 2000.

Looking at the options provided, Rs. 2000 is option 1.

Revision Table: Simple Interest Calculation

Additional Information on Simple Interest

Simple interest is one of the easiest methods to calculate interest on a principal amount. It is widely used in short-term loans and easy calculations. Here are some key points about simple interest:

• Constant Interest Amount: The amount of interest earned each year is the same because it is always calculated on the original principal amount.
• Comparison with Compound Interest: Unlike compound interest, simple interest does not add the earned interest back to the principal for subsequent calculations. Compound interest leads to faster growth of money over time because interest is earned on both the principal and the accumulated interest.
• Applications: Simple interest is typically used for personal loans between individuals, short-term borrowings, and some types of bonds.
• Total Amount (Maturity Value): The total amount to be repaid at the end of the loan period is the sum of the Principal and the Simple Interest: \(\text{Amount} = \text{P} + \text{SI}\). In this problem, the amount after 9 years would be \(2000 + 1440 = \text{Rs. } 3440\).

Understanding the relationship between Principal, Rate, Time, and Simple Interest is crucial for solving such problems. The key here was setting up the correct equation based on the problem statement: \(\text{SI} = \text{P} - 560\).