Understanding the Simple Interest Problem

The question asks us to find the annual interest rate, denoted by x%, at which Nirav and Mehul borrowed money. We are given their principal amounts, the time period, and the difference in the simple interest paid by them.

Let's break down the information provided:

• Nirav's Principal (P₁): Rs. 4000
• Mehul's Principal (P₂): Rs. 5000
• Time Period (T): 2.5 years
• Interest Rate (R): x% per annum
• Mehul paid Rs. 125 more interest than Nirav. This means Simple Interest for Mehul (SI₂) - Simple Interest for Nirav (SI₁) = Rs. 125.

We need to use the formula for calculating simple interest to find the value of x.

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

Calculating Simple Interest for Nirav and Mehul

Let's calculate the simple interest for both Nirav and Mehul using the given information and the rate x%.

For Nirav:

• Principal (P₁): 4000
• Rate (R): x
• Time (T): 2.5

Simple Interest for Nirav (SI₁):

$$SI_1 = \frac{4000 \times x \times 2.5}{100}$$

$$SI_1 = \frac{4000 \times x \times \frac{5}{2}}{100}$$

$$SI_1 = \frac{4000 \times x \times 5}{200}$$

$$SI_1 = \frac{20000x}{200}$$

$$SI_1 = 100x$$

For Mehul:

• Principal (P₂): 5000
• Rate (R): x
• Time (T): 2.5

Simple Interest for Mehul (SI₂):

$$SI_2 = \frac{5000 \times x \times 2.5}{100}$$

$$SI_2 = \frac{5000 \times x \times \frac{5}{2}}{100}$$

$$SI_2 = \frac{5000 \times x \times 5}{200}$$

$$SI_2 = \frac{25000x}{200}$$

$$SI_2 = 125x$$

Using the Interest Difference to Find x

We know that Mehul paid Rs. 125 more interest than Nirav. So, the difference between their simple interests is 125.

$$SI_2 - SI_1 = 125$$

Substitute the expressions we found for SI₁ and SI₂:

$$125x - 100x = 125$$

Combine the terms with x:

$$25x = 125$$

Now, solve for x by dividing both sides by 25:

$$x = \frac{125}{25}$$

$$x = 5$$

The value of x is 5. This means the interest rate is 5% per annum.

Verification of the Calculated Rate

Let's check if an interest rate of 5% results in the given interest difference.

• Rate (R) = 5%
• Time (T) = 2.5 years

Nirav's Interest:

$$SI_1 = \frac{4000 \times 5 \times 2.5}{100} = \frac{4000 \times 5 \times \frac{5}{2}}{100} = \frac{4000 \times 25}{200} = \frac{100000}{200} = 500$$

Nirav's interest is Rs. 500.

Mehul's Interest:

$$SI_2 = \frac{5000 \times 5 \times 2.5}{100} = \frac{5000 \times 5 \times \frac{5}{2}}{100} = \frac{5000 \times 25}{200} = \frac{125000}{200} = 625$$

Mehul's interest is Rs. 625.

Interest Difference:

$$SI_2 - SI_1 = 625 - 500 = 125$$

The difference in interest is Rs. 125, which matches the information given in the question. Thus, the calculated value of x = 5 is correct.

Revision Table: Key Simple Interest Concepts

Additional Information: Simple Interest vs. Compound Interest

It's important to distinguish between simple interest and compound interest.

• Simple Interest: Interest is calculated only on the original principal amount. The interest earned in previous periods is not added to the principal for calculating the current period's interest. This problem uses simple interest.
• Compound Interest: Interest is calculated on the principal amount plus any accumulated interest from previous periods. The interest "compounds" over time, leading to a higher total interest amount compared to simple interest over the same period and rate.

The formula for compound interest is different, and problems involving compound interest require a different approach.