We are tasked with dividing ₹6000 into two parts, such that the simple interest on the first part for 2 years at 6% per annum is equal to the simple interest on the second part for 3 years at 8% per annum.

Let the two parts be:

• The first part: ₹$x$
• The second part: ₹$6000 - x$

Simple Interest Formula:

The formula for 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 in years.

Step 1: Calculate the simple interest for the first part.

For the first part, the principal is ₹$x$, the rate is 6%, and the time is 2 years. The simple interest is: $SI_1 = \frac{x \times 6 \times 2}{100} = \frac{12x}{100} = \frac{3x}{25}$

Step 2: Calculate the simple interest for the second part.

For the second part, the principal is ₹$6000 - x$, the rate is 8%, and the time is 3 years. The simple interest is: $SI_2 = \frac{(6000 - x) \times 8 \times 3}{100} = \frac{24(6000 - x)}{100} = \frac{24(6000 - x)}{100} = \frac{144000 - 24x}{100} = \frac{1440 - 0.24x}{1}$

Step 3: Set up the equation for equality of interest.

We are given that the simple interest on both parts is equal: $\frac{3x}{25} = \frac{1440 - 0.24x}{1}$

Step 4: Solve for $x$.

1. Multiply both sides by 25 to eliminate the fraction on the left side: $3x = 25(1440 - 0.24x)$ $3x = 36000 - 6x$

2. Move all terms involving $x$ to one side: $3x + 6x = 36000$ $9x = 36000$

3. Solve for $x$: $x = \frac{36000}{9} = 4000$

Step 5: Determine the second part.

The second part will be: $6000 - x = 6000 - 4000 = 2000$

Final Answer:

The two parts of ₹6000 are ₹4000 and ₹2000.

Would you like a more detailed explanation or have any questions on the steps? Here are some related questions:

1. How can we use the compound interest formula to calculate interest for the same scenario?
2. How would the solution change if the interest rates were different for each part?
3. How does the time period affect the calculation of simple interest?
4. What happens to the distribution of ₹6000 if the interest rates are reversed (6% for 3 years and 8% for 2 years)?
5. How can we generalize this problem to work for more than two parts?

Tip: Always check the units of time and rate in the interest formula to make sure they are consistent (e.g., years and percentage per annum).