Divide rupees fifty five hundred in three parts so that their interest at the red fort percent simple interest 42 years to point five years and five years respectively are equal

To divide ₹5500 into three parts such that the simple interest earned by each part over 4 years, 2.5 years, and 5 years is the same, we can use the formula for simple interest and set up the necessary conditions.

### Simple Interest Formula:
\[
\text{Simple Interest (SI)} = \frac{P \times R \times T}{100}
\]
Where:
- \(P\) is the principal (the part of ₹5500)
- \(R\) is the rate of interest
- \(T\) is the time in years

### Given:
- The total sum is ₹5500.
- Time periods: 4 years, 2.5 years, and 5 years.
- Rate of interest \(R\) is the same for all three parts.
- The simple interest from each part is equal.

Let the three parts be \(P_1\), \(P_2\), and \(P_3\).

### Condition:
Since the interest from each part is the same, we have:
\[
\frac{P_1 \times R \times 4}{100} = \frac{P_2 \times R \times 2.5}{100} = \frac{P_3 \times R \times 5}{100}
\]
This simplifies to:
\[
P_1 \times 4 = P_2 \times 2.5 = P_3 \times 5
\]

Let the common interest factor be \(k\). So, we can express each principal in terms of \(k\):
\[
P_1 = \frac{k}{4}, \quad P_2 = \frac{k}{2.5}, \quad P_3 = \frac{k}{5}
\]

### Total Sum Equation:
The sum of the parts must be ₹5500:
\[
P_1 + P_2 + P_3 = 5500
\]
Substitute the expressions for \(P_1\), \(P_2\), and \(P_3\):
\[
\frac{k}{4} + \frac{k}{2.5} + \frac{k}{5} = 5500
\]

Now, we can solve for \(k\).The value of \( k \) is approximately 6470.59.

Now, using this \( k \), we can calculate each part of the total ₹5500 as follows:

\[
P_1 = \frac{k}{4}, \quad P_2 = \frac{k}{2.5}, \quad P_3 = \frac{k}{5}
\]

Let's calculate these values.The three parts of ₹5500 are approximately:

- \( P_1 = ₹1617.65 \) (for 4 years),
- \( P_2 = ₹2588.24 \) (for 2.5 years),
- \( P_3 = ₹1294.12 \) (for 5 years).

These amounts will generate the same simple interest over the respective time periods.

Would you like further details or have any questions?

Here are five related questions for further exploration:
1. How would the division change if the time periods were different?
2. What if the rate of interest was also different for each part?
3. How would the division be affected if compound interest was used instead?
4. Can you derive a general formula for dividing any amount in this way?
5. How does the interest change when the total amount is altered?

**Tip:** Simple interest depends directly on time, so varying time periods require different principal amounts for equal interest.

Divide rupees fifty five hundred in three parts so that their interest at the rate of simple interest over 4 years, 2.5 years, and 5 years respectively are equal.

Learn how to divide ₹5500 into three parts such that the simple interest earned over 4 years, 2.5 years, and 5 years is equal. This problem involves algebra and the simple interest formula. Suitable for students in Grades 9-10.

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