Solution:

We are given that a sum of Rs. 7,800 is lent in two parts such that:

1. The simple interest on one part at 10% p.a. for 5 years is equal to the simple interest on the second part at 9% p.a. for 6 years.
2. We need to determine the two sums.

Step 1: Key Variables

Let:

• $x$ = first part lent at 10% p.a. for 5 years
• $y$ = second part lent at 9% p.a. for 6 years
• Total sum = $x + y = 7800$

The condition states: $\text{Simple Interest on first part} = \text{Simple Interest on second part}.$

Step 2: Simple Interest Formula

The formula for simple interest is: $\text{SI} = \frac{P \cdot R \cdot T}{100}$ where $P$ = principal, $R$ = rate, $T$ = time.

For the first part: $\text{SI}_1 = \frac{x \cdot 10 \cdot 5}{100} = \frac{50x}{100} = 0.5x$

For the second part: $\text{SI}_2 = \frac{y \cdot 9 \cdot 6}{100} = \frac{54y}{100} = 0.54y$

Given that the two interests are equal: $0.5x = 0.54y$

Step 3: Solve for $x$ and $y$

From $0.5x = 0.54y$, rearrange to express $x$ in terms of $y$: $x = \frac{0.54y}{0.5} = 1.08y$

Now substitute into $x + y = 7800$: $1.08y + y = 7800$ $2.08y = 7800$ $y = \frac{7800}{2.08} = 3750$

Thus: $x = 1.08y = 1.08 \cdot 3750 = 4050$

Step 4: Verify the Result

The two parts are $x = 4050$ and $y = 3750$.

• For the first part ($4050$): $\text{SI}_1 = \frac{4050 \cdot 10 \cdot 5}{100} = 2025$

• For the second part ($3750$): $\text{SI}_2 = \frac{3750 \cdot 9 \cdot 6}{100} = 2025$

The interests are equal, satisfying the condition.

Final Answer:

The two sums are Rs. 4,050 and Rs. 3,750.

Correct Option: (a).

Do you need further clarifications or a detailed breakdown of any step?

Here are some related questions to explore:

1. What is the formula for simple interest, and how is it applied?
2. How can you split amounts based on interest conditions?
3. Can you solve similar problems using algebraic substitution?
4. How does changing the interest rate or time affect simple interest?
5. How is the proportionality method used in solving split amount problems?

Tip: Always verify your answers by plugging the values back into the original conditions. This ensures accuracy.