A man borrows a total sum of Rs. `10,000` from two sources. To one he pays `10% `and to the other` 5%` per annum simple interest. If the total interest paid by him is Rs. 700. How much did he borrow at each rate of interest?

Answer

To solve the problem, we need to find out how much the man borrowed at each interest rate, given that he borrowed a total of Rs. 10,000 and paid a total interest of Rs. 700.

### Step-by-Step Solution:

1. **Define Variables:**
Let \( X \) be the amount borrowed at 10% interest, and \( Y \) be the amount borrowed at 5% interest.

2. **Set Up the First Equation:**
Since the total amount borrowed is Rs. 10,000, we can write the first equation as:
\[ X + Y = 10,000 \quad \text{(Equation 1)} \]

3. **Set Up the Second Equation:**
The total interest paid is Rs. 700. The interest from the amount borrowed at 10% for one year is \( \frac{10}{100} \times X = 0.1X \), and the interest from the amount borrowed at 5% for one year is \( \frac{5}{100} \times Y = 0.05Y \). Therefore, we can write the second equation as:
\[ 0.1X + 0.05Y = 700 \quad \text{(Equation 2)} \]

4. **Eliminate Decimals in Equation 2:**
To eliminate the decimals, we can multiply the entire equation by 100:
\[ 10X + 5Y = 70,000 \quad \text{(Equation 3)} \]

5. **Rearrange Equation 1:**
From Equation 1, we can express \( Y \) in terms of \( X \):
\[ Y = 10,000 - X \quad \text{(Equation 4)} \]

6. **Substitute Equation 4 into Equation 3:**
Now, substitute \( Y \) from Equation 4 into Equation 3:
\[ 10X + 5(10,000 - X) = 70,000 \]
Simplifying this gives:
\[ 10X + 50,000 - 5X = 70,000 \]
\[ 5X + 50,000 = 70,000 \]
\[ 5X = 70,000 - 50,000 \]
\[ 5X = 20,000 \]
\[ X = \frac{20,000}{5} = 4,000 \]

7. **Find \( Y \):**
Now that we have \( X \), we can find \( Y \) using Equation 4:
\[ Y = 10,000 - X = 10,000 - 4,000 = 6,000 \]

8. **Conclusion:**
Therefore, the man borrowed Rs. 4,000 at 10% interest and Rs. 6,000 at 5% interest.

### Final Answer:
- Amount borrowed at 10%: Rs. 4,000
- Amount borrowed at 5%: Rs. 6,000