One says, ''Give me a hundred, friend I shall then become twice as rich as you''. The other replies, ''If you give me ten, I shall be six times as rich as you''. Tell me what is the amount of their (respective) capital?

Answer

To solve the problem, we need to set up a system of linear equations based on the statements made by the two friends. Let's denote the amount of money the first friend has as \( x \) and the amount the second friend has as \( y \).

### Step 1: Set up the equations based on the statements.

1. The first friend says, "Give me a hundred, friend I shall then become twice as rich as you."
- After receiving 100, the first friend will have \( x + 100 \).
- The second friend will have \( y - 100 \).
- The equation from this statement is:
\[ x + 100 = 2(y - 100) \]
- Simplifying this gives:
\[ x + 100 = 2y - 200 \implies x - 2y + 300 = 0 \quad \text{(Equation 1)} \]

2. The second friend replies, "If you give me ten, I shall be six times as rich as you."
- After receiving 10, the second friend will have \( y + 10 \).
- The first friend will have \( x - 10 \).
- The equation from this statement is:
\[ y + 10 = 6(x - 10) \]
- Simplifying this gives:
\[ y + 10 = 6x - 60 \implies 6x - y - 70 = 0 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations.

We have the following two equations:
1. \( x - 2y + 300 = 0 \)
2. \( 6x - y - 70 = 0 \)

Let's solve these equations simultaneously.

#### Rearranging Equation 1:
From Equation 1:
\[ x = 2y - 300 \]

#### Substituting into Equation 2:
Now substitute \( x \) from Equation 1 into Equation 2:
\[ 6(2y - 300) - y - 70 = 0 \]
Expanding this gives:
\[ 12y - 1800 - y - 70 = 0 \]
Combining like terms:
\[ 11y - 1870 = 0 \]
Thus:
\[ 11y = 1870 \implies y = \frac{1870}{11} \implies y = 170 \]

### Step 3: Find \( x \).

Now that we have \( y \), we can find \( x \):
Substituting \( y = 170 \) back into Equation 1:
\[ x = 2(170) - 300 = 340 - 300 = 40 \]

### Conclusion:
The respective capitals of the two friends are:
- First friend (x): \( 40 \) rupees
- Second friend (y): \( 170 \) rupees