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 six times as rich as you. "Find the amount of their capitals.

Options

A. ? 40, ? 170
B. ? 60, ?180
C. X 80, X200
D. Connot of the determined

Answer

To solve the problem, we will define the amounts of money that the two friends have and then set up equations based on their statements.

### Step 1: Define Variables
Let:
- \( X \) = the amount of money the first person has.
- \( Y \) = the amount of money the second person has.

### Step 2: Set Up the First Equation
The first person says, "Give me a hundred friend! I shall then become twice as rich as you."
This can be expressed mathematically as:
\[ X + 100 = 2(Y - 100) \]
This equation states that after receiving 100, the first person's total will be twice the second person's total after giving away 100.

### Step 3: Simplify the First Equation
Expanding the equation:
\[ X + 100 = 2Y - 200 \]
Rearranging gives:
\[ X - 2Y = -300 \quad \text{(Equation 1)} \]

### Step 4: Set Up the Second Equation
The second person replies, "If you give me ten, I shall be six times as rich as you."
This can be expressed as:
\[ Y + 10 = 6(X - 10) \]
This equation states that after receiving 10, the second person's total will be six times the first person's total after giving away 10.

### Step 5: Simplify the Second Equation
Expanding the equation:
\[ Y + 10 = 6X - 60 \]
Rearranging gives:
\[ 6X - Y = 70 \quad \text{(Equation 2)} \]

### Step 6: Solve the System of Equations
Now we have a system of two equations:
1. \( X - 2Y = -300 \)
2. \( 6X - Y = 70 \)

We can solve these equations simultaneously. From Equation 1, we can express \( X \) in terms of \( Y \):
\[ X = 2Y - 300 \]

Now, substitute \( X \) into Equation 2:
\[ 6(2Y - 300) - Y = 70 \]
Expanding gives:
\[ 12Y - 1800 - Y = 70 \]
Combining like terms:
\[ 11Y - 1800 = 70 \]
Adding 1800 to both sides:
\[ 11Y = 1870 \]
Dividing by 11:
\[ Y = 170 \]

### Step 7: Find \( X \)
Now substitute \( Y = 170 \) back into the equation for \( X \):
\[ X = 2(170) - 300 \]
Calculating gives:
\[ X = 340 - 300 = 40 \]

### Final Answer
Thus, the amounts of their capitals are:
- First person: \( X = 40 \)
- Second person: \( Y = 170 \)