One says, "Give me a hundred rupee, friend! I shall then become twice as rich as 
you are." The other replies, "If you give me ten rupees, I shall be six times as rich as 
you are." Tell me how much money both have initially?

Let the amount of money the first person has be \( x \), and the amount of money the second person has be \( y \).

### First Condition:
The first person says that if the second person gives them 100 rupees, they will become twice as rich as the second person. This can be written as:
\[
x + 100 = 2(y - 100)
\]
Simplifying the equation:
\[
x + 100 = 2y - 200
\]
\[
x = 2y - 300 \quad \text{(Equation 1)}
\]

### Second Condition:
The second person says that if the first person gives them 10 rupees, they will be six times as rich as the first person. This can be written as:
\[
y + 10 = 6(x - 10)
\]
Simplifying the equation:
\[
y + 10 = 6x - 60
\]
\[
y = 6x - 70 \quad \text{(Equation 2)}
\]

### Solving the System of Equations:
Now, substitute the expression for \( x \) from Equation 1 into Equation 2:

\[
y = 6(2y - 300) - 70
\]
Expanding:
\[
y = 12y - 1800 - 70
\]
\[
y = 12y - 1870
\]
Rearranging the terms:
\[
1870 = 11y
\]
\[
y = \frac{1870}{11} = 170
\]

Now substitute \( y = 170 \) into Equation 1 to find \( x \):
\[
x = 2(170) - 300 = 340 - 300 = 40
\]

### Final Answer:
- The first person has 40 rupees.
- The second person has 170 rupees.

Would you like a detailed explanation or further clarifications?

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Here are 5 relative questions for further exploration:
1. How would the problem change if the first person asked for 200 rupees instead of 100?
2. Can this problem be generalized to a similar one involving three people?
3. How would the equations change if both people had the same amount of money at the start?
4. What would happen if we solved this problem with different currencies instead of rupees?
5. How does this system of linear equations relate to real-life financial transactions?

**Tip:** Always verify your solutions by substituting them back into the original conditions.

One says, 'Give me a hundred rupee, friend! I shall then become twice as rich as you are.' The other replies, 'If you give me ten rupees, I shall be six times as rich as you are.' Tell me how much money both have initially?

Learn how to solve a system of linear equations to find the initial amounts of money two people have. One person asks for 100 rupees and claims they will be twice as rich, while the other person claims they would be six times as rich if given 10 rupees. This problem involves algebraic concepts and solving linear equations, suitable for students in Grades 9-11.

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