A person P asks one of his three friends X as to how much money he had. X replied, “If Y gives me Rs 40, then Y will have half of as much as Z, but if Z gives me @ 40, then three of us will have equal amount.” What is the total amount of money that X, Y and Z have?

Correct Option

Let the initial amounts of money with X, Y, and Z be represented by Rs. x, Rs. y, and Rs. z, respectively.

From the first condition, "If Y gives me Rs 40, then Y will have half of as much as Z":

• X's money becomes $x+40$
• Y's money becomes $y-40$
• Z's money remains $z$

The condition translates to the equation: $y-40=\frac{z}{2}$

This simplifies to: $2y-80=z$ (Equation 1)

From the second condition, "if Z gives me Rs 40, then three of us will have equal amount":

• X's money becomes $x+40$
• Y's money remains $y$
• Z's money becomes $z-40$

This condition implies that the amounts are equal: $x+40=y=z-40$

From $x+40=y$, we derive: $x=y-40$

From $y=z-40$, we derive: $z=y+40$ (Equation 2)

Now, substitute Equation 2 into Equation 1:

$y+40=2y-80$

Rearranging the terms to solve for y:

$40+80=2y-y$

$120=y$

Thus, Y has Rs. 120.

Using the value of y, calculate x and z:

• $x=y-40=120-40=80$
• $z=y+40=120+40=160$

Therefore, X has Rs. 80, Y has Rs. 120, and Z has Rs. 160.

The total amount of money X, Y, and Z have is the sum of their individual amounts:

Total = $x+y+z=80+120+160=360$

The total amount is Rs. 360.

Incorrect Options

Options 1 (Rs 420), 3 (Rs 300), and 4 (Rs 270) are incorrect. The unique solution derived from the given conditions establishes the total amount as Rs. 360. Any other total would contradict the mathematical relationships specified in the problem statement.