A person P asks one of his three friends X as to how much money he had. X replied, “If Y gives me ` 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?

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`). 5. State the correct answer clearly at the beginning. Let's format the solution carefully following all rules. ```html ₹360

Correct Answer: ₹360

Step-by-Step Explanation:

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

Step 1: Formulate the equations from the given conditions.

Condition 1: "If Y gives me ₹ 40, then Y will have half of as much as Z."

When Y gives ₹ 40 to X, Y's remaining amount becomes (y - 40).

According to the statement:

$y-40=\frac{z}{2}$

Multiplying both sides by 2:

$2y-80=z$

$z=2y-80$    --- (Equation 1)

Condition 2: "If Z gives me ₹ 40, then three of us will have equal amount."

When Z gives ₹ 40 to X:

• X's new amount = x + 40

• Y's amount remains = y

• Z's new amount = z - 40

Since all three amounts become equal:

$x+40=y=z-40$

From $y=z-40$, we get:

$z=y+40$    --- (Equation 2)

From $x+40=y$, we get:

$x=y-40$    --- (Equation 3)

Step 2: Solve for y.

Equating the expressions for z from Equation 1 and Equation 2:

$2y-80=y+40$

$2y-y=40+80$

$y=120$

Step 3: Calculate the individual amounts for X and Z.

Substitute $y=120$ into Equation 2:

$z=120+40=160$

Substitute $y=120$ into Equation 3:

$x=120-40=80$

Step 4: Find the total amount of money.

Total amount = x + y + z

$\text{Total amount}=80+120+160=360$

Thus, the total amount of money that X, Y, and Z have altogether is ₹360.