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=2z
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.