Let's break down the problem and solve it step by step.

Step 1: Calculate the total effective investments

Each person invests money for different periods, so we need to calculate the effective investments by multiplying the amount of money they invest by the number of months they invest for.

• X's effective investment:
  $50,000 \times 5 = 250,000$ rupees

• Y's effective investment:
  $70,000 \times 5 = 350,000$ rupees

• Z's effective investment:
  $40,000 \times 10 = 400,000$ rupees

Now, the total effective investment is: $250,000 + 350,000 + 400,000 = 1,000,000 \text{ rupees}$

Step 2: Set up profit sharing ratio

The profit is to be divided in the ratio of their effective investments. Thus, the ratio in which the profit is divided among X, Y, and Z is: $\text{X : Y : Z} = 250,000 : 350,000 : 400,000 = 25 : 35 : 40$

Let the total profit be denoted by $P = 2,00,000$ rupees.

The part of the profit that is divided among X, Y, and Z is given in the ratio of their investments. Also, it is mentioned that Z receives 4 times the amount that X receives.

Step 3: Set up an equation for the amounts received by X, Y, and Z

Let the amount received by X be $x$.
Then, according to the problem, the amount received by Z is $4x$.

The ratio of X to Z is $25:40$, which simplifies to $5:8$. Therefore: $\frac{Z}{X} = \frac{8}{5}$ Thus, if $X$ receives $x$, then $Z$ must receive $8x$.