The total distance between warehouse M and warehouse N is 900 km. Courier Y and Courier Z can travel the entire route between M and N in ‘X’ hours and (X + 4) hours respectively. If Courier Z and Courier Y begin their journeys from warehouse M at 6.00 am and 8.00 am respectively and both couriers cross paths at 10.30 am, what is the distance between warehouse M and the location where they meet?

Correct Answer: The correct option is 450 km.

Step-by-step Explanation:

1. Understand the travel duration of each courier until they cross paths:

Courier Z starts from warehouse M at 6:00 am and both couriers cross paths at 10:30 am.

Time taken by Courier Z = 10:30 am - 6:00 am = 4.5 hours = $\frac{9}{2}$ hours.

Courier Y starts from warehouse M at 8:00 am and both couriers cross paths at 10:30 am.

Time taken by Courier Y = 10:30 am - 8:00 am = 2.5 hours = $\frac{5}{2}$ hours.

2. Express the speeds of the couriers in terms of X:

The total distance between warehouse M and warehouse N is 900 km.

Speed of Courier Y, S_Y = $\frac{900}{\mathrm{X}}$ km/h

Speed of Courier Z, S_Z = $\frac{900}{\mathrm{X}+4}$ km/h

3. Calculate the distance traveled by each courier to the meeting point:

Since both Courier Y and Courier Z started from warehouse M and travel along the same route to the point where they cross paths, the distance covered by both couriers from warehouse M to the meeting point must be equal.

Distance covered by Courier Z = S_Z × Time_Z

$\text{Distance}=\frac{900}{\mathrm{X}+4}\times \frac{9}{2}=\frac{4050}{\mathrm{X}+4}$

Distance covered by Courier Y = S_Y × Time_Y

$\text{Distance}=\frac{900}{\mathrm{X}}\times \frac{5}{2}=\frac{2250}{\mathrm{X}}$

4. Equate the two distance expressions to solve for X:

$\frac{4050}{\mathrm{X}+4}=\frac{2250}{\mathrm{X}}$

Divide both sides by 450 to simplify:

$\frac{9}{\mathrm{X}+4}=\frac{5}{\mathrm{X}}$

Cross-multiplying gives:

$9\mathrm{X}=5(\mathrm{X}+4)$

$9\mathrm{X}=5\mathrm{X}+20$

$4\mathrm{X}=20\Rightarrow \mathrm{X}=5$

5. Find the meeting distance from warehouse M:

Now substitute X = 5 into the distance expression for Courier Y:

$\text{Distance}=\frac{2250}{5}=450\text{km}$

Thus, the distance between warehouse M and the meeting location is 450 km.