Question 1

Explanation

Determining Meeting Interval

It is known that the three vans meet for the $4$-th time $18$ hours after the initial meeting.

Meeting for the $4$-th time means that $3$ meeting intervals (LCM) have passed.

Let $T$ be the joint meeting interval (LCM of periods A, B, and C).

So, the three vans meet together every $6$ hours.

Determining Value of x

The joint meeting interval ($T$) is the Least Common Multiple (LCM) of the periods of each van.

Known periods:

• Van A: $2$ hours
• Van B: $3$ hours
• Van C: $x$ hours

Thus:

We know that $\text{LCM}(2, 3) = 6$. Therefore, we must find the value of $x$ such that $\text{LCM}(6, x) = 6$. For $\text{LCM}(6, x) = 6$, $x$ must be a factor of $6$.

From the available options, we are looking for a value of $x$ that is a factor of $6$.

The values $5$, $7$, $8$, and $9$ are not factors of $6$, so they do not satisfy the condition $\text{LCM}(6, x) = 6$.

For $x = 6$, we get $\text{LCM}(6, 6) = 6$, which satisfies the condition.

So, the satisfying value of $x$ is $6$.