Question

Matt is a tour guide who gives several tours over the course of a morning shift and an evening shift. His morning shift tours are given to groups of 10 people while his evening shift tours are given to groups of 5 people. If the total number of tourists in the morning shift is the same as the total number of tourists in the evening shift, what is the minimum number of tourists that Matt guides in each shift?

Answer 1

Let \( m \) be the number of tours Matt gives during the morning shift and \( e \) be the number of tours he gives during the evening shift.

In the morning shift, each tour has 10 people, so the total number of tourists Matt guides in the morning is given by:

\[
10m
\]

In the evening shift, each tour has 5 people, so the total number of tourists in the evening is:

\[
5e
\]

According to the problem, the total number of tourists in the morning shift equals the total number of tourists in the evening shift, which gives us the equation:

\[
10m = 5e
\]

To simplify this equation, we can divide both sides by 5:

\[
2m = e
\]

This means that the number of evening tours \( e \) is twice the number of morning tours \( m \).

Now, we need to find the minimum number of tourists guided in each shift. The total number of tourists guided during the morning shift is:

\[
10m
\]

And during the evening shift:

\[
5e = 5(2m) = 10m
\]

To minimize the number of tourists, we need to find the smallest integer values for \( m \) and \( e \). The smallest positive integer for \( m \) is 1. Thus:

- If \( m = 1 \), then from \( e = 2m \):

\[
e = 2 \cdot 1 = 2
\]

Next, we calculate the total number of tourists for both shifts:

- In the morning shift:

\[
10m = 10 \cdot 1 = 10
\]

- In the evening shift:

\[
5e = 5 \cdot 2 = 10
\]

Both shifts yield a total of 10 tourists, satisfying the equality \( 10m = 5e \).

Thus, the minimum number of tourists that Matt guides in each shift is:

\[
\boxed{10}
\]