CEMC Gauss (Grade 8), 2015

To find the image of $PQ$, we reflect points $P$ and $Q$ across the $x$-axis, then join them.

Since $P$ is 3 units above the $x$-axis, then the reflection of $P$ across the $x$-axis is 3 units below the $x$-axis at the same $x$-coordinate.

That is, point $T$ is the image of $P$ after it is reflected across the $x$-axis.

Similarly, after a reflection across the $x$-axis, the image of point $Q$ will be 6 units below the $x$-axis but have the same $x$-coordinate as $Q$.

That is, point $U$ is the image of $Q$ after it is reflected across the $x$-axis.
Therefore, the line segment $TU$ is the image of $PQ$ after it is reflected across the $x$-axis.

Solution 1

The multiples of 5 between 1 and 99 are: $$5,10,15,20,25,30,35,40,45,50,55,60,65,70, 75,80,85,90,95.$$ Of these, only $10,20,30,40,50,60,70,80,$ and $90$ are even.
Therefore, there are 9 even whole numbers between 1 and 99 that are multiples of 5.

Solution 2

To create an even multiple of 5, we must multiply 5 by an even whole number (since 5 is odd, multiplying 5 by an odd whole number creates an odd result).

The smallest positive even multiple of 5 is $5\times2=10$.

The largest even multiple of 5 less than 99 is $5\times18=90$.

That is, multiplying 5 by each of the even numbers from 2 to 18 results in the only even multiples of 5 between 1 and 99.
Since there are 9 even numbers from 2 to 18 (inclusive), then there are 9 even whole numbers between 1 and 99 that are multiples of 5.