A student solves the following problem. Timothy follows a trail that is 6 miles up a mountain. He takes the same trail down the mountain. It takes him half as long to get down the mountain as it took him to hike up the mountain. If it takes him 6 hours to complete the total hike, what was his rate, in miles/hour, while hiking up the mountain?

The student’s solution is shown.

Student work showing x is the time up the mountain, 0.5x is the time down the mountain which create the equations x + 0.5x = 6 hours and 1.5x = hours. Student divides 6 by 1.5 to get x = 4 which means it took Timothy 4 hours to hike up the mountain and since the total hike is 6 miles, 6 divided by 4 means his rate is 1.5 miles per hour

Use the student’s solution to complete the statements.

The student
when he related x to the total time of 6 hours.

To write his equation, the student used the idea of
.

To solve his equation for x, the student used
, which is the opposite of
.

The student justified his solution by
.