# Combine Rate Problem

I sometimes would give a combined rate problem to my Precalculus students and my Calculus students when I was teaching that course as a warm up problem before class. Here is the problem:

It takes Juan 2 hour to complete a job. It takes his brother Tim 3 hours to complete the same job. How long would it take them if they did the same job together?

We know that Rate x Time = Distance. We can set up the equation in this manner.

(R)(T) + (R)(T) = 1 job

Juan: Rate to do the job is 1/2 work per hour because it takes him 2 hours to do the 1 job.

Tim: Rate to do the job is 1/3 work per hour because it takes him 3 hours to do the 1 job.

(1/2)T + (1/3)T = 1. Solve for T for time it takes to do 1 job.

6[(1/2)T + (1/3)T] = 1(6) Multiply both sides of the equation by the LCD = 6.

3T + 2T = 6 Distributive Property

5T = 6

T = 6/5 = 1.2 hours or 1 hour and 12 minutes

What I found was that about half of my Precalculus students were not able to do this problem. The reason for this is because after Algebra 1, there are certain problems that students don't see until they take the SAT or ACT math portion of the test. This is like any other skill set. If you don't use the skill set, you lose the skill set.