Rational Equations

Question

Joe and Bob had to pick rocks from the baseball field. Joe could do it alone in 3 hours. Bob could finish alone in 2 hours. How long would it take them to finish the job together?

Answer 1

Let's start by finding Joe and Bob's hourly rates of rock-picking.

 

Since Joe can pick rocks in 3 hours, every hour he picks 1/3 of the rocks. Similarly, since Bob can pick rocks in 2 hours, every hour he picks 1/2 of the rocks. So, Joe's rate is 1/3 of the field per hour; Bob's rate is 1/2 of the field per hour.

 

Now, we'll set up an equation using the rates. Let h equal the number of hours worked. That means that the portion of the field Joe finishes in h hours is 1/3h. For example, if he worked for 2 hours, he would have cleaned (1/3)×2 = 2/3 of the field. The portion of the field Bob finishes in h hours is 1/2h. For example, if he worked for 3/2 hours, he would have cleaned (1/2)×(3/2) = 3/4 of the field.

 

We want the portion that Joe finishes plus the portion that Bob finishes to equal 1 (the whole field). That gives us the following equation:

 

1/3h + 1/2h = 1

 

Now, let's solve for h:

 

(1/3 + 1/2)h = 1

(2/6 + 3/6)h = 1

5/6h = 1

h = 1 × 6/5

h = 6/5

 

So, together, Joe and Bob's work will take only 6/5 hours, which is also equal to 1 1/5 hours or 1 hour and 12 minutes.