#### Question

# Help with a rates problem?

Joe and Moe can mow the lawn in 30 minutes if they work together. If Moe works twice as fast as Joe, how long does it take Joe to mow the lawn alone?

Please explain how to solve this! Thanks!

(The answer isn't 10 minutes)

#### Answer

Let J be the time it takes Joe to mow the lawn, and M be the time it takes Moe to mow the lawn. For a parallel work problem, the time it takes them to work together, 30 minutes, is given by:

1/J + 1/M = 1/30

We also know that Moe works twice as fast as Joe (i.e., Joe takes twice as long as Moe):

J = 2M

Substitute J = 2M into the first equation and solve for Moe's time, M:

1/2M + 1/M = 1/30

Get everything over the lowest common denominator, 30M:

15/30M + 30/30M = M/30M

Now cancel the denominator for all terms:

15 + 30 = M

M = 45 minutes

...and:

J = 2M = 2(45) = 90 minutes

Moe takes 45 minutes to mow the lawn alone, and Joe takes 90 minutes to mow the lawn alone.