# If Sam can do a job in 4 days that Lisa can do in 6 days and Tom can do in 2 days, how long would the job take if Sam, Lisa, and Tom worked together to complete it?

Oct 25, 2016

They could complete the job together in $\frac{12}{11}$ days.

#### Explanation:

Sam: 4 days
Lisa: 6 days
Tom: 2 days

This is a "rate" problem. The rate is jobs per day, or job/day.

Sam's rate is one job completed in 4 days, or $\frac{1}{4}$, i.e. in one day Sam could complete $\frac{1}{4}$th of the job.

Lisa's rate is one job completed in 6 days, or $\frac{1}{6}$. She could complete $\frac{1}{6}$th of the job in one day.

Tom's rate is one job completed in 2 days, or $\frac{1}{2}$. He could complete $\frac{1}{2}$ of the job in one day.

Together, they could complete $\frac{1}{4} + \frac{1}{6} + \frac{1}{2}$ of the job in one day.

We are trying to find the rate at which Sam, Lisa and Tom could complete the job together, or one job in $x$ days, for a rate of $\frac{1}{x}$.

$\frac{1}{4} + \frac{1}{6} + \frac{1}{2} = \frac{1}{x}$

The least common denominator is $12 x$.

Multiplying through by the LCD gives:

$12 x \left(\frac{1}{4}\right) + 12 x \left(\frac{1}{6}\right) + 12 x \left(\frac{1}{2}\right) = 12 x \left(\frac{1}{x}\right)$

$3 x + 2 x + 6 x = 12$

$11 x = 12$

$11 \frac{x}{11} = \frac{12}{11}$

$x = \frac{12}{11}$ days