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?

Question

Ethan Adkins · Accepted Answer

They could complete the job together in #12/11# days.

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 #1/4#, i.e. in one day Sam could complete #1/4#th of the job. Lisa's rate is one job completed in 6 days, or #1/6#. She could complete #1/6#th of the job in one day. Tom's rate is one job completed in 2 days, or #1/2#. He could complete #1/2# of the job in one day. Together, they could complete #1/4 + 1/6 +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 #1/x#. #1/4 + 1/6+1/2 =1/x# The least common denominator is #12x#. Multiplying through by the LCD gives: #12x ( 1/4) +12x (1/6) + 12x ( 1/2) = 12x ( 1/x)# #3x + 2x + 6x=12# #11x=12# #11x/11 =12/11# #x = 12/11# days

Avery Alexander · Answer

undefined To find out how long it would take for Sam, Lisa, and Tom to complete the job together, you can use the formula:

1 / (Sam's rate) + 1 / (Lisa's rate) + 1 / (Tom's rate) = 1 / (Combined rate)

Sam's rate = 1 job / 4 days = 1/4
Lisa's rate = 1 job / 6 days = 1/6
Tom's rate = 1 job / 2 days = 1/2

1 / (1/4) + 1 / (1/6) + 1 / (1/2) = 1 / (Combined rate)

Solve for the combined rate:

1 / (1/4) + 1 / (1/6) + 1 / (1/2) = 6/24 + 4/24 + 12/24 = 22/24 = 11/12

So, the combined rate of Sam, Lisa, and Tom working together is 11/12.

Now, find out how long it would take for them to complete the job together:

Combined rate = 1 / (Time taken together)

11/12 = 1 / (Time taken together)

Time taken together = 12 / 11

So, it would take Sam, Lisa, and Tom $ \frac{12}{11} $ days to complete the job together.