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Sam can do a job in 4 days, and Lisa can do it in 6 days












































Rate of work problem
time to complete job working together



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

Comments for Sam can do a job in 4 days, and Lisa can do it in 6 days

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Jul 01, 2014
time to complete job working together
by: Staff


Answer

Part I


Work-Rate-Problem:  how long would the job take if Sam., Lisa,  and Tom worked together? 

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1. The equation showing the “Total Work Completed” is:

(work completed by Sam) + (work completed by Lisa) + (work completed by Tom) = Total Work Completed

For the sake of brevity, this can be written:


Sam_work + Lisa_work + Tom_work = Total_work




Work-Rate-Problem:  total work completed by Sam, Lisa,  and Tom. 

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2. THIS IS THE KEY. Convert the equation shown above into FRACTIONS.

Divide each side of the equation by Total_work

(Sam_work + Lisa_work + Tom_work = Total_work) / (Total_work) = (Total_work) / (Total_work)


(Sam_work) / (Total_work) + (Lisa_work) / (Total_work) + (Tom_work) / (Total_work) = 1



As you can see, the fraction of the entire job completed by “Sam” plus the fraction of the entire job completed by “Lisa” plus the fraction of the entire job completed by “Tom” = 1

This can be written:


Sam_work_fraction + Lisa_work_fraction + Tom_work_fraction = 1





convert equation into fractions 

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a. What is Sam’s fraction of the job ?


The portion of the job completed by Sam depends upon two things: 1) How fast Sam works, and 2) how long Sam works.


How fast Sam works: Since Sam can complete the job in 4 days; he completes 1/4 of the job per day. Sam’s speed is 1/4 per day.

How long Sam works: time in days = unknown = t₁

Sam’s fraction of the job = (Sam’s Speed) * (Sam’s time spent working)

Sam_work_fraction = (1/4)*t₁

(explanation: if Sam works 1 day, his fraction of the job completed is 1/4; if Sam works 2 days, his fraction of the job completed is 2/4, and so on.)



work rate problem:  Sam’s fraction of the job. 

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Jul 01, 2014
time to complete job working together
by: Staff


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Part II


b. What is Lisa’s fraction of the job ?


The portion of the job completed by Lisa depends upon two things: 1) How fast Lisa works, and 2) how long Lisa works.


How fast Lisa works: Since Lisa can complete the job in 6 days, she completes 1/6 of the job per day. Lisa’s speed is 1/6 per day.

How long Lisa works: time in days = unknown = t₂

Lisa’s fraction of the job = (Lisa’s Speed) * (Lisa’s time spent working)

Lisa_work_fraction = (1/6)*t₂

(explanation: if Lisa works 1 day, her fraction of the job completed is 1/6; if Lisa works 2 days, her fraction of the job completed is 2/6, and so on.)



work rate problem:  Lisa’s fraction of the job. 

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c. What is Tom’s fraction of the job ?


The portion of the job completed by Tom depends upon two things: 1) How fast Tom works, and 2) how long Tom works.


How fast Tom works: Since Tom can complete the job in 2 days, he completes 1/2 of the job per day. Tom’s speed is 1/2 per day.

How long Tom works: time in days = unknown = t₃

Tom’s fraction of the job = (Tom’s Speed) * (Tom’s time spent working)

Tom_work_fraction = (1/2)*t₃

(explanation: if Tom works 1 day, his fraction of the job completed is 1/2; if Tom works 2 days, his fraction of the job completed is 2/2, or, job completed.)





work rate problem:  Tom’s fraction of the job. 

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3. substitute the values for Sam’s fraction, Lisa’s fraction, and Tom’s fraction of the job into the original equation:


Sam_work_fraction + Lisa_work_fraction + Tom_work_fraction = 1



Sam_work_fraction = (1/4)*t₁

Lisa_work_fraction = (1/6)*t₂

Tom_work_fraction = (1/2)*t₃



(1/4)*t₁ + (1/6)*t₂ + (1/2)*t₃ = 1




work rate problem:  substitute known values into the original equation 

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Evaluation of the times worked: t₁, t₂, and t₃


Sam, Lisa, and Tom start the job at the same time and work together until the job is completed:

t₁ = t₂ = t₃ = t = total time to complete job working together


substitute t for t₁, t₂, and t₃

(1/4)*t₁ + (1/6)*t₂ + (1/2)*t₃ = 1


(1/4)*t + (1/6)*t + (1/2)*t = 1



work rate problem:  Sam, Lisa, and Tom all work exactly the same amount of time.   

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4. Solve for t (the total time to complete the job):


(1/4)*t + (1/6)*t + (1/2)*t = 1



Convert each fraction on the left side of the equation to the same common denominator.


The new common denominator can be the least common multiple (LCM) of the three original denominators: 4, 6, and 2


To find the LCM, you can use the prime factorization method

- Factor the denominators 4, 6, and 2 into prime factors. Use exponents to show how many times each prime factor is listed.


4 = 2²

6 = 2¹ * 3¹

2 = 2¹



- List every prime number listed as a factor of 4, 6, and 2.


Prime Number Factors Listed: 2, 3


- List these same prime factors again, but include the highest exponent listed for each


Prime Number Factors with highest exponent: 2², 3¹

Multiply the factors with the highest exponent to compute the LCM: LCM = 2² * 3¹ = 12



- Convert the denominator of all three fractions on the left side of the equation to 12


(1/4)*t + (1/6)*t + (1/2)*t = 1

(3/3)*(1/4)*t + (2/2)*(1/6)*t + (6/6)* (1/2)*t = 1

(3/12)*t + (2/12)*t + (6/12)*t = 1







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Jul 01, 2014
time to complete job working together
by: Staff


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Part III


work rate problem:  solve for time

convert each fraction to a common denominator 

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Add the three fractions on the left side of the equation


(3/12)*t + (2/12)*t + (6/12)*t = 1

(3t/12) + (2t/12) + (6t/12) = 1

(3t + 2t + 6t)/12 = 1

11t/12 = 1





work rate problem:  add the three fractions on the left side of the equation 

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Multiply both sides of the equation by 12 to remove the 12 as the denominator for the fraction on the left side of the equation


12*(11t)/(12) = 12*1

(11t)*(12/12) = 12*1

(11t)*(1) = 12*1

11t = 12




work rate problem:  multiply both sides of the equation by twelve to remove the 12 as the denominator for the fraction on the left side of the equation 

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Divide each side of the equation by 11


11t = 12

11t/11 = 12/11

t * (11/11) = 12/11

t * (1) = 12/11

t = 12/11

t = 1 and 1/11 days (or 1.090909091 days)



 work rate problem:  divide both sides of the equation by eleven 

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if one day is eight hours of work

t = 1 day, 1 hour, and 13 minutes



The final answer to the question is: t = 1 day, 1 hour, and 13 minutes; or 12/11 days



work rate problem:  final answer 

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5. check your answer by substituting 12/11 days for t in the original equation:



(1/4)*t + (1/6)*t + (1/2)*t = 1

(1/4)*(12/11) + (1/6) *(12/11) + (1/2) *(12/11) = 1


(3/11) + (2/11) + (6/11) = 1

(3 + 2 + 6) /11 = 1

(11) /11 = 1

1 = 1


1 = 1, OK → t = 12/11 days is a valid solution







Thanks for writing.

Staff
www.solving-math-problems.com


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