# 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

 Jul 01, 2014 time to complete job working together by: Staff Answer Part I 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 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 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.) -----------------------------------

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

 Jul 01, 2014 time to complete job working together by: Staff ----------------------------------- Part III 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 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 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) 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 -------------------------------------------------------- 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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