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Time required to Pour Concrete Walkway










































Person A, working alone, can pour a concrete walkway in 9 hours. Person B, working alone, can pour the same walkway in 6 hours. How long will it take both people to pour the concrete walkway working together?

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Sep 12, 2012
Pour the concrete walkway working together
by: Staff


Answer:

Part I


The solution


1. Start with an equation showing the “Total Work Completed”:

     (work completed by Person “A”) + (work completed by Person “B”) = Total Work Completed


         For the sake of brevity:

         A_work + B_work = Total_work


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

         A_work + B_work = Total_work


         Divide each side of the equation by Total_work

         (A_work + B_work = Total_work) / (Total_work) = (Total_work) / (Total_work)

         (A_work) / (Total_work) + (B_work) / (Total_work) = 1


     As you can see, the fraction of the entire job completed by “A” plus the fraction of the entire job completed by “B” = 1

         Again, for the sake of brevity:

         A_work_fraction + B_work_fraction = 1


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

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

         How fast A works: Since A can complete the job in 9 hours; he completes 1/9 of the job per hour. A’s speed is 1/9 per hour.

         How long A works: time in hours = unknown = t₁

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

             A_work_fraction = (1/9)*t₁

         (explanation: if A works 1 hour, his fraction of the job completed is 1/9; if A works 2 hours, his fraction of the job completed is 2/9, and so on.)

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

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Sep 12, 2012
Pour the concrete walkway working together
by: Staff

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Part II

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

         How fast B works: Since B can complete the job in 6 hours, he completes 1/6 of the job per hour. B’s speed is 1/6 per hour.

         How long B works: time in hours = unknown = t₂

         B’s fraction of the job = Speed * time

             B_work_fraction = (1/6)*t₂

         (explanation: if B works 1 hour, his fraction of the job completed is 1/6; if B works 2 hours, his fraction of the job completed is 2/6, and so on.)


3. the equation now becomes:

             A_work_fraction + B_work_fraction = 1

             (1/9)*t₁ + (1/6)*t₂ = 1


         Evaluation of t₁ and t₂

         Since both person A and person B work start the job at the same time and work together until the job is completed:

             t₁ = t₂ = t = total time to complete job


     substitute t for t₁ and t₂

         (1/9)*t₁ + (1/6)*t₂ = 1

         (1/9)*t + (1/6)*t = 1


     Solve for t (the total time to complete the job)

         (1/9)*t + (1/6)*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 two original denominators: 9 and 6

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Sep 12, 2012
Pour the concrete walkway working together
by: Staff


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Part III


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

     - Factor the numbers 9 and 6 into prime factors. Use exponents to show how many times each prime factor is listed.

         9 = 3²
         6 = 2¹*3¹

     - List every prime number listed as a factor of either 9 or 6.
         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: 3², 2¹
         Multiply the factors with the highest exponent to compute the LCM:
         LCM = 3² * 2¹ = 18
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     Convert the denominator of both fractions on the left side of the equation to 18


(2/2) * (1/9)*t + (3/3) * (1/6)*t = 1

(2*1*t) / (2*9) + (3*1*t) / (3*6) = 1

(2t) / (18) + (3t) / (18) = 1

(2t) / (18) + (3t) / (18) = 1

     Add the two fractions on the left side of the equations

(2t) / (18) + (3t) / (18) = 1

(5t) / (18) = 1

5t / 18 = 1

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Sep 12, 2012
Pour the concrete walkway working together
by: Staff


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Part IV
     Multiply both sides of the equation by 18 to remove the 18 as the denominator for the fraction on the left side of the equation

         18*(5t)/(18) = 18*1

         (5t)*(18/18) = 18*1

         (5t)*(1) = 18*1

         5t = 18


     Divide each side of the equation by 5

         5t = 18

         5t/5 = 18/5

         t * (5/5) = 18/5

         t * (1) = 18/5

         t = 18/5

         t = 3.6 hours

         t = 3 hours 36 minutes


The final answer to the question is: t = 3 hours 36 minutes

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4. check your answer by substituting 3.6 hours for t in the original equation:


         (1/9)*t + (1/6)*t = 1

         (1/9)*3.6 + (1/6)*3.6 = 1

         0.4 + 0.6 = 1

         1 = 1, OK → t = 3.6 hours is a valid solution





Thanks for writing.

Staff
www.solving-math-problems.com



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