I need to know the number of hours it will take Jessica and Ian to clean the house together?

Question

In one hour we see that Jessica can clean 1/3 of the house, and Ian can clean 1/2 of the house.

If x = number of hours then we can use the equation x/3 + x/2 = 1

Solve for x to find the number of hours it will take them to clean the house together.

Answer 1

The unusual thing about “combination of work” problems is that we are given units such as “hours per job,” but must use “jobs per hour” to solve the problem.

For example, “If it takes Sue 4 hours to mow the lawn and it takes Joe 6 hours to mow the same lawn, how long will it take them when working together [each with a lawn mower]?”

Sue works at 4 hours per lawn, or (4 hrs)/(1 lawn). This is also ((1 lawn)/(4 hours)).

Now, if x is the time it takes the two of them working together, then Sue does:
                  (x hrs)*((1 lawn)/(4 hrs)) = (x/4) lawn
           Joe, working slower, does:
                  (x hrs)(*(1 lawn)/(6 hrs)) = (x/6) lawn

When they are combined [again, PLZ notice the very important units]:
        Sue’s portion       +         Joe’s portion    =     1 lawn
          (x/4) lawn         +           (x/6) lawn     =     1 lawn


          x/4     +     x/6    =    1                 [lawn cancels]
           3x     +      2x    =   12                [multiply by LCM of 12]
                   5x            =    12
                   x    =     12/5
                   x    = 2 2/5

They work together for 2 2/5 hours [that is, 2.4 hours, or 2 hours 24 minutes].

 

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For this problem, Jessica can clean the whole house in 3 hours.  Ian can clean the same whole house in 2 hours.

If they work together for x hours:

    Amount Jessica Cleaned       +     Amount Ian Cleaned             =      1 house

    (x hrs)*((1 house)/(3 hrs)   +     (x hrs)*((1 house)/(2 hrs))    =     1 house

 

       x/3                                 +             x/2                               =  1      [hrs and house cancel]

 

   2x + 3x = 6                [multiply by LCM=6)

     5x = 6

      x = 6/5

      x = 1 1/5

 

 Working together, it will take Jessica and Ian 1 1/5 hours = 1.2 hours = 1 hour 12 minutes.

Answer 2

To solve the equation for x, multiply both sides by the common denominator, which is 6

 

2x + 3x = 6

5x = 6

x = 6/5 = 1.2

 

It will take 1.2 hours or 1 hour and 12 minutes for the two of them to clean the house if they work together