word problem

Question

pastry chef lauren and her assistant chef mike are preparing fancy desserts for a banquet. the job would take mike working alone, 9 hours longer than it would
take lauren, working alone. working together lauren and mike can complete the job in 20 hours. how long would it take each chef to do the job working alone

Answer 1

Let it take Lauren L hours to complete the task.  Then, it takes Mike L+9 hours.

 

The key is understanding how to get from here to how long it takes them working together.

 

If Lauren works L hours, her rate per hour is 1/L

 

Mike's rate is 1/(L+9)

 

Then, their combined rate is 1/L + 1/(L+9) = ((L+9) + L)/(L(L+9)) = (2L+9)/(L^2+9L)

 

Thus, they complete the deserts in the reciprocal of the rate, or (L^2+9L)/ (2L+9)

 

Since it takes 20 hours for them to complete the deserts, (L^2+9L)/ (2L+9) = 20

 

Multiplying both sides by (2L+9),

 

(L^2+9L) =  (2L+9)(20)

L^2+9L = 40L + 180

L^2+9L - 40L - 180 = 0

L^2- 31L - 180 = 0

(L - 36)(L+5) = 0

L = -5, 36

Ignore the negative solution

The solution is 36.

It takes Lauren 36 hours and Mike 45 hours.

Note that this answer immediately passes the plausibility test, because if they both took 40 hours, working together, it would take half as long, or 20 hours, and while 36 < 40, 45 > 40

 

Now, check the answer

1/36 + 1/45 = 1/(9*4) + 1/(9*5) = 5/(9*4*5) + 4/(9*4*5) = 5/180 + 4/180 = 9/180 = 1/20

 

 

Answer 2

If Lauren takes x hours alone, then Mike takes (x + 9) hours alone.

So in 1 hour the fraction of the whole job Lauren does = 1/x, and Mike's fraction = 1/(x + 9). Since it takes 20 hours together, then in 1 hour together they get 1/20 of the job done.

 

So: 1/x  +  1/(x + 9) = 1/20

Multiply both sides by 20x(x + 9) to give:

20(x + 9) + 20x =  x(x + 9) or 40x + 180 = x² + 9x

 

Rearranging gives: x² - 31x - 180 = 0  This factors into:

(x + 5)(x - 36) = 0, so x = 36 (ignore x = -5)

 

So alone, Lauren takes 36 hours, and Mike takes 45 hours.