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Any help on this word problem would be appreciated.

It takes Ann 4 fewer minutes to clean the chalkboards than it takes Bill. They started cleaning the chalkboards together at 9:32 A.M. They should have finished the job in 360/19 minutes but after 360/19 minutes, the job was only 5/6 done because Ann worked continuously, while Bill quit early. Find the number of minutes after 9:32 A.M. that Bill quit working. Express your answer as an improper fraction reduced to lowest terms.

Questioner:   Isaac
Country:   Illinois, United States
Private:   Yes  <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< Changed
Subject:   Difficult word problem
Question:   Any help on this word problem would be appreciated.

It takes Ann 4 fewer minutes to clean the chalkboards than it takes Bill.

>>>> excuse me, but we used to call them blackboards.

They started cleaning the chalkboards together at 9:32 A.M. They should have finished the job in 360/19 minutes but after 360/19 minutes, the job was only 5/6 done because Ann worked continuously, while Bill quit early. Find the number of minutes after 9:32 A.M. that Bill quit working. Express your answer as an improper fraction reduced to lowest terms.
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Let

A = time needed by Ann to clean one BB
B = time needed by Bill to clean one BB

then

1/A = Ann's rate of cleaning in BB/minute
1/B = Bill's rate of cleaning in BB/minute

1/A + 1/B is the rate of cleaning when they work together.

"They should have finished the job in 360/19 minutes"
means
"They clean 19/360 of a BB per minute."

Or:  1/A + 1/B = 19/360

"It takes Ann 4 fewer minutes to clean the chalkboards than it takes Bill."

means A = B - 4

Then we have:

1/(B-4) + 1/B = 19/360  <<<<  a fractional equation.

Solve it  [ LCD = 360B(B-4) ]   <<<< multiply through

360B + 360(B- 4) = 19B(B - 4)

360B + 360B - 1440 = 19B^2 - 76B

720B - 1440 = 19B^2 - 76B

0 = 19B^2 - 796B + 1440   <<<< a quadratic

0 = (19B -  36)(B - 40)   <<<< An easy factoring. [Yeah, right!]

B = 36/19, not possible (Ann would clean up in a negative time)

B = 40;  then A = 36

Ann cleans in 36 minutes, Bill in 40.

Let t = the time Bill worked.

In those 360/19 minutes,

Ann worked all 360/19.

Bill worked t minutes.

Ann cleaned (360/19)(1/36) BB's. = 10/19
Bill cleaned t(1/40)          =  t/40

The total cleaning was 5/6 of a BB.

10/19 + t/40 = 5/6

Take it from there.

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#### Paul Klarreich

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