You are here:

Advanced Math/Rate-time problem

Advertisement


Question
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.

Answer
Questioner:   Isaac
Country:   Illinois, United States
Category:   Advanced Math
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.
.........................................................

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.

Now how about your 5/6 of the job business?

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.

Your equation is:

10/19 + t/40 = 5/6

Take it from there.

Advanced Math

All Answers


Answers by Expert:


Ask Experts

Volunteer


Paul Klarreich

Expertise

I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

Experience

I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.

Education/Credentials
-----------

©2016 About.com. All rights reserved.