Problem Solving

BTGModeratorVI wrote: ↑

Tue Mar 31, 2020 5:10 am

A professional janitor can clean a certain high school in 8 hours, working at a constant rate. A student sentenced to detention can clean that same high school in 20 hours, also working at a constant rate. If the student is paid $7 total per hour and the janitor is paid $21 per hour, how much more would it cost the school to pay the janitor to do the job himself than it would to pay the student and the janitor to do the job together?

A. -$56
B. -$6
C. $0
D. $6
E. $8

Answer: E
Source: Veritas Prep

For work questions, there are two useful rules:

Rule #1: If a person can complete an entire job in k hours, then in one hour, the person can complete 1/k of the job
Example: If it takes Sue 5 hours to complete a job, then in one hour, she can complete 1/5 of the job. In other words, her work rate is 1/5 of the job per hour

Rule #2: If a person completes a/b of the job in one hour, then it will take b/a hours to complete the entire job
Example: If Sam can complete 1/8 of the job in one hour, then it will take him 8/1 hours to complete the job.
Likewise, if Joe can complete 2/3 of the job in one hour, then it will take him 3/2 hours to complete the job.

Let’s use these rules to solve the question. . . .
A professional janitor can clean a certain high school in 8 hours
So (applying rule #1), the janitor can clean 1/8 of the school in ONE HOUR

A student sentenced to detention can clean that same high school in 20 hours
So (applying rule #1), the student can clean 1/20 of the school in ONE HOUR

So, COMBINED, the student and janitor can clean (1/8 + 1/20) of the school in ONE HOUR
1/8 + 1/20 = 5/40 + 2/40 = 7/40
So, in ONE HOUR they can clean 7/40 of the school.

Applying rule #2, it will take them 40/7 hours to clean the ENTIRE school.

The janitor earns $21/hour and the student earns $7/hour, so their combined rate is $28/hour.
Their combined wages = (pay rate)(time) = ( $28/hour)(40/7 hours) = $160

Working ALONE, the janitor takes 8 hours and earns $21/hour
So, working alone, the janitor's earnings = (pay rate)(time) = ($21/hour)(8 hours) = $168

$168 - $160 = $8, so the answer is E

Cheers,
Brent

BTGModeratorVI wrote: ↑

Tue Mar 31, 2020 5:10 am

A professional janitor can clean a certain high school in 8 hours, working at a constant rate. A student sentenced to detention can clean that same high school in 20 hours, also working at a constant rate. If the student is paid $7 total per hour and the janitor is paid $21 per hour, how much more would it cost the school to pay the janitor to do the job himself than it would to pay the student and the janitor to do the job together?

A. -$56
B. -$6
C. $0
D. $6
E. $8

Answer: E
Source: Veritas Prep

If the janitor cleans the school by himself, it will cost the school 21 x 8 = $168.

The janitor’s rate is 1/8, and the student’s rate is 1/20. If we let t = the number of hours it takes if the janitor works together with the student to clean the school, we can create the equation:

(1/8 + 1/20) x t = 1

t/8 + t/20 = 1

5t + 2t = 40

7t = 40

t = 40/7

Therefore, if they work together, it will cost the school 21 x 40/7 + 7 x 40/7 = 120 + 40 = $160. We see that it will cost the school 168 - 160 = $8 more if the janitor works alone.

Answer: E