Two musicians, Maria and Perry, work at independent constant rates to tune a warehouse full of instruments. If both

M7MBA wrote: ↑

Wed May 20, 2020 7:12 am

Two musicians, Maria and Perry, work at independent constant rates to tune a warehouse full of instruments. If both musicians start at the same time and work at their normal rates, they will complete the job in 45 minutes. However, if Perry were to work at twice Maria’s rate, they would take only 20 minutes. How long would it take Perry, working alone at his normal rate, to tune the warehouse full of instruments?

A. 1 hr 20 min
B. 1 hr 45 min
C. 2 hr
D. 2 hr 20 min
E. 3 hr

[spoiler]OA=E[/spoiler]

Solution:

We can let the time it takes Perry to complete the job alone = p and the time it takes Maria to complete the job alone = m. Thus, Perry’s rate = 1/p and Maria’s rate = 1/m. Since they complete the job in 45 minutes, we use the formula work = rate x time to get:

(1/m)45 + (1/p)45 = 1

45/m + 45/p = 1

Multiplying the entire equation by mp, we have:

45p + 45m = mp

We are also given that if Perry were to work at twice Maria’s rate, they would take only 20 minutes. Since Maria’s rate is 1/m, Perry’s rate would be 2/m. We can create the following equation to determine p:

(2/m)20 + (1/m)20 = 1

40/m + 20/m = 1

Multiplying the entire equation by m, we have:

40 + 20 = m

m = 60

Recalling that 45p + 45m = mp, we can substitute m = 60 in the equation and solve for p:

45p + 45(60) = 60p

3p + 3(60) = 4p

180 = p

Since Perry’s time is 180 minutes, and 60 minutes = 1 hour, it takes him 3 hours to complete the job at his normal rate.

Alternate Solution:

Mary working together with Perry working at twice Mary’s rate is equivalent to three people working together at Mary’s rate. Since the job is finished in 20 minutes under these conditions, it takes Mary 3 x 20 = 60 minutes to finish the job alone.

Since Mary and Perry can finish the job together in 45 minutes, we have:

1/60 + 1/P = 1/45

1/P = 1/45 - 1/60

1/P = 4/180 - 3/180 = 1/180

So, working alone, Perry can finish the job in 180 minutes = 3 hours.

Answer: E