Nijo wrote: 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?

In the future, please include the answer choices.

One option is to assign a value to the total job.
Since the Least Common Multiple of 45 and 20 is 180, let's say that there are 180 instruments in the warehouse.

Let M = the number of instruments that Maria can tune PER MINUTE
Let P = the number of instruments that Perry can tune PER MINUTE

Both musicians working TOGETHER complete the job in 45 minutes
180/45 = 4
So, working TOGETHER, they can tune 4 instruments PER MINUTE
In other words, (Mary's rate) + (Perry's rate) = 4
We can write: M + P = 4

If Perry were to work at twice Maria's rate, they would take only 20 minutes.
180/20 = 9
So, in this scenario, they can tune 9 instruments PER MINUTE
In other words, (Mary's rate) + (Perry's rate) = 9
In this scenario, Perry's rate = 2M
So, we can write: M + 2M = 9
Simplify: 3M = 9
So, M = 3 (Maria can tune 3 instruments per minute)

Now that we know the value of M, we can use the equation M + P = 4 to conclude that P = 1
In other words, Perry can tune 1 instrument per minute

If there are 180 instruments to tune, it will take Perry 180 minutes to complete the job.

Cheers,
Brent

Hi Nijo,

Here's how you can solve this problem with the Work Formula:

(M)(P)/(M + P)

M = Maria's Time
P = Perry's Time

Working together, we're told that it takes them 45 minutes to complete the job....

(M)(P)/(M + P) = 45

If Perry works at TWICE Maria's rate, it takes them 20 minutes to complete the job....

**Mathematically, working at twice Maria's rate means that Perry works TWICE AS FAST, so we have to HALVE the variable.

P = 1/2M

(M)(.5M)/(M + .5M) = 20

We can solve this equation:

.5M^2/(1.5M) = 20

.5M^2 = 30M

M^2 = 60M

M^2 - 60M = 0
M(M - 60) = 0
M = 0, 60

Since 0 minutes isn't a possibility, we now know that Maria takes 60 minutes to complete the job on her own. We can plug THAT value back into the original equation:

60(P)/(60 + P) = 45

60P = 2700 + 45P
15P = 2700
P = 180

Now we know that Perry takes 180 minutes to complete the job on his own.

Final Answer: 180

GMAT assassins aren't born, they're made,
Rich

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?

Another approach:

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. . . .

Let M = the FRACTION of the total job that Maria can complete (working alone) in 1 MINUTE.
Let P = the FRACTION of the total job that Perry can complete (working alone) in 1 MINUTE

Both musicians working TOGETHER complete the job in 45 minutes
By Rule #1, we can conclude that, working together, Maria and Perry can complete 1/45 of the total job in 1 MINUTE
So, in 1 MINUTE, we can says that (Maria's contribution) + (Perry's contribution) = 1/45 of the total job
We can write: M + P = 1/45

If Perry were to work at twice Maria's rate, they would take only 20 minutes.
By Rule #1, we can conclude that, working together, Maria and Perry can complete 1/20 of the total job in 1 MINUTE
So, in 1 MINUTE, we can says that (Maria's contribution) + (Perry's contribution) = 1/20 of the total job
If Perry's rate is twice Maria's, then in 1 MINUTE, the fraction of the job that Perry can complete = 2M
So, we can write: M + 2M = 1/20
Simplify: 3M = 1/20
Solve: M = 1/60 (In 1 MINUTE, Maria can complete 1/60 of the job)

Now that we've solved for M, we can take the equation M + P = 1/45 and replace M with 1/60 to get: 1/60 + P = 1/45
Rewrite using common denominator: 3/180 + P = 4/180
Solve: P = 1/80
So, in 1 MINUTE, Perry can complete 1/180 of the job
By Rule #2, we can conclude that Perry can complete the ENTIRE job in 180 minutes.

Cheers,
Brent