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Two gardeners, Burton and Philips

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Two gardeners, Burton and Philips

by Uva@90 » Tue Aug 04, 2015 8:15 am

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Two gardeners, Burton and Philips, work at independent constant rates to prune a garden full of roses. If both gardeners start at the same time and work at their normal rates, they will complete the job in 45 minutes. However, if Philips were to work at twice Burton's rate, they would take only 20 minutes. How long would it take Philip, working alone at his normal rate , to tune the garden full of roses.

A) 1 hour and 20 minutes
B) 1 Hour and 45 Min
C) 2 Hour
D) 2 hour and 20 Min
E) 3 Hour

OA E
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by GMATGuruNY » Tue Aug 04, 2015 8:41 am
Uva@90 wrote:Two gardeners, Burton and Philips, work at independent constant rates to prune a garden full of roses. If both gardeners start at the same time and work at their normal rates, they will complete the job in 45 minutes. However, if Philips were to work at twice Burton's rate, they would take only 20 minutes. How long would it take Philip, working alone at his normal rate , to tune the garden full of roses.

A) 1 hour and 20 minutes
B) 1 Hour and 45 Min
C) 2 Hour
D) 2 hour and 20 Min
E) 3 Hour
Let the job = the LCM of 20 and 45 = 180 units.

If both gardeners start at the same time and work at their normal rates, they will complete the job in 45 minutes.
Since the time for Burton and Philip together is 45 minutes, the combined normal rate for Burton and Philip = w/t = 180/45 = 4 units per minute.

If Philip were to work at twice Burton's rate, they would take only 20 minutes.
Here, the combined faster rate for Burton and Philip = w/t = 180/20 = 9 units per minute.
Let B = Burton's rate.
When Philip works at twice Burton's rate, Philip's faster rate = 2B.
Implication:
Here, the combined faster rate for Burton and Philip = B + 2B = 3B.
Since the combined faster rate is 9 units per minute, we get:
3B = 9
B = 3 units per minute.

Since the combined normal rate for Burton and Philip = 4 units per minute, and Burton's rate alone = 3 units per minute, Philip's normal rate alone = 4-3 = 1 unit per minute.
Thus:
At his normal rate of 1 unit per minute, the time for Philip to produce 180 units = w/r = 180/1 = 180 minutes = 3 hours.

The correct answer is E.
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by OptimusPrep » Tue Aug 04, 2015 7:51 pm
As Mitch explained, the best way to do such problems is to assume the total work as the LCM for the rates. This simplifies the calculations.

Another way can be:

Let the time required by Philips and Burton be P and B minutes

So,

1/P + 1/B = 1/45
and 2/B + 1/B = 1/20 (Since it is given that Philips works at twice Burton's rate)

On solving, 3/B = 1/20 or B = 60

Putting this in equation 1, we get

1/P = 1/45 - 1/60
1/P = (4 - 3)/15*3*4

or P = 180 = 3 hours.

Hence the answer is E
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by fskilnik@GMATH » Sun Sep 16, 2018 6:54 pm
Uva@90 wrote:Two gardeners, Burton and Philips, work at independent constant rates to prune a garden full of roses. If both gardeners start at the same time and work TOGETHER at their normal rates, they will complete the job in 45 minutes. However, if Philips were to work at twice Burton's rate, TOGETHER they would take only 20 minutes. How long would it take Philip, working alone at his normal rate , to tune the garden full of roses?

A) 1 hour and 20 minutes
B) 1 Hour and 45 Min
C) 2 Hour
D) 2 hour and 20 Min
E) 3 Hour
Prune a garden full of roses = 1 job

Burton takes (say) 2x minutes to do 1 job alone.

If Philip takes x minutes to do 1 job alone (to work at twice Burton's rate), together they would do the job in 20min, hence:
\[ \frac{1}{{20}} = \frac{1}{{2x}} + \frac{{1 \cdot \boxed2}}{{x \cdot \boxed2}} = \frac{3}{{2x}}\,\,\,\,\, \Rightarrow \,\,\,x = 30\,\,\,\left[ {\min } \right] \]
Conclusion: Burton takes 2x = 60 minutes to do this job alone.

If Philip takes y minutes to do 1 job alone (our FOCUS!), from the fact that together they would do it in 45min, we have:
\[ \frac{1}{{45}} = \frac{1}{{60}} + \frac{1}{y}\,\,\,\,\, \Rightarrow \,\,\,\frac{1}{y} = \frac{{1 \cdot \boxed4}}{{3 \cdot 15 \cdot \boxed4}} - \frac{{1 \cdot \boxed3}}{{4 \cdot 15 \cdot \boxed3}} = \,\,\frac{1}{{3 \cdot 4 \cdot 15}}\,\,\,\,\left[ {\frac{1}{{\min }}} \right] \]
\[? = y = 3 \cdot 60\,\,\min = 3{\text{h}}\]

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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by [email protected] » Mon Sep 17, 2018 3:22 pm
Hi All,

We're told that two gardeners, Burton and Philips, work at independent constant rates to prune a garden full of roses. If both gardeners start at the same time and work at their normal rates, they will complete the job in 45 minutes. However, if Philips were to work at TWICE Burton's rate, they would take only 20 minutes. We're asked how long it would take Philip, working alone at his normal rate , to tune the garden full of roses. This question can be solved using the Work Formula, but it comes with a 'twist' that requires some extra math.

Work Formula = (A)(B)/(A+B) where A and B are the individual times it takes the two workers to complete the task alone. For this question, I'm going to use B and P for the two variables.

(B)(P)/(B+P) = 45 minutes
(B)P) = 45B + 45P

For the second equation, we're told that if Philips worked at TWICE Burton's rate, the total time would be 20 minutes. With Work Formula questions, the 'twist' is that if you DOUBLE a rate, you actually have to HALVE the number (since it actually takes HALF the time to do the job). Thus, instead of using "P", we're going to us "B/2" (since B/2 would be TWICE Burton's rate when using the Work Formula equation).

(B/2)(B)/(B/2 + B) = 20 minutes
B^2/2 = 10B + 20B
B^2/2 = 30B
B^2 = 60B
B^2 - 60B = 0
B(B - 60) = 0
B = 0, 60

Since the rate CANNOT be 0, we know that Burton's normal rate is 60 minutes to do the job alone. To find Phillip's rate, we have to plug B=60 back into the first equation:

(60)(P)/(60 + P) = 45
60P = 45(60+P)
60P = 2700 + 45P
15P = 2700
P = 180

Phillip's normal rate is 180 minutes to do the job alone.

Final Answer: E

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by Scott@TargetTestPrep » Tue Sep 18, 2018 5:42 pm
Uva@90 wrote:Two gardeners, Burton and Philips, work at independent constant rates to prune a garden full of roses. If both gardeners start at the same time and work at their normal rates, they will complete the job in 45 minutes. However, if Philips were to work at twice Burton's rate, they would take only 20 minutes. How long would it take Philip, working alone at his normal rate , to tune the garden full of roses.

A) 1 hour and 20 minutes
B) 1 Hour and 45 Min
C) 2 Hour
D) 2 hour and 20 Min
E) 3 Hour
Let's let B = the number of minutes for Burton to do the job alone. Thus, Burton's rate is 1/B. We also let P = the number of minutes for Philip to do the job alone; his rate is 1/P. Since they are working together, we can combine their rates and create the following equation:
1/B + 1/P = 1/45

If Philip were to work at twice Burton's rate, his new rate would be 2/B, and we would have:

2/B + 1/B = 1/20

3/B = 1/20

60 = B

Substituting, we have:

1/60 + 1/P = 1/45

Multiplying by 180P, we have:

3P + 180 = 4P

180 = P

Thus, it will take Philip 180 minutes, or 3 hours, to prune the garden, working alone.

Answer: E

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