Two gardeners, Burton and Philip, 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 Philip 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 20 minutes
B. 1 hour 45 minutes
C. 2 hours
D. 2 hours 20 minutes
E. 3 hours

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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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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