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.
