enigma123 wrote:
What is the distance between Harry’s home and his office?
(1) Harry’s average speed on his commute to work this Monday was 30 miles per hour.
(2) If Harry’s average speed on his commute to work this Monday had been twice as fast, his trip would have been 15 minutes shorter.
Excellent opportunity to practice UNITS CONTROL and BIFURCATION, two of our most powerful tools!
\(? = d\,\,\,\,\left[ {{\text{miles}}} \right]\)
\(\left( 1 \right)\,\,\,V = 30\,\,{\text{mph}}\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\,{\text{T = }}\,\,{\text{1}}\,\,{\text{h}}\,\,\,\, \Rightarrow \,\,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{30}}\,\, \hfill \\
\,{\text{Take}}\,\,\,{\text{T = }}\,\,2\,\,{\text{h}}\,\,\,\, \Rightarrow \,\,\,\,{\text{?}}\,\,{\text{ = }}\,\,60\,\,\, \hfill \\
\end{gathered} \right.\,\,\,\left[ {{\text{miles}}} \right]\)
\(\left( 2 \right)\,\,\,\,\frac{{d \cdot \boxed2}}{{V \cdot \boxed2}}\, - \frac{d}{{2V}}\,\,\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,\,\frac{1}{4}\,\,\,\,\left[ {\text{h}} \right]\,\,\,\,\,\, \Rightarrow \,\,\,\,\underleftrightarrow {\frac{d}{{2V}} = \frac{1}{4}}\,\,\,\,\,\, \Rightarrow \,\,\,\,V = 2d\,\,\,\left( {**} \right)\)
\(\left( * \right)\,\,\,\frac{{\left[ {{\text{miles}}} \right]}}{{\left[ {{\text{mph}}} \right]}} = \left[ {\text{h}} \right]\)
\(\left\{ \begin{gathered}
\,{\text{Take}}\,\,\,V = 2\,{\text{mph}}\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,{\text{d = }}\,\,{\text{1}}\,\,{\text{mile}}\,\,\,\,\left( {T = 0.5\,\,{\text{h}}} \right)\,\,\,\,::\,\,\,\,{\text{viable!}}\,\,\,\,\, \hfill \\
\,\,{\text{Take}}\,\,\,V = 4\,{\text{mph}}\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,{\text{d = }}\,\,{\text{2}}\,\,{\text{miles}}\,\,\,\,\left( {T = 0.5\,\,{\text{h}}} \right)\,\,\,\,::\,\,\,\,{\text{viable!}}\,\,\,\, \hfill \\
\end{gathered} \right.\)
\(\left( {1 + 2} \right)\,\,\,\,V = 30\,{\text{mph}}\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\boxed{{\text{d = }}\,\,{\text{15}}\,\,{\text{miles}}}\,\,\,\,\,\,\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
fskilnik.
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