swerve wrote:Mary and Kate are running clockwise around a circular track with a circumference of 500 meters, each at her own constant speed. Mary runs 1000 meters every five minutes and Kate runs 1000 meters every six minutes. If Mary and Kate start opposite one another on the circular track, how many minutes must Mary run in order to pass Kate and catch her again?
A. 7.5
B. 22.5
C. 750
D. 7.5Ï€
E. 45Ï€
Source: Magoosh
Let´s use immediately RELATIVE VELOCITY (speed) and UNITS CONTROL, two powerful tools covered in our course!
$$?\,\,\,:\,\,\,{\text{minutes}}\,\,{\text{for}}\,\,\left( {{\text{catch}}\,\, + 1\,\,{\text{lap}}\,\,{\text{ahead}}} \right)$$
$$\left. \matrix{
{V_M} = {{1000\,\,{\rm{m}}} \over {5\,\,\min }}\,\,\,\, \hfill \cr
{V_K} = {{1000\,\,{\rm{m}}} \over {6\,\,\min }} \hfill \cr} \right\}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{V_{M \to K}}\,\, = \,\,1000\,\,\underbrace {\left( {{1 \over 5} - {1 \over 6}} \right)}_{ = \,\,{1 \over {30}}}\,\,\,\, = \,\,\,\,{{100\,\,{\rm{m}}} \over {3\,\,\min }}$$
$$\left( {{\text{relative}}} \right)\,\,{\text{distance}}\,\,\,\,\,{\text{ = }}\,\,\left( {\frac{1}{2} + 1} \right) \cdot 500\,\,{\text{m}}$$
$${\text{?}}\,\,\,{\text{ = }}\,\,\,\frac{{3 \cdot 500}}{2}\,\,{\text{m}}\,\,\, \cdot \,\,\,\left( {\frac{{3\,\,\min }}{{100\,\,{\text{m}}}}} \right)\,\,\,\, = \,\,\,\,\frac{{9 \cdot 5}}{2}\,\,\min \,\,\, = \,\,\,22.5\,\,\min $$
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
