BTGmoderatorLU wrote:Source: Veritas Prep
3 people, Amy, Beth, and Cassie, have speeds of 3 mph, 4 mph and 6 mph respectively. They run a race in which Beth gives Amy a head start of 2 hrs. If both Beth and Cassie overtake Amy at the same time, what head start did Cassie give Amy?
A. 3 hours
B. 4 hours
C. 5 hours
D. 9 miles
E. 10 miles
Excellent opportunity for RELATIVE VELOCITY (speed) and UNITS CONTROL , two powerful tools covered in our course!
$$\left( * \right)\,\,\,2{\rm{h}}\,\, \cdot \,\,{{3\,\,{\rm{miles}}} \over {1\,\,{\rm{h}}}}\,\,\, = \,\,\,6\,\,{\rm{miles}}\,\,\,\,\,\,\,\,\,\,\left[ {\,{\rm{distance}}\,\,{\rm{A}}\,\,{\rm{starts}}\,\,{\rm{ahead}}\,\,{\rm{of}}\,\,{\rm{B}}\,} \right]$$
$${{\rm{V}}_{{\rm{B}} \to {\rm{A}}}} = {{4 - 3\,\,{\rm{miles}}} \over {1\,\,{\rm{h}}}}\,\,\, = \,\,\,{{6\,\,{\rm{miles}}} \over {{T_B}}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{T_B} = 6\,{\rm{h}}\,\,\,\,\,\,\,\,\,\left[ {\,{\rm{B}}\,\,{\rm{to}}\,\,{\rm{overtake}}\,\,A\,} \right]$$
$$? = x\,\,{\rm{h}}$$
$$\left( {**} \right)\,\,\,x\,\,{\rm{h}}\,\, \cdot \,\,{{3\,\,{\rm{miles}}} \over {1\,\,{\rm{h}}}}\,\,\, = \,\,\,3x\,\,{\rm{miles}}\,\,\,\,\,\,\,\,\,\,\left[ {\,{\rm{distance}}\,\,{\rm{A}}\,\,{\rm{starts}}\,\,{\rm{ahead}}\,\,{\rm{of}}\,\,{\rm{C}}\,} \right]$$
$${{\rm{V}}_{{\rm{C}} \to {\rm{A}}}} = {{6 - 3\,\,{\rm{miles}}} \over {1\,\,{\rm{h}}}}\,\,\, = \,\,\,{{3x\,\,{\rm{miles}}} \over {{T_C}}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{T_C} = x\,{\rm{h}}\,\,\,\,\,\,\,\,\,\left[ {\,{\rm{C}}\,\,{\rm{to}}\,\,{\rm{overtake}}\,\,A\,} \right]$$
$${\rm{Stem}}\,\,\,\, \Rightarrow \,\,\,\,\, {T_C} + x = {T_B} + 2\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,? = x = 4$$
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
