# GMAT Challenge Series (700+): Q21

Question:

Train A leaves New York for Boston at 3 PM and travels at the constant speed of 100 mph. An hour later, it passes Train B, which is making the trip from Boston to New York at a constant speed. If Train B left Boston at 3:50 PM and if the combined travel time of the two trains is 2 hours, what time did Train B arrive in New York?

(1) Train B arrived in New York before Train A arrived in Boston.
(2) The distance between New York and Boston is greater than 140 miles.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.

$$\text{Train A:}$$ $$\text{At 4:00 PM Train A has traveled 100 miles}$$ $$\text{Let the time that it takes Train A to get to Boston be: } \textit{A} \text{ hours}$$  $$\text{Train B:}$$ $$\text{Let the distance that Train B has traveled from 3:50 PM to 4:00 PM be: } \textit{x} \text{ miles}$$ $$\text{Given this, the speed of Train B is: } {6}\textit{x} \text{ mph}$$ $$\text{Let the time that it takes Train B to get to NY be: } \textit{B} \text{ hours}$$  $$\text{Situation at 4:00 PM:}$$ $$\text{NY} \text{ ------------- 100 miles ------------- A > < B ------- x miles ------- }\text{Boston}$$ $$\text{Total distance between NY and Boston is: } {100} + \textit{x} \text{ miles}$$  $$\textit{time} = {\textit{distance} \over \textit{speed}}$$ $$\textit{A} = {{{100} + \textit{x}} \over {100}}$$ $$\textit{B} = {{{100} + \textit{x}} \over {6}\textit{x}}$$ $$\text{To work out when Train B arrived in NY we need to know } \textit{x}$$  $$\textit{A} + \textit{B} = {2}$$ $${{{100} + \textit{x}} \over {100}} + {{{100} + \textit{x}} \over {6}\textit{x}} = {2}$$ $${600}\textit{x} + {6}\textit{x}^2 + {10,000} + {100}\textit{x} = {1,200}\textit{x}$$ $${6}\textit{x}^2 - {500}\textit{x} + {10,000} = {0}$$ $${3}\textit{x}^2 - {250}\textit{x} + {5,000} = {0}$$ $$\left( \textit{x} - {50} \right) \left( {3}\textit{x} - {100} \right)$$ $$\textit{x} = {50} \textit{ or } {33}\frac{1}{3}$$  $$\text{To work out when Train B arrived in NY we need to know if } \textit{x} \text{ is } {50} \text{ or } {33}\frac{1}{3}$$  $$\text{Statement 1:}$$ $$\text{If } \textit{x} \text{ is } {50} \text{: } \textit{A} = {1.5} \text{, }\textit{B} = {0.5} \text{, } \textit{A - B} = {1}$$ $$\text{If } \textit{x} \text{ is } {33}\frac{1}{3} \text{: } \textit{A} = {1}\frac{1}{3} \text{, }\textit{B} = \frac{2}{3} \text{, } \textit{A - B} = \frac{2}{3}$$ $$\text{If Train B arrived before Train A, } \textit{A - B} \text{ must be } > \frac{5}{6}$$ $$\text{Thus, } \textit{x} \text{ must be } {50}$$ $$\text{Statement 1 is SUFFICIENT}$$  $$\text{Statement 2:}$$ $${100} + \textit{x} > {140}$$ $$\textit{x} > {40}$$ $$\text{Thus, } \textit{x} \text{ must be } {50}$$ $$\text{Statement 2 is SUFFICIENT}$$