Question

The distance between Town X and Town Y is 150 miles. Car A leaves Town X for Town Y, and sometime later, Car B leaves Town Y for Town X. If the two cars meet exactly halfway between Town X and Town Y, what is the speed of Car B?
(1) Car B leaves Town Y exactly 45 minutes after Car A leaves Town X.
(2) Car B travels at a speed 10 mph faster than Car A.

Hint:

In data sufficiency, you don't need to find the final numerical answer. You only need to determine if you *can* find it. Once you establish a solvable system of equations (e.g., two unique equations for two variables), you can conclude that the information is sufficient.

Answer 1

Step 1: Understanding the Concept:
This is a data sufficiency problem involving relative speed and the relationship between distance, speed, and time (\(D=S \times T\)). We need to determine if we can find the speed of Car B.
Step 2: Analyze the Main Question:
Total distance = 150 miles.
Meeting point is exactly halfway, so each car travels 75 miles.
Let \(S_A\) and \(T_A\) be the speed and time of Car A.
Let \(S_B\) and \(T_B\) be the speed and time of Car B.
Distance traveled by Car A: \(S_A \times T_A = 75\).
Distance traveled by Car B: \(S_B \times T_B = 75\).
We need to find the value of \(S_B\).
Step 3: Analyze the Statements:
Statement (1): Car B leaves Town Y exactly 45 minutes after Car A leaves Town X.
45 minutes = 0.75 hours.
The travel time of Car A until they meet is 0.75 hours longer than the travel time of Car B. \[ T_A = T_B + 0.75 \] Substituting \(T_A = 75/S_A\) and \(T_B = 75/S_B\): \[ \frac{75}{S_A} = \frac{75}{S_B} + 0.75 \] This is one equation with two unknown variables (\(S_A\) and \(S_B\)). We cannot solve for \(S_B\). Thus, Statement (1) is NOT sufficient.
Statement (2): Car B travels at a speed 10 mph faster than Car A.
\[ S_B = S_A + 10 \] This gives a relationship between the speeds, but provides no information about the times. We cannot find a unique value for \(S_B\). Thus, Statement (2) is NOT sufficient.
Combining Statements (1) and (2):
We now have a system of two equations with two variables:

\( \frac{75}{S_A} = \frac{75}{S_B} + 0.75 \)
\( S_B = S_A + 10 \implies S_A = S_B - 10 \)
Substitute the second equation into the first: \[ \frac{75}{S_B - 10} = \frac{75}{S_B} + 0.75 \] This is a single equation with only one variable, \(S_B\). It can be solved to find a unique positive value for \(S_B\). We do not need to solve it, but just confirm that a solution exists. \[ \frac{75}{S_B - 10} - \frac{75}{S_B} = 0.75 \] \[ 75\left(\frac{S_B - (S_B - 10)}{S_B(S_B - 10)}\right) = 0.75 \] \[ \frac{750}{S_B^2 - 10S_B} = 0.75 \] \[ 1000 = S_B^2 - 10S_B \] \[ S_B^2 - 10S_B - 1000 = 0 \] This quadratic equation can be solved for \(S_B\). Since we can determine a unique speed for Car B, the statements together are sufficient.
Step 4: Final Answer:
Neither statement alone is sufficient, but both statements together are sufficient.