Driving at their respective constant speeds along the same route, Alfred passed a certain landmark 1 hour after Violet did....
GMAT Data Sufficiency : (DS) Questions
Driving at their respective constant speeds along the same route, Alfred passed a certain landmark \(1\) hour after Violet did. Both Alfred and Violet continued driving along the same route in the same direction at their respective constant speeds. If Alfred's speed was \(24\) kilometers per hour greater than Violet's, what was Violet's speed?
- Alfred overtook Violet \(4\) hours after she passed the landmark.
- Alfred's speed was \(\frac{4}{3}\) of Violet's speed.
Understanding the Question
We're asked to find Violet's speed. Let's break down what we know:
Given Information:
- Both drivers maintain constant speeds on the same route in the same direction
- Alfred passed a landmark 1 hour after Violet did
- Alfred's speed is \(24 \text{ km/hr}\) greater than Violet's speed
What We Need to Determine:
A specific value for Violet's speed in km/hr.
Key Insight from the Question:
The crucial setup here is that Alfred starts 1 hour behind Violet at the landmark. Since he's traveling \(24 \text{ km/hr}\) faster, he's gradually catching up to her. This is a classic "catch-up" problem where the faster person closes the gap over time.
For this to be sufficient, we need to be able to determine exactly one value for Violet's speed.
Analyzing Statement 1
Statement 1: Alfred overtook Violet 4 hours after she passed the landmark.
Let's visualize what this means:
- Hour 0: Violet passes the landmark
- Hour 1: Alfred passes the landmark (1 hour behind)
- Hour 4: Alfred catches up to Violet
So Alfred needed 3 hours of driving (from hour 1 to hour 4) to make up for being 1 hour behind.
The key insight: In those 3 hours, Alfred gains exactly enough ground to overcome Violet's 1-hour head start. Since Alfred travels \(24 \text{ km/hr}\) faster:
- In 3 hours, Alfred gains: \(3 \times 24 = 72 \text{ km}\) on Violet
- This \(72 \text{ km}\) must equal the distance Violet traveled in her 1-hour head start
- Therefore, Violet's speed = \(72 \text{ km/hr}\)
[STOP - Sufficient!] We can determine Violet's exact speed.
Statement 1 is SUFFICIENT.
This eliminates choices B, C, and E.
Analyzing Statement 2
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: Alfred's speed was \(\frac{4}{3}\) of Violet's speed.
This gives us a direct speed ratio. Let's connect this with what we already know:
- From the question: Alfred's speed = Violet's speed + 24
- From Statement 2: Alfred's speed = \(\frac{4}{3} \times \text{Violet's speed}\)
The logical insight: If Alfred's speed is \(\frac{4}{3}\) of Violet's, he's traveling \(\frac{1}{3}\) faster than her. This means:
- The extra \(\frac{1}{3}\) of Violet's speed = \(24 \text{ km/hr}\)
- So: \(\frac{1}{3} \times \text{Violet's speed} = 24 \text{ km/hr}\)
- Therefore: Violet's speed = \(3 \times 24 = 72 \text{ km/hr}\)
[STOP - Sufficient!] We can determine Violet's exact speed.
Statement 2 is SUFFICIENT.
This eliminates choice A (and confirms we've already eliminated C and E).
The Answer: D
Both statements independently give us enough information to determine that Violet's speed is \(72 \text{ km/hr}\).
Answer Choice D: "Each statement alone is sufficient."