Driving at their respective constant speeds along the same route, Alfred passed a certain landmark 1 hour after Violet did....

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."