A cyclist covers his first three miles at an average speed of 8 m.p.h. Anoter two miles at 3m.p.h. and the last two miles at 2m.p.h. The average speed for the entire journey is : (in m.p.h)

Options

A. 3
B. 2.4
C. 3.8
D. 3.43

Answer

To find the average speed for the entire journey of the cyclist, we can follow these steps:

### Step 1: Calculate the total distance
The cyclist covers three segments:
- First segment: 3 miles
- Second segment: 2 miles
- Third segment: 2 miles

**Total distance = 3 + 2 + 2 = 7 miles**

### Step 2: Calculate the time taken for each segment
1. **Time for the first segment (3 miles at 8 mph)**:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{3 \text{ miles}}{8 \text{ mph}} = \frac{3}{8} \text{ hours} \]

2. **Time for the second segment (2 miles at 3 mph)**:
\[ \text{Time} = \frac{2 \text{ miles}}{3 \text{ mph}} = \frac{2}{3} \text{ hours} \]

3. **Time for the third segment (2 miles at 2 mph)**:
\[ \text{Time} = \frac{2 \text{ miles}}{2 \text{ mph}} = 1 \text{ hour} \]

### Step 3: Calculate the total time
Now, we add the times for all three segments:
\[ \text{Total time} = \frac{3}{8} + \frac{2}{3} + 1 \]

To add these fractions, we need a common denominator. The least common multiple of 8 and 3 is 24.

- Convert \(\frac{3}{8}\) to a fraction with a denominator of 24:
\[ \frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24} \]

- Convert \(\frac{2}{3}\) to a fraction with a denominator of 24:
\[ \frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24} \]

- Convert 1 to a fraction with a denominator of 24:
\[ 1 = \frac{24}{24} \]

Now we can add:
\[ \text{Total time} = \frac{9}{24} + \frac{16}{24} + \frac{24}{24} = \frac{49}{24} \text{ hours} \]

### Step 4: Calculate the average speed
The average speed is given by the formula:
\[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{7 \text{ miles}}{\frac{49}{24} \text{ hours}} = 7 \times \frac{24}{49} = \frac{168}{49} \text{ mph} \]

### Step 5: Simplify the average speed
Now we simplify \(\frac{168}{49}\):
\[ \frac{168}{49} = 3.42857 \approx 3.43 \text{ mph} \]

### Final Answer
The average speed for the entire journey is approximately **3.43 mph**.

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