A cyclist covers his first three miles at an average speed of 8 m.p.h., another two miles at 3 m.p.h and the last two miles at 3 m.p.h. Find his average speed for the entire journey.

Answer

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

### Step 1: Calculate the total distance
The total distance covered by the cyclist is the sum of the distances of each segment of the journey.

- First segment: 3 miles
- Second segment: 2 miles
- Third segment: 2 miles

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

### Step 2: Calculate the time taken for each segment
Next, we will calculate the time taken for each segment using the formula:

\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

**For the first segment:**
- Distance = 3 miles
- Speed = 8 mph

\[ \text{Time}_1 = \frac{3 \text{ miles}}{8 \text{ mph}} = \frac{3}{8} \text{ hours} \]

**For the second segment:**
- Distance = 2 miles
- Speed = 3 mph

\[ \text{Time}_2 = \frac{2 \text{ miles}}{3 \text{ mph}} = \frac{2}{3} \text{ hours} \]

**For the third segment:**
- Distance = 2 miles
- Speed = 3 mph

\[ \text{Time}_3 = \frac{2 \text{ miles}}{3 \text{ mph}} = \frac{2}{3} \text{ hours} \]

### Step 3: Calculate the total time taken
Now, we will sum up the time taken for each segment to find the total time.

**Total Time** = Time for first segment + Time for second segment + Time for third segment

\[ \text{Total Time} = \frac{3}{8} + \frac{2}{3} + \frac{2}{3} \]

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:
\[ \frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24} \]

Now we can add:
\[ \text{Total Time} = \frac{9}{24} + \frac{16}{24} + \frac{16}{24} = \frac{9 + 16 + 16}{24} = \frac{41}{24} \text{ hours} \]

### Step 4: Calculate the average speed
Finally, we can find the average speed using the formula:

\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]

Substituting the values we found:

\[ \text{Average Speed} = \frac{7 \text{ miles}}{\frac{41}{24} \text{ hours}} \]

To divide by a fraction, we multiply by its reciprocal:

\[ \text{Average Speed} = 7 \times \frac{24}{41} = \frac{168}{41} \text{ mph} \]

Calculating this gives approximately:

\[ \text{Average Speed} \approx 4.1 \text{ mph} \]

### Final Answer:
The average speed for the entire journey is approximately **4.1 mph**.

---