A bus averages. 2miles per hour faster than a motorcycle. If the bus travels 165miles in the same time it takes the motorcycle to travel 155miles, then what is the speed of each?

1. Let v_m be the speed of the motorcycle in miles per hour.
2. The speed of the bus is therefore v_b = v_m + 2 miles per hour.
3. The time it takes for the motorcycle to travel 155 miles is \frac{155}{v_m} hours.
4. The time it takes for the bus to travel 165 miles is \frac{165}{v_b} hours.

Since these times are equal:
\frac{155}{v_m} = \frac{165}{v_b}

5. Substitute v_b = v_m + 2 into the equation:
\frac{155}{v_m} = \frac{165}{v_m + 2}

6. Cross-multiply to solve for v_m :
155(v_m + 2) = 165v_m
155v_m + 310 = 165v_m
310 = 10v_m
v_m = 31

7. Therefore, the speed of the motorcycle is v_m = 31 mph.
8. The speed of the bus is:
v_b = v_m + 2 = 31 + 2 = 33 mph

Answer:
v_m = 31 \text{ mph}, \quad v_b = 33 \text{ mph}