Question

a bicyclist travels $6\frac{1}{2}$ miles in $\frac{2}{3}$ hour. what is the average speed, in miles per hour, of the bicyclist?

Explanation:

Step1: Recall the formula for speed

Speed is calculated by the formula \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \). Here, the distance is \( 6\frac{1}{2} \) miles and the time is \( \frac{2}{3} \) hour. First, convert the mixed number \( 6\frac{1}{2} \) to an improper fraction. \( 6\frac{1}{2}=\frac{6\times2 + 1}{2}=\frac{13}{2} \) miles.

Step2: Substitute values into the formula

Now, substitute the distance \( \frac{13}{2} \) miles and time \( \frac{2}{3} \) hour into the speed formula. So, \( \text{Speed}=\frac{\frac{13}{2}}{\frac{2}{3}} \). When dividing by a fraction, we multiply by its reciprocal. So, \( \frac{\frac{13}{2}}{\frac{2}{3}}=\frac{13}{2}\times\frac{3}{2} \).

Step3: Multiply the fractions

Multiply the numerators and denominators: \( \frac{13\times3}{2\times2}=\frac{39}{4} \). Convert this improper fraction to a mixed number: \( \frac{39}{4}=9\frac{3}{4} \) miles per hour.

Answer:

The average speed of the bicyclist is \( 9\frac{3}{4} \) miles per hour.