To find Harvey the wonder hamster's average speed, we need to determine his speed in kilometers per hour (km/h). Let's break this down step-by-step.
1. Convert the distance to an improper fraction:
Harvey runs [tex]\(3 \frac{1}{6}\)[/tex] kilometers. To work with this distance more easily, let's convert the mixed number to an improper fraction.
[tex]\[
3 \frac{1}{6} = \frac{3 \times 6 + 1}{6} = \frac{18 + 1}{6} = \frac{19}{6} \text{ kilometers}
\][/tex]
2. Identify the time taken:
Harvey takes [tex]\(\frac{1}{4}\)[/tex] hour to cover this distance.
3. Calculate the average speed:
To find the average speed, we use the formula:
[tex]\[
\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}
\][/tex]
Plugging in the distance and time, we get:
[tex]\[
\text{Average speed} = \frac{\frac{19}{6} \text{ kilometers}}{\frac{1}{4} \text{ hour}}
\][/tex]
4. Divide the fractions:
When dividing by a fraction, we multiply by its reciprocal:
[tex]\[
\text{Average speed} = \frac{19}{6} \times \frac{4}{1} = \frac{19 \times 4}{6 \times 1} = \frac{76}{6}
\][/tex]
5. Simplify the result:
[tex]\(\frac{76}{6}\)[/tex] can be simplified by dividing the numerator and the denominator by their greatest common divisor, which is 2:
[tex]\[
\frac{76}{6} = \frac{76 \div 2}{6 \div 2} = \frac{38}{3}
\][/tex]
If we convert [tex]\(\frac{38}{3}\)[/tex] to a decimal, we get:
[tex]\[
\frac{38}{3} \approx 12.6667
\][/tex]
Therefore, Harvey's average speed is approximately [tex]\(12.67\)[/tex] kilometers per hour.
In summary, Harvey the wonder hamster's average speed is:
[tex]\[
\boxed{12.67} \text{ kilometers per hour}
\][/tex]
