To find Harvey the wonder hamster's average speed in kilometers per hour, we will follow these steps:
1. Convert the distance to an improper fraction:
Harvey runs a distance of [tex]\(3 \frac{1}{6}\)[/tex] kilometers. This can be written as an improper fraction. First, we convert the mixed fraction:
[tex]\[
3 \frac{1}{6} = 3 + \frac{1}{6}
\][/tex]
We can express the integer part 3 as a fraction with a denominator of 6:
[tex]\[
3 = \frac{18}{6}
\][/tex]
Now, add it to the fractional part:
[tex]\[
3 + \frac{1}{6} = \frac{18}{6} + \frac{1}{6} = \frac{19}{6}
\][/tex]
Therefore, the distance Harvey runs is [tex]\( \frac{19}{6} \)[/tex] kilometers.
2. Time given in hours:
The time taken is given as [tex]\( \frac{1}{4} \)[/tex] hour.
3. Calculate the average speed:
The formula for average speed is:
[tex]\[
\text{Average Speed} = \frac{\text{Distance}}{\text{Time}}
\][/tex]
Using the values we have:
[tex]\[
\text{Average Speed} = \frac{\frac{19}{6} \text{ km}}{\frac{1}{4} \text{ hr}}
\][/tex]
To divide by a fraction, we multiply by its reciprocal. The reciprocal of [tex]\( \frac{1}{4} \)[/tex] is 4:
[tex]\[
\text{Average Speed} = \frac{19}{6} \times 4
\][/tex]
Simplify the multiplication:
[tex]\[
\frac{19 \times 4}{6} = \frac{76}{6} = 12 \frac{4}{6} = 12 \frac{2}{3}
\][/tex]
Finally, expressing [tex]\( 12 \frac{2}{3} \)[/tex] as a decimal:
[tex]\[
12 \frac{2}{3} = 12.666666666666666
\][/tex]
Therefore, Harvey’s average speed is [tex]\( 12.666666666666666 \)[/tex] kilometers per hour.
[tex]\[
\boxed{12.666666666666666}
\][/tex]
