To determine who will raise [tex]$100 with the fewest number of laps, we need to calculate the rate of money raised per lap, or the slope of the lines described by the data points in both Michelle's and Andrea's tables. The slope is calculated as the change in money ($[/tex]y[tex]$) divided by the change in laps ($[/tex]x[tex]$).
### Michelle's Data:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
10 & 15 \\
\hline
30 & 45 \\
\hline
60 & 90 \\
\hline
\end{array}
\]
The slope between the points (10, 15) and (30, 45) can be calculated as follows:
\[
\text{slope} = \frac{45 - 15}{30 - 10} = \frac{30}{20} = 1.5
\]
### Andrea's Data:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
20 & 25 \\
\hline
32 & 40 \\
\hline
40 & 50 \\
\hline
\end{array}
\]
The slope between the points (20, 25) and (32, 40) can be calculated as follows:
\[
\text{slope} = \frac{40 - 25}{32 - 20} = \frac{15}{12} = 1.25
\]
With the slopes found, we can determine how many laps each girl needs to walk to raise $[/tex]100.
### Calculating Laps Needed:
Given the slope (rate) is the amount of money raised per lap, the number of laps needed to raise [tex]$100 can be calculated by:
\[
\text{laps needed} = \frac{100}{\text{slope}}
\]
#### Michelle:
\[
\text{laps needed} = \frac{100}{1.5} \approx 66.67
\]
#### Andrea:
\[
\text{laps needed} = \frac{100}{1.25} = 80
\]
### Conclusion:
Comparing the number of laps needed, Michelle will require approximately 66.67 laps, whereas Andrea will need 80 laps to raise $[/tex]100. Therefore:
Michelle will raise $100 with the fewest number of laps, because the slope of the line described by the data in her table is the greatest.
