from the grid on Diane's swim you can see that, for every two squares on the grid over the x-axis, it goes up 3 squares over the y-axis, it moves 2 to the right and then 3 up, and you get the next point. What does that mean? well, is a constant speed and thus the graph is a line, with a slope of 3 meters per 2 seconds, so her slope is 3/2 m/s.
now, for Rick's slope, we can just pick two points off of it, say, hmmm 10, 12.5 and 20, 25, and get the slope,
[tex]\bf \begin{array}{ccccccccc}
&&x_1&&y_1&&x_2&&y_2\\
% (a,b)
&&(~{{ 10}} &,&{{ 12.5}}~)
% (c,d)
&&(~{{ 20}} &,&{{ 25}}~)
\end{array}
\\\\\\
% slope = m
slope = {{ m}}\implies
\cfrac{\stackrel{rise}{{{ y_2}}-{{ y_1}}}}{\stackrel{run}{{{ x_2}}-{{ x_1}}}}\implies \cfrac{25-12.5}{20-10}\implies \cfrac{12.5}{10}\implies \cfrac{\frac{125}{10}}{10}
\\\\\\
\cfrac{\frac{125}{10}}{\frac{10}{1}}\implies \cfrac{125}{10}\cdot \cfrac{1}{10}\implies \cfrac{125}{100}\implies \cfrac{5}{4}\cdot \cfrac{meters}{second}[/tex]
so Diane is doing 3 meters for every 2 seconds, and Rick is doing 5 meters for every 4 seconds.
how long will it be for each to do the 150 meters anyway?
[tex]\bf ~~~~~~~~~~~~~~Diane's\\\\
\begin{array}{ccll}
meters&seconds\\
\text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\
3&2\\
150&x
\end{array}\implies \cfrac{3}{150}=\cfrac{2}{x}\implies x=\cfrac{150\cdot 2}{3}\implies x=100\\\\
-------------------------------\\\\[/tex]
[tex]\bf ~~~~~~~~~~~~~~Rick's\\\\
\begin{array}{ccll}
meters&seconds\\
\text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\
5&4\\
150&y
\end{array}\implies\cfrac{5}{150}=\cfrac{4}{y}\implies y=\cfrac{150\cdot 4}{5}\implies y=120[/tex]
what's their difference? well y - x.
