Which pair of ratios form a proportion? My daughter is stuck on this question and we're not sure what the answer is. I'm rusty with math skills, so I don't understand how to help her, or what approaches she's tried. Please help us!
10.5/12 and 2/5
13/7 and 7/13
9/30 and 1.5/5
7/5 and 10/8

A proportion is just a statement that two ratios (i.e. fractions) are equal. So, for example, $\frac{1}{2}$ and $\frac{2}{4}$ form a proportion since $\frac{2}{4} = \frac{1}{2}$ (you can see this by reducing $\frac{2}{4}$).
One way to check to see if two fractions form a proportion is to "cross-multiply" them. In the example I just gave, we multiply the numerator of the first fraction by 4, and the numerator of the second fraction by 2, and compare the two sides:
\begin{gather*}
\frac{1}{2}  \stackrel{?}{=} \frac{2}{4} \\
4 \cdot 1  \stackrel{?}{=} 2 \cdot 2\\
4 \stackrel{?}{=} 4
\end{gather*}
Since the equality we get at the end is true, the fractions do, indeed, form a proportion.
Contrast that with the fractions $\frac{1}{3}$ and $\frac{3}{4}$. Clearly these are not equal, so they don't form a proportion, but let's check using the method we just learned:
\begin{gather*}
\frac{1}{3} \stackrel{?}{=} \frac{3}{4} \\
4 \cdot 1 \stackrel{?}{=} 3 \cdot 3 \\
4 \stackrel{?}{=} 9
\end{gather*}
Four is certainly not equal to nine, so the fractions are not in proportion.
I'd be happy to talk a little more about why this procedure works, if you would like, but I hope that this is at least enough to get you started. Good luck, and have fun!
