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Equivalent Ratios

Suppose you have a recipe meant to feed $5$ people, but you need to feed $10$ people. How will you change the recipe? You will need to double each quantity. Similarly, if you need to feed $15$ people, you'll need to triple each quantity. But the ratios of the ingredients will be the same no matter how many people you're feeding. That is, the ratios are equivalent.

Consider the ratios $2:5$ and $4:10$ . The second ratio is obtained by multiplying each number of the first ratio by $2$ . So, they are equivalent.

If you consider two ratios as fractions, they are equivalent if their cross products are equal.

Consider the multiplication table shown.

$\begin{array}{l}1\times 3=3\\ 2\times 3=6\\ 3\times 3=9\\ 4\times 3=12\\ 5\times 3=15\end{array}$

The ratio of the first and last number is the same in any of the rows. That is, $\frac{1}{3}=\frac{2}{6}=\frac{3}{9}=\frac{4}{12}=\frac{5}{15}$ . The cross products of any two ratios are equal. For example, taking the first and last ratios, $1\times 15=5\times 3$ . Therefore, the ratios are equivalent.

Now, consider plotting the numbers in these ratios on a coordinate plane , with the numerators along the $x$ -axis and the denominators along the $y$ -axis. The line joining the plotted points will pass through the origin.

In general, if the line joining two points on a coordinate plane passes through the origin, then the ratios of the coordinates of the points are equivalent. You can also say that the coordinates of the points are in proportion.