To show that two ratios are proportional, you can check if the cross products of the ratios are equal. If you have two ratios \(a:b\) and \(c:d\), they are proportional if:
\[ \frac{a}{b} = \frac{c}{d} \quad \text{if and only if} \quad a \cdot d = b \cdot c \]
This means that if you cross-multiply the ratios, the products should be equal.
Example:
Consider the ratios \(2:3\) and \(4:6\).
- Set up the equation:
- Here, \(a = 2\), \(b = 3\), \(c = 4\), and \(d = 6\).
- Cross multiply:
- Calculate \(a \cdot d\) and \(b \cdot c\): \[ 2 \cdot 6 = 12 \] \[ 3 \cdot 4 = 12 \]
- Check if they are equal:
- Since \(12 = 12\), the ratios \(2:3\) and \(4:6\) are proportional.
Summary of Method:
Cross-multiplication is a straightforward way to demonstrate that two ratios are proportional by equating the cross products. If they match, the two ratios are indeed proportional.