10.1 Proportionality: A Quick Recap

In a previous chapter we studied how quantities relate to one another and used ratio notation to capture those links. Whenever two related quantities scale up or down by the same factor, we say they sit in a proportional relationship.

Here is an appetising example. Dosa batter is prepared by blending rice and urad dal. A frequently used proportion is 3 cups of rice for every 1 cup of urad dal, written 3 : 1.

Suppose Mehul stirs together 12 cups of rice and 4 cups of urad dal, while Farida uses 9 cups of rice with 3 cups of urad dal. Cooked identically, would their dosas taste alike? Mehul’s blend is 12 : 4 and Farida’s is 9 : 3. To test whether these two ratios are proportional, we cross-multiply:

12 : 4 \quad\text{and}\quad 9 : 3 \qquad\Rightarrow\qquad 12 \times 3 \;=\; 4 \times 9.

Both products work out to 36, so the ratios are proportional, and the dosas should taste the same (provided every other ingredient is in step too).

The Key Idea Two ratios a : b and c : d are proportional when their cross-products match: a \times d = b \times c, \qquad\text{equivalently}\qquad \frac{a}{b} = \frac{c}{d}.

Worked Example

Are 14 : 21 and 10 : 15 proportional?

Cross-multiply: 14 \times 15 = 210 and 21 \times 10 = 210. The cross-products agree, so yes, the ratios are proportional. (Both reduce to 2 : 3.)

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