Explaining why cross multiplication works using a pie

Imagine 9 cakes with 6 pieces each. Both taking 3 of 9 cakes, or 2 pieces of each of the 9 cakes result in 18 pieces of cake.

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Suppose you cut the pie into $6 \cdot 9 = 54$ slices, and call each of those a "unit slice". Then both $2 \cdot 9=3 \cdot 6$ would denote the same amount of "unit slices", which is $\dfrac{18}{54}=\dfrac{1}{3}$ of the full pie.

Without the pie analogy, just note that $\,\dfrac{2}{6} \color{blue}{\cdot \dfrac{9}{9}}=\dfrac{18}{54}=\dfrac{3}{9} \color{blue}{\cdot \dfrac{6}{6}}\,$.


Imagine 2 cakes with 18 strawberries on top.

  1. Take the first cake, cut it into 6, and take 2 slices. (2/6) Each slice has 3 stawberries
  2. Take the second cake, cut it into 9 and take 3 slices. (3/9) Each slice has 2 strawberries.

Now, you have 9 slices with 2 strawberries each (9∗2), and 6 slices with 3 strawberries each (6∗3), for a total of 18 strawberries per cake

Tags:

Arithmetic

Intuition

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