Question

in parts (a) and (b) of problem 2-89, the rectangular models for 1/2 of 1/4 and 1/4 of 1/2 looked different, but the answers were the same. read the steps for the example given in math notes 2.2.5 on canvas for how to calculate 2/3 of 2/5. then follow the steps and draw a similar rectangle to show 2/5 of 2/3. are the answers the same? why or why not?

Explanation:

Step1: Recall fraction - of - a - fraction rule

To find $a$ of $b$ (where $a$ and $b$ are fractions), we multiply the two fractions. So, to find $\frac{2}{3}$ of $\frac{2}{5}$, we calculate $\frac{2}{3}\times\frac{2}{5}$.
$$\frac{2}{3}\times\frac{2}{5}=\frac{2\times2}{3\times5}=\frac{4}{15}$$

Step2: Calculate $\frac{2}{5}$ of $\frac{2}{3}$

Using the same rule, to find $\frac{2}{5}$ of $\frac{2}{3}$, we multiply the two fractions: $\frac{2}{5}\times\frac{2}{3}$.
$$\frac{2}{5}\times\frac{2}{3}=\frac{2\times2}{5\times3}=\frac{4}{15}$$

Step3: Analyze the results

The commutative property of multiplication for fractions states that for any two fractions $\frac{a}{b}$ and $\frac{c}{d}$, $\frac{a}{b}\times\frac{c}{d}=\frac{c}{d}\times\frac{a}{b}$. Since $\frac{2}{3}\times\frac{2}{5}=\frac{2}{5}\times\frac{2}{3}=\frac{4}{15}$, the answers are the same.

Answer:

The answers for $\frac{2}{3}$ of $\frac{2}{5}$ and $\frac{2}{5}$ of $\frac{2}{3}$ are the same. They are both $\frac{4}{15}$ because of the commutative property of multiplication for fractions.