Question

1.
a. to find 40% of 75, priya calculates \\(\frac{2}{5} \cdot 75\\). does her calculation give the correct value for 40% of 75? explain or show how you know.
b. if \\(x\\) represents a number, does \\(\frac{2}{5} \cdot x\\) always represent 40% of that number? explain your reasoning.

Explanation:

Step1: Convert 40% to fraction

$40\% = \frac{40}{100} = \frac{2}{5}$

Step2: Calculate 40% of 75 two ways

First way: $0.4 \times 75 = 30$
Second way: $\frac{2}{5} \times 75 = 30$

Step3: Generalize for any number $x$

$40\% \text{ of } x = 0.4x = \frac{2}{5}x$

Answer:

a. Yes, her calculation gives the correct value. 40% is equivalent to $\frac{2}{5}$, so $\frac{2}{5} \times 75 = 30$, which is the same as $0.4 \times 75 = 30$.
b. Yes, $\frac{2}{5} \cdot x$ always represents 40% of $x$. Since 40% simplifies to the fraction $\frac{2}{5}$, multiplying any number $x$ by $\frac{2}{5}$ is the same as finding 40% of that number.