Question

assessment practice

1. select all the expressions equivalent to \\(\frac{8}{7} + \frac{2}{7}y\\). 7.ar.1.1, 7.ar.1.2

\\(\square\\) \\(\frac{1}{7}(8 + 2y)\\)
\\(\square\\) \\(8(\frac{1}{7} + 2y)\\)
\\(\square\\) \\(\frac{2}{7}(4 + y)\\)
\\(\square\\) \\(4(\frac{2}{7} + y)\\)
\\(\square\\) \\(\frac{2}{7}(2 + 2y)\\)

Explanation:

Step1: Analyze the first option

Factor out $\frac{1}{7}$ from $\frac{8}{7}+\frac{2}{7}y$. We get $\frac{1}{7}(8 + 2y)$ (using the distributive property $a\times c + b\times c=(a + b)\times c$, here $c=\frac{1}{7}$, $a = 8$, $b = 2y$). So this option is equivalent.

Step2: Analyze the second option

Expand $8(\frac{1}{7}+2y)$ using the distributive property: $8\times\frac{1}{7}+8\times2y=\frac{8}{7}+16y$, which is not equal to $\frac{8}{7}+\frac{2}{7}y$. So this option is not equivalent.

Step3: Analyze the third option

Expand $\frac{2}{7}(4 + y)$ using the distributive property: $\frac{2}{7}\times4+\frac{2}{7}\times y=\frac{8}{7}+\frac{2}{7}y$. So this option is equivalent.

Step4: Analyze the fourth option

Expand $4(\frac{2}{7}+y)$ using the distributive property: $4\times\frac{2}{7}+4\times y=\frac{8}{7}+4y$, which is not equal to $\frac{8}{7}+\frac{2}{7}y$. So this option is not equivalent.

Step5: Analyze the fifth option

Expand $\frac{2}{7}(2 + 2y)$ using the distributive property: $\frac{2}{7}\times2+\frac{2}{7}\times2y=\frac{4}{7}+\frac{4}{7}y$, which is not equal to $\frac{8}{7}+\frac{2}{7}y$. So this option is not equivalent.

Answer:

$\frac{1}{7}(8 + 2y)$, $\frac{2}{7}(4 + y)$