Question

1. critique reasoning explain any mistakes in the renaming of the fractions below. show the correct renaming. \\(\frac{3}{4} = \frac{9}{12}\\) \\(\frac{2}{3} = \frac{6}{12}\\)

Explanation:

for $\boldsymbol{\frac{3}{4}=\frac{9}{12}}$:

Step1: Recall fraction renaming (equivalent fractions)

To rename a fraction, we multiply numerator and denominator by the same non - zero number. For $\frac{3}{4}$, if we want to get a denominator of 12, we find what number we multiply 4 by to get 12. Let that number be $x$. So $4\times x = 12$, which means $x=\frac{12}{4}=3$.

Step2: Multiply numerator and denominator by 3

Multiply the numerator 3 by 3: $3\times3 = 9$. Multiply the denominator 4 by 3: $4\times3 = 12$. So $\frac{3\times3}{4\times3}=\frac{9}{12}$. So $\frac{3}{4}=\frac{9}{12}$ is correct.

for $\boldsymbol{\frac{2}{3}=\frac{6}{12}}$:

Step1: Recall fraction renaming (equivalent fractions)

To rename $\frac{2}{3}$ to a fraction with denominator 12, we find the multiplier. Let the multiplier be $y$. We have $3\times y=12$, so $y = \frac{12}{3}=4$.

Step2: Multiply numerator and denominator by 4

Multiply the numerator 2 by 4: $2\times4 = 8$. Multiply the denominator 3 by 4: $3\times4 = 12$. So the correct equivalent fraction should be $\frac{2\times4}{3\times4}=\frac{8}{12}$, not $\frac{6}{12}$. The mistake here is that when renaming $\frac{2}{3}$ to a denominator of 12, the numerator was multiplied by 3 (since $2\times3 = 6$) instead of 4 (the number needed to multiply the denominator 3 to get 12).

Answer:

• $\frac{3}{4}=\frac{9}{12}$ is correct. The reasoning is that we multiply the numerator and denominator of $\frac{3}{4}$ by 3 (since $4\times3 = 12$) to get $\frac{9}{12}$.
• $\frac{2}{3}=\frac{6}{12}$ is incorrect. The correct renaming is $\frac{2}{3}=\frac{2\times4}{3\times4}=\frac{8}{12}$ (we should multiply numerator and denominator by 4, not 3, to get a denominator of 12).