Question

choose all fractions equivalent to each given fraction.

(a) given fraction: \\(\frac{-3}{-7}\\)
check all fractions that are equivalent to the given fraction:
\\(\square -\frac{3}{7}\\) \\(\square \frac{-3}{7}\\) \\(\square \frac{3}{-7}\\) \\(\square \frac{3}{7}\\)

(b) given fraction: \\(-\frac{4}{5}\\)
check all fractions that are equivalent to the given fraction:
\\(\square \frac{-4}{5}\\) \\(\square \frac{4}{5}\\) \\(\square \frac{4}{-5}\\) \\(\square \frac{-4}{-5}\\)

Explanation:

Step1: Recall the rule of signs for fractions

The rule is $\frac{-a}{-b}=\frac{a}{b}$, $\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}$.

Step2: Solve part (a)

Given fraction is $\frac{-3}{-7}$. Using the rule, $\frac{-3}{-7}=\frac{3}{7}$. Also, $\frac{-3}{7}=-\frac{3}{7}$, $\frac{3}{-7}=-\frac{3}{7}$, and $\frac{-3}{-7}=\frac{3}{7}$. Wait, let's re - evaluate:

• For $\frac{-3}{-7}$: The numerator and denominator are both negative. When we divide two negative numbers, the result is positive. So $\frac{-3}{-7}=\frac{3}{7}$.
• For $-\frac{3}{7}$: This is equal to $\frac{-3}{7}$ or $\frac{3}{-7}$.
• For $\frac{-3}{7}$: Numerator negative, denominator positive, so it is $-\frac{3}{7}$.
• For $\frac{3}{-7}$: Numerator positive, denominator negative, so it is $-\frac{3}{7}$.
• For $\frac{3}{7}$: Positive fraction.

Since $\frac{-3}{-7}=\frac{3}{7}$, and also, let's check the signs again. $\frac{-3}{-7}=\frac{3}{7}$, and also, if we consider the sign rules:
$\frac{-3}{-7}=\frac{3}{7}$, and also, $-\frac{3}{7}=\frac{-3}{7}=\frac{3}{-7}$, but $\frac{-3}{-7}$ is equal to $\frac{3}{7}$. Wait, no, let's do it step by step. The fraction $\frac{-3}{-7}$: divide numerator and denominator by - 1, we get $\frac{(-3)\div(-1)}{(-7)\div(-1)}=\frac{3}{7}$. Also, if we multiply numerator and denominator of $\frac{-3}{-7}$ by - 1, we get $\frac{3}{7}$. Now, let's check each option:

• $-\frac{3}{7}$: $\frac{-3}{7}$, which is not equal to $\frac{-3}{-7}$ (since $\frac{-3}{-7}=\frac{3}{7}$)
• $\frac{-3}{7}$: Not equal to $\frac{-3}{-7}$
• $\frac{3}{-7}$: Not equal to $\frac{-3}{-7}$
• $\frac{3}{7}$: Equal to $\frac{-3}{-7}$ (because $\frac{-3}{-7}=\frac{3}{7}$)

Wait, I made a mistake earlier. The rule is $\frac{-a}{-b}=\frac{a}{b}$, $\frac{-a}{b}=-\frac{a}{b}$, $\frac{a}{-b}=-\frac{a}{b}$. So for $\frac{-3}{-7}$, applying $\frac{-a}{-b}=\frac{a}{b}$, we have $\frac{-3}{-7}=\frac{3}{7}$. Also, let's check the other way:
$\frac{-3}{-7}=\frac{3}{7}$, $-\frac{3}{7}=\frac{-3}{7}$, $\frac{-3}{7}
eq\frac{3}{7}$, $\frac{3}{-7}=\frac{-3}{7}
eq\frac{3}{7}$, $\frac{3}{7}=\frac{3}{7}$. So the fraction equivalent to $\frac{-3}{-7}$ is $\frac{3}{7}$.

Step3: Solve part (b)

Given fraction is $-\frac{4}{5}$.

• For $\frac{-4}{5}$: This is equal to $-\frac{4}{5}$ (since numerator is negative, denominator is positive)
• For $\frac{4}{5}$: Positive fraction, not equal to $-\frac{4}{5}$
• For $\frac{4}{-5}$: Numerator positive, denominator negative, so $\frac{4}{-5}=-\frac{4}{5}$
• For $\frac{-4}{-5}$: Numerator and denominator negative, so $\frac{-4}{-5}=\frac{4}{5}$, which is positive, not equal to $-\frac{4}{5}$

So for part (b), the equivalent fractions are $\frac{-4}{5}$ and $\frac{4}{-5}$.

Answer:

(a) The fraction equivalent to $\frac{-3}{-7}$ is $\frac{3}{7}$.
(b) The fractions equivalent to $-\frac{4}{5}$ are $\frac{-4}{5}$ and $\frac{4}{-5}$.

Wait, let's re - check part (a) with the sign rule $\frac{-a}{-b}=\frac{a}{b}$, $\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}$.

Part (a):
Given $\frac{-3}{-7}$. Using $\frac{-a}{-b}=\frac{a}{b}$, so $\frac{-3}{-7}=\frac{3}{7}$. Also, let's check the other options:

• $-\frac{3}{7}$: $\frac{-3}{7}$, which is $-\frac{3}{7}

eq\frac{3}{7}$

• $\frac{-3}{7}$: $-\frac{3}{7}

eq\frac{3}{7}$

• $\frac{3}{-7}$: $-\frac{3}{7}

eq\frac{3}{7}$

• $\frac{3}{7}$: Equal to $\frac{-3}{-7}$

Part (b):
Given $-\frac{4}{5}$.

• $\frac{-4}{5}$: $-\frac{4}{5}$, equal to the given fraction.
• $\frac{4}{5}$: $\frac{4}{5}

eq-\frac{4}{5}$

• $\frac{4}{-5}$: $\frac{4}{-5}=-\frac{4}{5}$, equal to the given fraction.
• $\frac{-4}{-5}$: $\frac{-4}{-5}=\frac{4}{5}

eq-\frac{4}{5}$

So for part (a), the equivalent fraction is $\frac{3}{7}$. For part (b), the equivalent fractions are $\frac{-4}{5}$ and $\frac{4}{-5}$.