Question

choose all fractions equivalent to each given fraction.

given fraction check all fractions that are equivalent to the given fraction.
(a) \\(\frac{4}{9}\\) \\(\square \frac{-4}{-9}\\) \\(\square \frac{4}{-9}\\) \\(\square \frac{-4}{9}\\) \\(\square -\frac{4}{9}\\)

given fraction check all fractions that are equivalent to the given fraction.
(b) \\(\frac{-10}{7}\\) \\(\square \frac{10}{7}\\) \\(\square \frac{10}{-7}\\) \\(\square \frac{-10}{-7}\\) \\(\square -\frac{10}{7}\\)

Explanation:

Step1: Recall the rule of signs for fractions

The rule is that \(\frac{a}{b}=\frac{-a}{-b}=-\frac{a}{-b}=-\frac{-a}{b}\), and \(-\frac{a}{b}=\frac{a}{-b}=\frac{-a}{b}\).

Step2: Analyze part (a) with given fraction \(\frac{4}{9}\)

• For \(\frac{-4}{-9}\): Using the rule, \(\frac{-4}{-9}=\frac{4}{9}\) (since negative divided by negative is positive), so this is equivalent.
• For \(\frac{4}{-9}\): \(\frac{4}{-9}=-\frac{4}{9}\), which is not equivalent to \(\frac{4}{9}\).
• For \(\frac{-4}{9}\): \(\frac{-4}{9}=-\frac{4}{9}\), not equivalent.
• For \(-\frac{4}{9}\): This is the negative of \(\frac{4}{9}\), not equivalent. Wait, no, wait: Wait, the given fraction is \(\frac{4}{9}\). Wait, \(\frac{-4}{-9}\): numerator and denominator both negative, so it's \(\frac{4}{9}\) (because \((-4)\div(-9) = 4\div9\)). Then \(\frac{4}{-9}\) is \(-\frac{4}{9}\), \(\frac{-4}{9}\) is \(-\frac{4}{9}\), and \(-\frac{4}{9}\) is \(-\frac{4}{9}\). Wait, no, I made a mistake. Wait, the given fraction is \(\frac{4}{9}\). Let's re - check:

The sign rules for fractions: \(\frac{a}{b}=\frac{-a}{-b}\), \(\frac{a}{-b}=-\frac{a}{b}\), \(\frac{-a}{b}=-\frac{a}{b}\), and \(-\frac{a}{b}=\frac{a}{-b}=\frac{-a}{b}\).

So for \(\frac{4}{9}\):

• \(\frac{-4}{-9}\): Since both numerator and denominator are negative, dividing two negatives gives a positive, so \(\frac{-4}{-9}=\frac{4}{9}\), so this is equivalent.
• \(\frac{4}{-9}\): Here, denominator is negative, so \(\frac{4}{-9}=-\frac{4}{9}\), not equivalent to \(\frac{4}{9}\).
• \(\frac{-4}{9}\): Numerator is negative, so \(\frac{-4}{9}=-\frac{4}{9}\), not equivalent.
• \(-\frac{4}{9}\): This is the negative of \(\frac{4}{9}\), not equivalent. Wait, no, wait, maybe I messed up. Wait, the given fraction is \(\frac{4}{9}\). Let's check each option:

1. \(\frac{-4}{-9}\): \((-4)\div(-9)=\frac{4}{9}\) (because negative ÷ negative = positive), so this is equal to \(\frac{4}{9}\).
2. \(\frac{4}{-9}\): \(4\div(-9)=-\frac{4}{9}

eq\frac{4}{9}\)

1. \(\frac{-4}{9}\): \((-4)\div9 = -\frac{4}{9}

eq\frac{4}{9}\)

1. \(-\frac{4}{9}\): This is \(-\frac{4}{9}

eq\frac{4}{9}\)

So for part (a), the equivalent fraction is \(\frac{-4}{-9}\).

Step3: Analyze part (b) with given fraction \(\frac{-10}{7}\)

• For \(\frac{10}{7}\): This is positive, while \(\frac{-10}{7}\) is negative, so not equivalent.
• For \(\frac{10}{-7}\): \(\frac{10}{-7}=-\frac{10}{7}\), and \(\frac{-10}{7}=-\frac{10}{7}\), so this is equivalent.
• For \(\frac{-10}{-7}\): \(\frac{-10}{-7}=\frac{10}{7}\), not equivalent to \(\frac{-10}{7}\) (which is negative).
• For \(-\frac{10}{7}\): \(-\frac{10}{7}=\frac{-10}{7}\), so this is equivalent.

Wait, let's re - do part (b) properly. The given fraction is \(\frac{-10}{7}\).

Using the sign rules:

• \(\frac{10}{7}\): positive, \(\frac{-10}{7}\) is negative, so not equivalent.
• \(\frac{10}{-7}\): \(\frac{10}{-7}=-\frac{10}{7}\), and \(\frac{-10}{7}=-\frac{10}{7}\), so equivalent.
• \(\frac{-10}{-7}\): \(\frac{-10}{-7}=\frac{10}{7}\), positive, not equivalent to \(\frac{-10}{7}\) (negative).
• \(-\frac{10}{7}\): \(-\frac{10}{7}=\frac{-10}{7}\), so equivalent.

Answer:

Part (a)

The fraction equivalent to \(\frac{4}{9}\) is \(\frac{-4}{-9}\).

Part (b)

The fractions equivalent to \(\frac{-10}{7}\) are \(\frac{10}{-7}\) and \(-\frac{10}{7}\).

(In box - checking terms:

For part (a): Check the box for \(\frac{-4}{-9}\)

For part (b): Check the boxes for \(\frac{10}{-7}\) and \(-\frac{10}{7}\))