Certainly! Let's go through each part of the question step-by-step.
### 1. Draw a Number Line and Mark the Rational Numbers [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{-5}{3}\)[/tex] on it.
Here's a representation of the number line with [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{-5}{3}\)[/tex] marked on it:
[tex]\[
\text{Number Line: } ... -2, \frac{-5}{3}, -1.333, -1, \frac{-2}{3}, -0.666, 0, \frac{1}{3}, 0.333, \frac{2}{3}, 0.666, 1, ...
\][/tex]
### 2. Write Two Equivalent Rational Numbers for Each of the Following:
(i) [tex]\(\frac{6}{-7}\)[/tex]
Equivalent fractions can be found by multiplying or dividing the numerator and the denominator by the same non-zero number.
[tex]\[
\frac{6}{-7} = \frac{-6}{7} \quad \text{(Multiplying numerator and denominator by -1)}
\][/tex]
[tex]\[
\frac{6}{-7} = \frac{12}{-14} \quad \text{(Multiplying numerator and denominator by 2)}
\][/tex]
So two equivalent fractions for [tex]\(\frac{6}{-7}\)[/tex] are:
[tex]\(-0.8571428571428571 \quad (\frac{-6}{7})\)[/tex]
[tex]\(-0.8571428571428571 \quad (\frac{12}{-14})\)[/tex]
(ii) [tex]\(\frac{1}{20}\)[/tex]
[tex]\[
\frac{1}{20} = \frac{2}{40} \quad \text{(Multiplying numerator and denominator by 2)}
\][/tex]
[tex]\[
\frac{1}{20} = \frac{3}{60} \quad \text{(Multiplying numerator and denominator by 3)}
\][/tex]
So two equivalent fractions for [tex]\(\frac{1}{20}\)[/tex] are:
[tex]\(0.05 \quad (\frac{2}{40})\)[/tex]
[tex]\(0.05 \quad (\frac{3}{60})\)[/tex]
(iii) [tex]\(\frac{-2}{5}\)[/tex]
[tex]\[
\frac{-2}{5} = \frac{2}{-5} \quad \text{(Multiplying numerator and denominator by -1)}
\][/tex]
[tex]\[
\frac{-2}{5} = \frac{-4}{10} \quad \text{(Multiplying numerator and denominator by 2)}
\][/tex]
So two equivalent fractions for [tex]\(\frac{-2}{5}\)[/tex] are:
[tex]\(-0.4 \quad (\frac{2}{-5})\)[/tex]
[tex]\(-0.4 \quad (\frac{-4}{10})\)[/tex]
### 3. Identify the Greater Fraction:
(iv) [tex]\(\frac{-4}{-8}\)[/tex]
To simplify this, we first note that:
[tex]\[
\frac{-4}{-8} = \frac{4}{8} = \frac{1}{2}
\][/tex]
So the fraction simplifies to [tex]\(\frac{1}{2}\)[/tex]. Comparing this with [tex]\(\frac{1}{1}\)[/tex], we can see that:
[tex]\[
\frac{1}{2} = 0.5 \quad \text{and} \quad \frac{1}{1} = 1.0
\][/tex]
Clearly, [tex]\(\frac{1}{1}\)[/tex] is greater than [tex]\(\frac{1}{2}\)[/tex].
Therefore, the greater fraction between [tex]\(\frac{-4}{-8}\)[/tex] (which simplifies to [tex]\(\frac{1}{2}\)[/tex]) and [tex]\(\frac{1}{1}\)[/tex] is:
[tex]\(1.0\)[/tex]
This completes the detailed solution to the problem given.
