Certainly! Let's tackle each part of the question step-by-step:
### (i) Find three rational numbers between [tex]\( \frac{3}{7} \)[/tex] and [tex]\( \frac{2}{3} \)[/tex]
To find three rational numbers between [tex]\( \frac{3}{7} \)[/tex] and [tex]\( \frac{2}{3} \)[/tex], we convert these fractions to like fractions with a common denominator. The least common multiple (LCM) of 7 and 3 is 21.
[tex]\[
\frac{3}{7} = \frac{3 \times 3}{7 \times 3} = \frac{9}{21}
\][/tex]
[tex]\[
\frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21}
\][/tex]
Now, we choose three rational numbers between [tex]\( \frac{9}{21} \)[/tex] and [tex]\( \frac{14}{21} \)[/tex]:
[tex]\[
\frac{10}{21}, \frac{11}{21}, \frac{12}{21}
\][/tex]
So the three rational numbers are:
[tex]\[
\frac{10}{21} \approx 0.476, \quad \frac{11}{21} \approx 0.524, \quad \frac{12}{21} \approx 0.571
\][/tex]
### (ii) Find [tex]\( \frac{3}{7} + \left( -\frac{6}{11} \right) + \left( -\frac{8}{21} \right) + \frac{5}{22} \)[/tex]
To add these fractions, we first find a common denominator. The least common multiple of 7, 11, 21, and 22 is 462.
We convert each fraction to have the common denominator:
[tex]\[
\frac{3}{7} = \frac{3 \times 66}{7 \times 66} = \frac{198}{462}
\][/tex]
[tex]\[
-\frac{6}{11} = \frac{-6 \times 42}{11 \times 42} = \frac{-252}{462}
\][/tex]
[tex]\[
-\frac{8}{21} = \frac{-8 \times 22}{21 \times 22} = \frac{-176}{462}
\][/tex]
[tex]\[
\frac{5}{22} = \frac{5 \times 21}{22 \times 21} = \frac{105}{462}
\][/tex]
Adding these fractions:
[tex]\[
\frac{198}{462} + \frac{-252}{462} + \frac{-176}{462} + \frac{105}{462} = \frac{198 - 252 - 176 + 105}{462} = \frac{-125}{462} \approx 0.0038
\][/tex]
### (iii) Represent [tex]\( -\frac{2}{11}, -\frac{5}{11}, -\frac{9}{11} \)[/tex] on the number line
To represent these fractions on the number line, we should note their approximate decimal values and place them accordingly between -1 and 0.
[tex]\[
-\frac{2}{11} \approx -0.182, \quad -\frac{5}{11} \approx -0.455, \quad -\frac{9}{11} \approx -0.818
\][/tex]
### (iv) What should be added to [tex]\( -\frac{16}{3} \)[/tex] to make it [tex]\( \frac{1}{9} \)[/tex]?
Let [tex]\( x \)[/tex] be the number we need to add to [tex]\( -\frac{16}{3} \)[/tex] to get [tex]\( \frac{1}{9} \)[/tex].
[tex]\[
-\frac{16}{3} + x = \frac{1}{9}
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{1}{9} - \left( -\frac{16}{3} \right) = \frac{1}{9} + \frac{16}{3}
\][/tex]
Convert [tex]\( \frac{16}{3} \)[/tex] to have a common denominator of 9:
[tex]\[
\frac{16}{3} = \frac{16 \times 3}{3 \times 3} = \frac{48}{9}
\][/tex]
Now add the fractions:
[tex]\[
x = \frac{1}{9} + \frac{48}{9} = \frac{49}{9} \approx 5.444
\][/tex]
### (v) What is the additive inverse of [tex]\( \frac{3}{5} \)[/tex]?
The additive inverse of a number is the number that, when added to it, results in zero. For [tex]\( \frac{3}{5} \)[/tex]:
[tex]\[
\text{Additive Inverse} = -\frac{3}{5} = -0.6
\][/tex]
So the answers are:
[tex]\[
\begin{aligned}
& (i) & \quad & \left( \frac{10}{21}, \frac{11}{21}, \frac{12}{21} \right) \approx (0.476, 0.524, 0.571) \\
& (ii) & \quad & 0.0038 \\
& (iii) & \quad & \text{Representing } -0.182, -0.455, -0.818 \text{ on the number line} \\
& (iv) & \quad & 5.444 \\
& (v) & \quad & -0.6 \\
\end{aligned}
\][/tex]
