Answer :
Certainly! Let's go through each part of the question step-by-step.
### Section D
#### Part (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 first convert these fractions to equivalent fractions with a common denominator. The least common multiple (LCM) of 7 and 3 is 21. Therefore, we have:
[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 need to find three rational numbers between [tex]\(\frac{9}{21}\)[/tex] and [tex]\(\frac{14}{21}\)[/tex]. These could be:
[tex]\[ \frac{10}{21}, \frac{11}{21}, \frac{12}{21} \][/tex]
However, the actual rational numbers between [tex]\( \frac{3}{7} \)[/tex] and [tex]\( \frac{2}{3} \)[/tex] are slightly different:
[tex]\[ 0.47619047619047616, 0.5238095238095238, 0.5714285714285714, 0.6190476190476191 \][/tex]
#### Part (ii): Find the sum of [tex]\( \frac{3}{7} + \left(-\frac{6}{11}\right) + \left(-\frac{8}{21}\right) + \frac{5}{22} \)[/tex].
Given the fractions:
[tex]\[ \frac{3}{7}, -\frac{6}{11}, -\frac{8}{21}, \frac{5}{22} \][/tex]
Adding these fractions together:
[tex]\[ \frac{3}{7} + \left(-\frac{6}{11}\right) + \left(-\frac{8}{21}\right) + \frac{5}{22} = -0.27056277056277056 \][/tex]
#### Part (iii): Represent [tex]\( -\frac{2}{11}, -\frac{5}{11}, -\frac{9}{11} \)[/tex] on the number line.
The respective decimal representations are:
[tex]\[ -\frac{2}{11} = -0.18181818181818182 \][/tex]
[tex]\[ -\frac{5}{11} = -0.45454545454545453 \][/tex]
[tex]\[ -\frac{9}{11} = -0.8181818181818182 \][/tex]
These values can thus be placed on the number line as such:
[tex]\[ -0.1818, -0.4545, -0.8181 \][/tex]
#### Part (iv): What should be added to [tex]\( -\frac{16}{3} \)[/tex] to make it [tex]\( \frac{1}{9} \)[/tex]?
To find what should be added to [tex]\( -\frac{16}{3} \)[/tex] to get [tex]\( \frac{1}{9} \)[/tex]:
[tex]\[ \text{Let the number to be added be } x. \\ \text{So, } -\frac{16}{3} + x = \frac{1}{9} \\ x = \frac{1}{9} - \left(-\frac{16}{3}\right) = \frac{1}{9} + \frac{16}{3} \][/tex]
Solving this:
[tex]\[ x = 5.444444444444444 \][/tex]
#### Part (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 the original number, results in zero.
The additive inverse of [tex]\( \frac{3}{5} \)[/tex] is:
[tex]\[ -\frac{3}{5} = -0.6 \][/tex]
### Summary of Results
[tex]\[ \begin{array}{l} \text{(i) Three rational numbers between } \frac{3}{7} \text{ and } \frac{2}{3} \text{ are:} \\ 0.47619047619047616, 0.5238095238095238, 0.5714285714285714, 0.6190476190476191 \\ \text{(ii) The sum of the given rational numbers is:} \\ -0.27056277056277056 \\ \text{(iii) Representation of } -\frac{2}{11}, -\frac{5}{11}, -\frac{9}{11} \text{ on the number line is:} \\ -0.18181818181818182, -0.45454545454545453, -0.8181818181818182 \\ \text{(iv) Number to be added to } -\frac{16}{3} \text{ to make it } \frac{1}{9} \text{ is:} \\ 5.444444444444444 \\ \text{(v) The additive inverse of } \frac{3}{5} \text{ is:} \\ -0.6 \end{array} \][/tex]
### Section D
#### Part (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 first convert these fractions to equivalent fractions with a common denominator. The least common multiple (LCM) of 7 and 3 is 21. Therefore, we have:
[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 need to find three rational numbers between [tex]\(\frac{9}{21}\)[/tex] and [tex]\(\frac{14}{21}\)[/tex]. These could be:
[tex]\[ \frac{10}{21}, \frac{11}{21}, \frac{12}{21} \][/tex]
However, the actual rational numbers between [tex]\( \frac{3}{7} \)[/tex] and [tex]\( \frac{2}{3} \)[/tex] are slightly different:
[tex]\[ 0.47619047619047616, 0.5238095238095238, 0.5714285714285714, 0.6190476190476191 \][/tex]
#### Part (ii): Find the sum of [tex]\( \frac{3}{7} + \left(-\frac{6}{11}\right) + \left(-\frac{8}{21}\right) + \frac{5}{22} \)[/tex].
Given the fractions:
[tex]\[ \frac{3}{7}, -\frac{6}{11}, -\frac{8}{21}, \frac{5}{22} \][/tex]
Adding these fractions together:
[tex]\[ \frac{3}{7} + \left(-\frac{6}{11}\right) + \left(-\frac{8}{21}\right) + \frac{5}{22} = -0.27056277056277056 \][/tex]
#### Part (iii): Represent [tex]\( -\frac{2}{11}, -\frac{5}{11}, -\frac{9}{11} \)[/tex] on the number line.
The respective decimal representations are:
[tex]\[ -\frac{2}{11} = -0.18181818181818182 \][/tex]
[tex]\[ -\frac{5}{11} = -0.45454545454545453 \][/tex]
[tex]\[ -\frac{9}{11} = -0.8181818181818182 \][/tex]
These values can thus be placed on the number line as such:
[tex]\[ -0.1818, -0.4545, -0.8181 \][/tex]
#### Part (iv): What should be added to [tex]\( -\frac{16}{3} \)[/tex] to make it [tex]\( \frac{1}{9} \)[/tex]?
To find what should be added to [tex]\( -\frac{16}{3} \)[/tex] to get [tex]\( \frac{1}{9} \)[/tex]:
[tex]\[ \text{Let the number to be added be } x. \\ \text{So, } -\frac{16}{3} + x = \frac{1}{9} \\ x = \frac{1}{9} - \left(-\frac{16}{3}\right) = \frac{1}{9} + \frac{16}{3} \][/tex]
Solving this:
[tex]\[ x = 5.444444444444444 \][/tex]
#### Part (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 the original number, results in zero.
The additive inverse of [tex]\( \frac{3}{5} \)[/tex] is:
[tex]\[ -\frac{3}{5} = -0.6 \][/tex]
### Summary of Results
[tex]\[ \begin{array}{l} \text{(i) Three rational numbers between } \frac{3}{7} \text{ and } \frac{2}{3} \text{ are:} \\ 0.47619047619047616, 0.5238095238095238, 0.5714285714285714, 0.6190476190476191 \\ \text{(ii) The sum of the given rational numbers is:} \\ -0.27056277056277056 \\ \text{(iii) Representation of } -\frac{2}{11}, -\frac{5}{11}, -\frac{9}{11} \text{ on the number line is:} \\ -0.18181818181818182, -0.45454545454545453, -0.8181818181818182 \\ \text{(iv) Number to be added to } -\frac{16}{3} \text{ to make it } \frac{1}{9} \text{ is:} \\ 5.444444444444444 \\ \text{(v) The additive inverse of } \frac{3}{5} \text{ is:} \\ -0.6 \end{array} \][/tex]
