Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Discover a wealth of knowledge from professionals across various disciplines on our user-friendly Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
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]
### (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]
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
