Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Discover detailed solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To find three rational numbers between [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] using the method of finding the mean of two numbers, we will perform the following steps:
1. Compute the First Rational Number (Mean of [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex])
First, we add [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] and then divide by 2 to find their mean.
[tex]\[ \text{Mean}_1 = \frac{\frac{-2}{3} + \frac{1}{6}}{2} \][/tex]
We need a common denominator to add these fractions, which is 6 in this case:
[tex]\[ \frac{-2}{3} = \frac{-2 \times 2}{3 \times 2} = \frac{-4}{6} \][/tex]
Now add the two fractions:
[tex]\[ \frac{-4}{6} + \frac{1}{6} = \frac{-4+1}{6} = \frac{-3}{6} = \frac{-1}{2} \][/tex]
Then find the mean:
[tex]\[ \text{Mean}_1 = \frac{\frac{-1}{2}}{2} = \frac{-1}{4} = -0.25 \][/tex]
So, the first rational number is [tex]\(-0.25\)[/tex].
2. Compute the Second Rational Number (Mean of [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(-0.25\)[/tex])
Find the mean of [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(-0.25\)[/tex]:
[tex]\[ \text{Mean}_2 = \frac{\frac{-2}{3} + (-0.25)}{2} \][/tex]
Convert [tex]\(-0.25\)[/tex] into fraction form:
[tex]\[ -0.25 = \frac{-1}{4} \][/tex]
Find a common denominator (12 in this case):
[tex]\[ \frac{-2}{3} = \frac{-2 \times 4}{3 \times 4} = \frac{-8}{12}, \quad \frac{-1}{4} = \frac{-1 \times 3}{4 \times 3} = \frac{-3}{12} \][/tex]
Add the fractions:
[tex]\[ \frac{-8}{12} + \frac{-3}{12} = \frac{-11}{12} \][/tex]
Then find the mean:
[tex]\[ \text{Mean}_2 = \frac{\frac{-11}{12}}{2} = \frac{-11}{24} \approx -0.4583333333333333 \][/tex]
So, the second rational number is approximately [tex]\(-0.4583\)[/tex].
3. Compute the Third Rational Number (Mean of [tex]\(-0.25\)[/tex] and [tex]\(\frac{1}{6}\)[/tex])
Find the mean of [tex]\(-0.25\)[/tex] and [tex]\(\frac{1}{6}\)[/tex]:
[tex]\[ \text{Mean}_3 = \frac{-0.25 + \frac{1}{6}}{2} \][/tex]
Convert [tex]\(-0.25\)[/tex] to fraction form:
[tex]\[ -0.25 = \frac{-1}{4} \][/tex]
Find a common denominator (12 in this case):
[tex]\[ \frac{-1}{4} = \frac{-1 \times 3}{4 \times 3} = \frac{-3}{12}, \quad \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \][/tex]
Add the fractions:
[tex]\[ \frac{-3}{12} + \frac{2}{12} = \frac{-1}{12} \][/tex]
Then find the mean:
[tex]\[ \text{Mean}_3 = \frac{\frac{-1}{12}}{2} = \frac{-1}{24} \approx -0.04166666666666667 \][/tex]
So, the third rational number is approximately [tex]\(-0.0417\)[/tex].
Finally, the three rational numbers between [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] are approximately:
- [tex]\(-0.25\)[/tex]
- [tex]\(-0.4583\)[/tex]
- [tex]\(-0.0417\)[/tex]
### Representing the Numbers on the Number Line
To represent these numbers on the number line, we mark:
1. [tex]\(\frac{-2}{3} \approx -0.6667\)[/tex]
2. [tex]\(\frac{1}{6} \approx 0.1667\)[/tex]
3. [tex]\(-0.25\)[/tex]
4. [tex]\(-0.4583\)[/tex]
5. [tex]\(-0.0417\)[/tex]
Visualise plotting these points on the number line from left to right:
[tex]\[ -0.6667 \quad -0.4583 \quad -0.25 \quad -0.0417 \quad 0.1667 \][/tex]
These points fall between [tex]\(\frac{-2}{3}\)[/tex] (approximately [tex]\(-0.6667\)[/tex]) and [tex]\(\frac{1}{6}\)[/tex] (approximately [tex]\(0.1667\)[/tex]), successfully identifying three rational numbers between the two given numbers.
1. Compute the First Rational Number (Mean of [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex])
First, we add [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] and then divide by 2 to find their mean.
[tex]\[ \text{Mean}_1 = \frac{\frac{-2}{3} + \frac{1}{6}}{2} \][/tex]
We need a common denominator to add these fractions, which is 6 in this case:
[tex]\[ \frac{-2}{3} = \frac{-2 \times 2}{3 \times 2} = \frac{-4}{6} \][/tex]
Now add the two fractions:
[tex]\[ \frac{-4}{6} + \frac{1}{6} = \frac{-4+1}{6} = \frac{-3}{6} = \frac{-1}{2} \][/tex]
Then find the mean:
[tex]\[ \text{Mean}_1 = \frac{\frac{-1}{2}}{2} = \frac{-1}{4} = -0.25 \][/tex]
So, the first rational number is [tex]\(-0.25\)[/tex].
2. Compute the Second Rational Number (Mean of [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(-0.25\)[/tex])
Find the mean of [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(-0.25\)[/tex]:
[tex]\[ \text{Mean}_2 = \frac{\frac{-2}{3} + (-0.25)}{2} \][/tex]
Convert [tex]\(-0.25\)[/tex] into fraction form:
[tex]\[ -0.25 = \frac{-1}{4} \][/tex]
Find a common denominator (12 in this case):
[tex]\[ \frac{-2}{3} = \frac{-2 \times 4}{3 \times 4} = \frac{-8}{12}, \quad \frac{-1}{4} = \frac{-1 \times 3}{4 \times 3} = \frac{-3}{12} \][/tex]
Add the fractions:
[tex]\[ \frac{-8}{12} + \frac{-3}{12} = \frac{-11}{12} \][/tex]
Then find the mean:
[tex]\[ \text{Mean}_2 = \frac{\frac{-11}{12}}{2} = \frac{-11}{24} \approx -0.4583333333333333 \][/tex]
So, the second rational number is approximately [tex]\(-0.4583\)[/tex].
3. Compute the Third Rational Number (Mean of [tex]\(-0.25\)[/tex] and [tex]\(\frac{1}{6}\)[/tex])
Find the mean of [tex]\(-0.25\)[/tex] and [tex]\(\frac{1}{6}\)[/tex]:
[tex]\[ \text{Mean}_3 = \frac{-0.25 + \frac{1}{6}}{2} \][/tex]
Convert [tex]\(-0.25\)[/tex] to fraction form:
[tex]\[ -0.25 = \frac{-1}{4} \][/tex]
Find a common denominator (12 in this case):
[tex]\[ \frac{-1}{4} = \frac{-1 \times 3}{4 \times 3} = \frac{-3}{12}, \quad \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \][/tex]
Add the fractions:
[tex]\[ \frac{-3}{12} + \frac{2}{12} = \frac{-1}{12} \][/tex]
Then find the mean:
[tex]\[ \text{Mean}_3 = \frac{\frac{-1}{12}}{2} = \frac{-1}{24} \approx -0.04166666666666667 \][/tex]
So, the third rational number is approximately [tex]\(-0.0417\)[/tex].
Finally, the three rational numbers between [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] are approximately:
- [tex]\(-0.25\)[/tex]
- [tex]\(-0.4583\)[/tex]
- [tex]\(-0.0417\)[/tex]
### Representing the Numbers on the Number Line
To represent these numbers on the number line, we mark:
1. [tex]\(\frac{-2}{3} \approx -0.6667\)[/tex]
2. [tex]\(\frac{1}{6} \approx 0.1667\)[/tex]
3. [tex]\(-0.25\)[/tex]
4. [tex]\(-0.4583\)[/tex]
5. [tex]\(-0.0417\)[/tex]
Visualise plotting these points on the number line from left to right:
[tex]\[ -0.6667 \quad -0.4583 \quad -0.25 \quad -0.0417 \quad 0.1667 \][/tex]
These points fall between [tex]\(\frac{-2}{3}\)[/tex] (approximately [tex]\(-0.6667\)[/tex]) and [tex]\(\frac{1}{6}\)[/tex] (approximately [tex]\(0.1667\)[/tex]), successfully identifying three rational numbers between the two given numbers.
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
