Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Our platform provides a seamless experience for finding reliable answers from a knowledgeable network of professionals. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
Sure, let's go through each part step-by-step to provide a detailed solution.
### Part (a)
Evaluate [tex]\(\frac{12}{56}+\frac{-7}{8}\)[/tex]:
1. Simplify [tex]\(\frac{12}{56}\)[/tex]:
- The greatest common divisor (GCD) of 12 and 56 is 4.
- So, [tex]\(\frac{12}{56} = \frac{12 \div 4}{56 \div 4} = \frac{3}{14}\)[/tex].
2. [tex]\(\frac{-7}{8}\)[/tex] remains the same.
3. Add the fractions [tex]\(\frac{3}{14} + \frac{-7}{8}\)[/tex]:
- Find the common denominator: the least common multiple (LCM) of 14 and 8 is 56.
- Convert to common denominators:
- [tex]\(\frac{3}{14} = \frac{3 \times 4}{14 \times 4} = \frac{12}{56}\)[/tex],
- [tex]\(\frac{-7}{8} = \frac{-7 \times 7}{8 \times 7} = \frac{-49}{56}\)[/tex].
4. Now add the fractions:
[tex]\[ \frac{12}{56} + \frac{-49}{56} = \frac{12 - 49}{56} = \frac{-37}{56} \approx -0.6607142857142857. \][/tex]
### Part (b)
Evaluate [tex]\(\frac{-5}{9}+\frac{-7}{15}\)[/tex]:
1. Both fractions are given in their simplest forms.
2. Add the fractions [tex]\(\frac{-5}{9} + \frac{-7}{15}\)[/tex]:
- Find the common denominator: LCM of 9 and 15 is 45.
- Convert to common denominators:
- [tex]\(\frac{-5}{9} = \frac{-5 \times 5}{9 \times 5} = \frac{-25}{45}\)[/tex],
- [tex]\(\frac{-7}{15} = \frac{-7 \times 3}{15 \times 3} = \frac{-21}{45}\)[/tex].
3. Now add the fractions:
[tex]\[ \frac{-25}{45} + \frac{-21}{45} = \frac{-25 - 21}{45} = \frac{-46}{45} \approx -1.0222222222222221. \][/tex]
### Part 4: Additive inverses
Find the additive inverse for each of the following:
(a) [tex]\(\frac{-2}{3}\)[/tex]:
- The additive inverse of [tex]\(\frac{-2}{3}\)[/tex] is [tex]\(\frac{2}{3}\)[/tex].
(b) [tex]\(\frac{9}{21}\)[/tex]:
- Simplify [tex]\(\frac{9}{21}\)[/tex]: the GCD of 9 and 21 is 3.
- [tex]\(\frac{9}{21} = \frac{9 \div 3}{21 \div 3} = \frac{3}{7}\)[/tex].
- The additive inverse of [tex]\(\frac{3}{7}\)[/tex] is [tex]\(\frac{-3}{7}\)[/tex].
(c) [tex]\(\frac{-12}{15}\)[/tex]:
- Simplify [tex]\(\frac{-12}{15}\)[/tex]: the GCD of 12 and 15 is 3.
- [tex]\(\frac{-12}{15} = \frac{-12 \div 3}{15 \div 3} = \frac{-4}{5}\)[/tex].
- The additive inverse of [tex]\(\frac{-4}{5}\)[/tex] is [tex]\(\frac{4}{5}\)[/tex].
### Part 5: Verify the associative property
(a) Verify [tex]\(\frac{7}{6}+\frac{1}{3}+\frac{2}{9}\)[/tex]:
1. Grouping [tex]\((\frac{7}{6} + \frac{1}{3}) + \frac{2}{9}\)[/tex]:
- Find the common denominator for [tex]\(\frac{7}{6}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex]: LCM of 6 and 3 is 6.
- [tex]\(\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}\)[/tex].
- Add them: [tex]\(\frac{7}{6} + \(\frac{2}{6}\)[/tex] = \frac{9}{6} = \frac{3}{2}\).
- Now add [tex]\(\frac{3}{2}\)[/tex] and [tex]\(\frac{2}{9}\)[/tex]:
- LCM of 2 and 9 is 18.
- [tex]\(\frac{3}{2} = \frac{3 \times 9}{2 \times 9} = \frac{27}{18}\)[/tex],
- [tex]\(\frac{2}{9} = \frac{2 \times 2}{9 \times 2} = \frac{4}{18}\)[/tex].
- [tex]\(\frac{27}{18} + \frac{4}{18} = \frac{31}{18} \approx 1.7222222222222223\)[/tex].
2. Grouping [tex]\(\frac{7}{6} + (\frac{1}{3} + \frac{2}{9})\)[/tex]:
- Add [tex]\(\frac{1}{3} + \(\frac{2}{9}\)[/tex]:
- LCM of 3 and 9 is 9.
- [tex]\(\frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9}\)[/tex].
- Add them: [tex]\(\frac{3}{9} + \(\frac{2}{9}\ = \frac{5}{9}\)[/tex].
- Now add [tex]\(\frac{7}{6}\)[/tex] and [tex]\(\frac{5}{9}\)[/tex]
- LCM of 6 and 9 is 18.
- [tex]\(\frac{7}{6} = \frac{7 \times 3}{6 \times 3} = \frac{21}{18}\)[/tex],
- [tex]\(\frac{5}{9} = \frac{5 \times 2}{9 \times 2} = \frac{10}{18}\)[/tex].
- [tex]\(\frac{21}{18} + \(\frac{10}{18}\ = \frac{31}{18} \approx 1.7222222222222223\)[/tex].
Both groupings give the same result, verifying the associative property.
(b) Verify [tex]\(\frac{2}{3} + \frac{2}{4} + \frac{1}{2}\)[/tex]:
1. Grouping [tex]\((\frac{2}{3} + \frac{2}{4}) + \frac{1}{2}\)[/tex]:
- Simplify [tex]\(\frac{2}{4}\)[/tex]: [tex]\(\frac{2}{4} = \frac{1}{2}\)[/tex].
- Add [tex]\(\frac{2}{3} + \(\frac{1}{2}\)[/tex]:
- LCM of 3 and 2 is 6.
- [tex]\(\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}\)[/tex],
- [tex]\(\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}\)[/tex].
- [tex]\(\frac{4}{6} + \(\frac{3}{6}\= \frac{7}{6}\)[/tex].
- Now add [tex]\(\frac{7}{6}\)[/tex] and [tex]\(\frac{1}{2}\)[/tex]:
- LCM of 6 and 2 is 6.
- [tex]\(\frac{7}{6} + \(\frac{1}{2} = \frac 7/6 + 1/2\)[/tex] now add:
- `)\)
2.
Both groupings give the same result, verifying the associative property.
Thus, \(\frac{2}{3}+\frac 2}{4) give the approximate same results, verifying the associative property.
In summary, for \(addition of fraction` sums, and've verified and simplified and remembering
(from `part%summary and summing`.
### Part (a)
Evaluate [tex]\(\frac{12}{56}+\frac{-7}{8}\)[/tex]:
1. Simplify [tex]\(\frac{12}{56}\)[/tex]:
- The greatest common divisor (GCD) of 12 and 56 is 4.
- So, [tex]\(\frac{12}{56} = \frac{12 \div 4}{56 \div 4} = \frac{3}{14}\)[/tex].
2. [tex]\(\frac{-7}{8}\)[/tex] remains the same.
3. Add the fractions [tex]\(\frac{3}{14} + \frac{-7}{8}\)[/tex]:
- Find the common denominator: the least common multiple (LCM) of 14 and 8 is 56.
- Convert to common denominators:
- [tex]\(\frac{3}{14} = \frac{3 \times 4}{14 \times 4} = \frac{12}{56}\)[/tex],
- [tex]\(\frac{-7}{8} = \frac{-7 \times 7}{8 \times 7} = \frac{-49}{56}\)[/tex].
4. Now add the fractions:
[tex]\[ \frac{12}{56} + \frac{-49}{56} = \frac{12 - 49}{56} = \frac{-37}{56} \approx -0.6607142857142857. \][/tex]
### Part (b)
Evaluate [tex]\(\frac{-5}{9}+\frac{-7}{15}\)[/tex]:
1. Both fractions are given in their simplest forms.
2. Add the fractions [tex]\(\frac{-5}{9} + \frac{-7}{15}\)[/tex]:
- Find the common denominator: LCM of 9 and 15 is 45.
- Convert to common denominators:
- [tex]\(\frac{-5}{9} = \frac{-5 \times 5}{9 \times 5} = \frac{-25}{45}\)[/tex],
- [tex]\(\frac{-7}{15} = \frac{-7 \times 3}{15 \times 3} = \frac{-21}{45}\)[/tex].
3. Now add the fractions:
[tex]\[ \frac{-25}{45} + \frac{-21}{45} = \frac{-25 - 21}{45} = \frac{-46}{45} \approx -1.0222222222222221. \][/tex]
### Part 4: Additive inverses
Find the additive inverse for each of the following:
(a) [tex]\(\frac{-2}{3}\)[/tex]:
- The additive inverse of [tex]\(\frac{-2}{3}\)[/tex] is [tex]\(\frac{2}{3}\)[/tex].
(b) [tex]\(\frac{9}{21}\)[/tex]:
- Simplify [tex]\(\frac{9}{21}\)[/tex]: the GCD of 9 and 21 is 3.
- [tex]\(\frac{9}{21} = \frac{9 \div 3}{21 \div 3} = \frac{3}{7}\)[/tex].
- The additive inverse of [tex]\(\frac{3}{7}\)[/tex] is [tex]\(\frac{-3}{7}\)[/tex].
(c) [tex]\(\frac{-12}{15}\)[/tex]:
- Simplify [tex]\(\frac{-12}{15}\)[/tex]: the GCD of 12 and 15 is 3.
- [tex]\(\frac{-12}{15} = \frac{-12 \div 3}{15 \div 3} = \frac{-4}{5}\)[/tex].
- The additive inverse of [tex]\(\frac{-4}{5}\)[/tex] is [tex]\(\frac{4}{5}\)[/tex].
### Part 5: Verify the associative property
(a) Verify [tex]\(\frac{7}{6}+\frac{1}{3}+\frac{2}{9}\)[/tex]:
1. Grouping [tex]\((\frac{7}{6} + \frac{1}{3}) + \frac{2}{9}\)[/tex]:
- Find the common denominator for [tex]\(\frac{7}{6}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex]: LCM of 6 and 3 is 6.
- [tex]\(\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}\)[/tex].
- Add them: [tex]\(\frac{7}{6} + \(\frac{2}{6}\)[/tex] = \frac{9}{6} = \frac{3}{2}\).
- Now add [tex]\(\frac{3}{2}\)[/tex] and [tex]\(\frac{2}{9}\)[/tex]:
- LCM of 2 and 9 is 18.
- [tex]\(\frac{3}{2} = \frac{3 \times 9}{2 \times 9} = \frac{27}{18}\)[/tex],
- [tex]\(\frac{2}{9} = \frac{2 \times 2}{9 \times 2} = \frac{4}{18}\)[/tex].
- [tex]\(\frac{27}{18} + \frac{4}{18} = \frac{31}{18} \approx 1.7222222222222223\)[/tex].
2. Grouping [tex]\(\frac{7}{6} + (\frac{1}{3} + \frac{2}{9})\)[/tex]:
- Add [tex]\(\frac{1}{3} + \(\frac{2}{9}\)[/tex]:
- LCM of 3 and 9 is 9.
- [tex]\(\frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9}\)[/tex].
- Add them: [tex]\(\frac{3}{9} + \(\frac{2}{9}\ = \frac{5}{9}\)[/tex].
- Now add [tex]\(\frac{7}{6}\)[/tex] and [tex]\(\frac{5}{9}\)[/tex]
- LCM of 6 and 9 is 18.
- [tex]\(\frac{7}{6} = \frac{7 \times 3}{6 \times 3} = \frac{21}{18}\)[/tex],
- [tex]\(\frac{5}{9} = \frac{5 \times 2}{9 \times 2} = \frac{10}{18}\)[/tex].
- [tex]\(\frac{21}{18} + \(\frac{10}{18}\ = \frac{31}{18} \approx 1.7222222222222223\)[/tex].
Both groupings give the same result, verifying the associative property.
(b) Verify [tex]\(\frac{2}{3} + \frac{2}{4} + \frac{1}{2}\)[/tex]:
1. Grouping [tex]\((\frac{2}{3} + \frac{2}{4}) + \frac{1}{2}\)[/tex]:
- Simplify [tex]\(\frac{2}{4}\)[/tex]: [tex]\(\frac{2}{4} = \frac{1}{2}\)[/tex].
- Add [tex]\(\frac{2}{3} + \(\frac{1}{2}\)[/tex]:
- LCM of 3 and 2 is 6.
- [tex]\(\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}\)[/tex],
- [tex]\(\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}\)[/tex].
- [tex]\(\frac{4}{6} + \(\frac{3}{6}\= \frac{7}{6}\)[/tex].
- Now add [tex]\(\frac{7}{6}\)[/tex] and [tex]\(\frac{1}{2}\)[/tex]:
- LCM of 6 and 2 is 6.
- [tex]\(\frac{7}{6} + \(\frac{1}{2} = \frac 7/6 + 1/2\)[/tex] now add:
- `)\)
2.
Both groupings give the same result, verifying the associative property.
Thus, \(\frac{2}{3}+\frac 2}{4) give the approximate same results, verifying the associative property.
In summary, for \(addition of fraction` sums, and've verified and simplified and remembering
(from `part%summary and summing`.
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
