Answer :
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`.
