Answer :
Sure, let’s go through each part of the question step-by-step.
### Part 1: Adding Fractions
#### (e) [tex]\(\frac{1}{3} + \frac{3}{4} + \frac{3}{6}\)[/tex]
To add these fractions, we need to find a common denominator.
1. Find the least common denominator (LCD):
- For [tex]\(\frac{1}{3}\)[/tex], [tex]\(\frac{3}{4}\)[/tex], and [tex]\(\frac{3}{6}\)[/tex], the least common multiple of [tex]\(3, 4,\)[/tex] and [tex]\(6\)[/tex] is [tex]\(12\)[/tex].
2. Convert each fraction:
- [tex]\(\frac{1}{3} = \frac{4}{12}\)[/tex]
- [tex]\(\frac{3}{4} = \frac{9}{12}\)[/tex]
- [tex]\(\frac{3}{6} = \frac{1}{2} = \frac{6}{12}\)[/tex]
3. Add the numerators:
- [tex]\(\frac{4}{12} + \frac{9}{12} + \frac{6}{12} = \frac{4 + 9 + 6}{12} = \frac{19}{12}\)[/tex]
Thus, [tex]\(\frac{1}{3} + \frac{3}{4} + \frac{3}{6} = \frac{19}{12}\)[/tex].
#### (f) [tex]\(\frac{1}{8} + \frac{5}{12}\)[/tex]
To add these fractions, we also need to find a common denominator.
1. Find the least common denominator (LCD):
- For [tex]\(\frac{1}{8}\)[/tex] and [tex]\(\frac{5}{12}\)[/tex], the least common multiple of [tex]\(8\)[/tex] and [tex]\(12\)[/tex] is [tex]\(24\)[/tex].
2. Convert each fraction:
- [tex]\(\frac{1}{8} = \frac{3}{24}\)[/tex]
- [tex]\(\frac{5}{12} = \frac{10}{24}\)[/tex]
3. Add the numerators:
- [tex]\(\frac{3}{24} + \frac{10}{24} = \frac{3 + 10}{24} = \frac{13}{24}\)[/tex]
Thus, [tex]\(\frac{1}{8} + \frac{5}{12} = \frac{13}{24}\)[/tex].
### Part 2: Subtracting Fractions
#### (a) [tex]\(\frac{1}{3} - \frac{1}{6}\)[/tex]
To subtract these fractions, we need a common denominator.
1. Find the least common denominator (LCD):
- For [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex], the least common multiple of [tex]\(3\)[/tex] and [tex]\(6\)[/tex] is [tex]\(6\)[/tex].
2. Convert each fraction:
- [tex]\(\frac{1}{3} = \frac{2}{6}\)[/tex]
- [tex]\(\frac{1}{6} = \frac{1}{6}\)[/tex]
3. Subtract the numerators:
- [tex]\(\frac{2}{6} - \frac{1}{6} = \frac{2 - 1}{6} = \frac{1}{6}\)[/tex]
Thus, [tex]\(\frac{1}{3} - \frac{1}{6} = \frac{1}{6}\)[/tex].
#### (b) [tex]\(\frac{7}{8} - \frac{3}{4}\)[/tex]
To subtract these fractions, we need a common denominator.
1. Find the least common denominator (LCD):
- For [tex]\(\frac{7}{8}\)[/tex] and [tex]\(\frac{3}{4}\)[/tex], the least common multiple of [tex]\(8\)[/tex] and [tex]\(4\)[/tex] is [tex]\(8\)[/tex].
2. Convert each fraction:
- [tex]\(\frac{7}{8} = \frac{7}{8}\)[/tex]
- [tex]\(\frac{3}{4} = \frac{6}{8}\)[/tex]
3. Subtract the numerators:
- [tex]\(\frac{7}{8} - \frac{6}{8} = \frac{7 - 6}{8} = \frac{1}{8}\)[/tex]
Thus, [tex]\(\frac{7}{8} - \frac{3}{4} = \frac{1}{8}\)[/tex].
### Part 3: Whole Numbers and Fractions
#### (e) [tex]\(1 - \frac{7}{11}\)[/tex]
1. Convert the whole number to a fraction:
- [tex]\(1 = \frac{11}{11}\)[/tex]
2. Subtract the numerators:
- [tex]\(\frac{11}{11} - \frac{7}{11} = \frac{11 - 7}{11} = \frac{4}{11}\)[/tex]
Thus, [tex]\(1 - \frac{7}{11} = \frac{4}{11}\)[/tex].
#### (f) [tex]\(2 - \frac{3}{8}\)[/tex]
1. Convert the whole number to a fraction:
- [tex]\(2 = \frac{16}{8}\)[/tex]
2. Subtract the numerators:
- [tex]\(\frac{16}{8} - \frac{3}{8} = \frac{16 - 3}{8} = \frac{13}{8}\)[/tex]
Thus, [tex]\(2 - \frac{3}{8} = \frac{13}{8}\)[/tex].
### Summary:
- [tex]\(\frac{1}{3} + \frac{3}{4} + \frac{3}{6} = \frac{19}{12}\)[/tex]
- [tex]\(\frac{1}{8} + \frac{5}{12} = \frac{13}{24}\)[/tex]
- [tex]\(\frac{1}{3} - \frac{1}{6} = \frac{1}{6}\)[/tex]
- [tex]\(\frac{7}{8} - \frac{3}{4} = \frac{1}{8}\)[/tex]
- [tex]\(1 - \frac{7}{11} = \frac{4}{11}\)[/tex]
- [tex]\(2 - \frac{3}{8} = \frac{13}{8}\)[/tex]
### Part 1: Adding Fractions
#### (e) [tex]\(\frac{1}{3} + \frac{3}{4} + \frac{3}{6}\)[/tex]
To add these fractions, we need to find a common denominator.
1. Find the least common denominator (LCD):
- For [tex]\(\frac{1}{3}\)[/tex], [tex]\(\frac{3}{4}\)[/tex], and [tex]\(\frac{3}{6}\)[/tex], the least common multiple of [tex]\(3, 4,\)[/tex] and [tex]\(6\)[/tex] is [tex]\(12\)[/tex].
2. Convert each fraction:
- [tex]\(\frac{1}{3} = \frac{4}{12}\)[/tex]
- [tex]\(\frac{3}{4} = \frac{9}{12}\)[/tex]
- [tex]\(\frac{3}{6} = \frac{1}{2} = \frac{6}{12}\)[/tex]
3. Add the numerators:
- [tex]\(\frac{4}{12} + \frac{9}{12} + \frac{6}{12} = \frac{4 + 9 + 6}{12} = \frac{19}{12}\)[/tex]
Thus, [tex]\(\frac{1}{3} + \frac{3}{4} + \frac{3}{6} = \frac{19}{12}\)[/tex].
#### (f) [tex]\(\frac{1}{8} + \frac{5}{12}\)[/tex]
To add these fractions, we also need to find a common denominator.
1. Find the least common denominator (LCD):
- For [tex]\(\frac{1}{8}\)[/tex] and [tex]\(\frac{5}{12}\)[/tex], the least common multiple of [tex]\(8\)[/tex] and [tex]\(12\)[/tex] is [tex]\(24\)[/tex].
2. Convert each fraction:
- [tex]\(\frac{1}{8} = \frac{3}{24}\)[/tex]
- [tex]\(\frac{5}{12} = \frac{10}{24}\)[/tex]
3. Add the numerators:
- [tex]\(\frac{3}{24} + \frac{10}{24} = \frac{3 + 10}{24} = \frac{13}{24}\)[/tex]
Thus, [tex]\(\frac{1}{8} + \frac{5}{12} = \frac{13}{24}\)[/tex].
### Part 2: Subtracting Fractions
#### (a) [tex]\(\frac{1}{3} - \frac{1}{6}\)[/tex]
To subtract these fractions, we need a common denominator.
1. Find the least common denominator (LCD):
- For [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex], the least common multiple of [tex]\(3\)[/tex] and [tex]\(6\)[/tex] is [tex]\(6\)[/tex].
2. Convert each fraction:
- [tex]\(\frac{1}{3} = \frac{2}{6}\)[/tex]
- [tex]\(\frac{1}{6} = \frac{1}{6}\)[/tex]
3. Subtract the numerators:
- [tex]\(\frac{2}{6} - \frac{1}{6} = \frac{2 - 1}{6} = \frac{1}{6}\)[/tex]
Thus, [tex]\(\frac{1}{3} - \frac{1}{6} = \frac{1}{6}\)[/tex].
#### (b) [tex]\(\frac{7}{8} - \frac{3}{4}\)[/tex]
To subtract these fractions, we need a common denominator.
1. Find the least common denominator (LCD):
- For [tex]\(\frac{7}{8}\)[/tex] and [tex]\(\frac{3}{4}\)[/tex], the least common multiple of [tex]\(8\)[/tex] and [tex]\(4\)[/tex] is [tex]\(8\)[/tex].
2. Convert each fraction:
- [tex]\(\frac{7}{8} = \frac{7}{8}\)[/tex]
- [tex]\(\frac{3}{4} = \frac{6}{8}\)[/tex]
3. Subtract the numerators:
- [tex]\(\frac{7}{8} - \frac{6}{8} = \frac{7 - 6}{8} = \frac{1}{8}\)[/tex]
Thus, [tex]\(\frac{7}{8} - \frac{3}{4} = \frac{1}{8}\)[/tex].
### Part 3: Whole Numbers and Fractions
#### (e) [tex]\(1 - \frac{7}{11}\)[/tex]
1. Convert the whole number to a fraction:
- [tex]\(1 = \frac{11}{11}\)[/tex]
2. Subtract the numerators:
- [tex]\(\frac{11}{11} - \frac{7}{11} = \frac{11 - 7}{11} = \frac{4}{11}\)[/tex]
Thus, [tex]\(1 - \frac{7}{11} = \frac{4}{11}\)[/tex].
#### (f) [tex]\(2 - \frac{3}{8}\)[/tex]
1. Convert the whole number to a fraction:
- [tex]\(2 = \frac{16}{8}\)[/tex]
2. Subtract the numerators:
- [tex]\(\frac{16}{8} - \frac{3}{8} = \frac{16 - 3}{8} = \frac{13}{8}\)[/tex]
Thus, [tex]\(2 - \frac{3}{8} = \frac{13}{8}\)[/tex].
### Summary:
- [tex]\(\frac{1}{3} + \frac{3}{4} + \frac{3}{6} = \frac{19}{12}\)[/tex]
- [tex]\(\frac{1}{8} + \frac{5}{12} = \frac{13}{24}\)[/tex]
- [tex]\(\frac{1}{3} - \frac{1}{6} = \frac{1}{6}\)[/tex]
- [tex]\(\frac{7}{8} - \frac{3}{4} = \frac{1}{8}\)[/tex]
- [tex]\(1 - \frac{7}{11} = \frac{4}{11}\)[/tex]
- [tex]\(2 - \frac{3}{8} = \frac{13}{8}\)[/tex]
