Simplify
(i)`2/3+5/6-1/9`
(ii)`8-4 1/2-2 1/4`
(iii)`8 5/6- 3 3/8+ 1 7/12`

Answer

Let's simplify each part of the question step by step.

### Part (i): Simplify \( \frac{2}{3} + \frac{5}{6} - \frac{1}{9} \)

**Step 1: Find the LCM of the denominators.**
The denominators are 3, 6, and 9. The LCM of these numbers is 18.

**Step 2: Convert each fraction to have the same denominator.**
- For \( \frac{2}{3} \):
\[ \frac{2}{3} = \frac{2 \times 6}{3 \times 6} = \frac{12}{18} \]
- For \( \frac{5}{6} \):
\[ \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18} \]
- For \( \frac{1}{9} \):
\[ \frac{1}{9} = \frac{1 \times 2}{9 \times 2} = \frac{2}{18} \]

**Step 3: Substitute back into the equation.**
\[ \frac{12}{18} + \frac{15}{18} - \frac{2}{18} \]

**Step 4: Combine the fractions.**
\[ \frac{12 + 15 - 2}{18} = \frac{25}{18} \]

**Step 5: Convert to a mixed number if necessary.**
\[ \frac{25}{18} = 1 \frac{7}{18} \]

### Final Answer for Part (i):
\[ 1 \frac{7}{18} \]

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### Part (ii): Simplify \( 8 - 4 \frac{1}{2} - 2 \frac{1}{4} \)

**Step 1: Convert mixed numbers to improper fractions.**
- For \( 4 \frac{1}{2} \):
\[ 4 \frac{1}{2} = \frac{9}{2} \]
- For \( 2 \frac{1}{4} \):
\[ 2 \frac{1}{4} = \frac{9}{4} \]

**Step 2: Find a common denominator.**
The denominators are 1, 2, and 4. The LCM is 4.

**Step 3: Convert all terms to have the same denominator.**
- For 8:
\[ 8 = \frac{32}{4} \]
- For \( 4 \frac{1}{2} \):
\[ \frac{9}{2} = \frac{18}{4} \]
- For \( 2 \frac{1}{4} \):
\[ \frac{9}{4} \]

**Step 4: Substitute back into the equation.**
\[ \frac{32}{4} - \frac{18}{4} - \frac{9}{4} \]

**Step 5: Combine the fractions.**
\[ \frac{32 - 18 - 9}{4} = \frac{5}{4} \]

**Step 6: Convert to a mixed number if necessary.**
\[ \frac{5}{4} = 1 \frac{1}{4} \]

### Final Answer for Part (ii):
\[ 1 \frac{1}{4} \]

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### Part (iii): Simplify \( 8 \frac{5}{6} - 3 \frac{3}{8} + 1 \frac{7}{12} \)

**Step 1: Convert mixed numbers to improper fractions.**
- For \( 8 \frac{5}{6} \):
\[ 8 \frac{5}{6} = \frac{53}{6} \]
- For \( 3 \frac{3}{8} \):
\[ 3 \frac{3}{8} = \frac{27}{8} \]
- For \( 1 \frac{7}{12} \):
\[ 1 \frac{7}{12} = \frac{19}{12} \]

**Step 2: Find a common denominator.**
The denominators are 6, 8, and 12. The LCM is 24.

**Step 3: Convert all terms to have the same denominator.**
- For \( \frac{53}{6} \):
\[ \frac{53}{6} = \frac{53 \times 4}{6 \times 4} = \frac{212}{24} \]
- For \( \frac{27}{8} \):
\[ \frac{27}{8} = \frac{27 \times 3}{8 \times 3} = \frac{81}{24} \]
- For \( \frac{19}{12} \):
\[ \frac{19}{12} = \frac{19 \times 2}{12 \times 2} = \frac{38}{24} \]

**Step 4: Substitute back into the equation.**
\[ \frac{212}{24} - \frac{81}{24} + \frac{38}{24} \]

**Step 5: Combine the fractions.**
\[ \frac{212 - 81 + 38}{24} = \frac{169}{24} \]

**Step 6: Convert to a mixed number if necessary.**
\[ \frac{169}{24} = 7 \frac{1}{24} \]

### Final Answer for Part (iii):
\[ 7 \frac{1}{24} \]

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