Simplify the following :
(i) `3/11+5/11+9/11 " (ii) " 3""2/3+1""3/5+2""4/5`.

Answer

Let's simplify the given fractions step by step.

### Question (i): Simplify `3/11 + 5/11 + 9/11`

**Step 1: Identify the denominators.**
- The denominators of all the fractions are the same, which is 11.

**Hint:** When the denominators are the same, you can add the numerators directly.

**Step 2: Add the numerators.**
- Add the numerators: \(3 + 5 + 9 = 17\).

**Step 3: Write the sum over the common denominator.**
- The sum is \( \frac{17}{11} \).

**Step 4: Convert to a mixed number (if necessary).**
- To convert \( \frac{17}{11} \) to a mixed number, divide 17 by 11.
- \( 17 \div 11 = 1 \) remainder \( 6 \).
- Thus, \( \frac{17}{11} = 1 \frac{6}{11} \).

**Final Answer for (i):** \( \frac{17}{11} \) or \( 1 \frac{6}{11} \).

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### Question (ii): Simplify `3 2/3 + 1 3/5 + 2 4/5`

**Step 1: Convert mixed numbers to improper fractions.**
- For \( 3 \frac{2}{3} \):
- \( 3 \times 3 + 2 = 9 + 2 = 11 \)
- So, \( 3 \frac{2}{3} = \frac{11}{3} \).

- For \( 1 \frac{3}{5} \):
- \( 1 \times 5 + 3 = 5 + 3 = 8 \)
- So, \( 1 \frac{3}{5} = \frac{8}{5} \).

- For \( 2 \frac{4}{5} \):
- \( 2 \times 5 + 4 = 10 + 4 = 14 \)
- So, \( 2 \frac{4}{5} = \frac{14}{5} \).

**Hint:** To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator.

**Step 2: Add the whole numbers.**
- Add the whole numbers: \( 3 + 1 + 2 = 6 \).

**Step 3: Add the fractions.**
- Now, we need to add \( \frac{11}{3} + \frac{8}{5} + \frac{14}{5} \).
- First, combine \( \frac{8}{5} + \frac{14}{5} = \frac{22}{5} \).
- Now we need to add \( \frac{11}{3} + \frac{22}{5} \).

**Step 4: Find a common denominator.**
- The LCM of 3 and 5 is 15.
- Convert \( \frac{11}{3} \) to a denominator of 15:
- \( \frac{11 \times 5}{3 \times 5} = \frac{55}{15} \).
- Convert \( \frac{22}{5} \) to a denominator of 15:
- \( \frac{22 \times 3}{5 \times 3} = \frac{66}{15} \).

**Step 5: Add the fractions.**
- Now add \( \frac{55}{15} + \frac{66}{15} = \frac{121}{15} \).

**Step 6: Convert to a mixed number.**
- Divide \( 121 \div 15 = 8 \) remainder \( 1 \).
- So, \( \frac{121}{15} = 8 \frac{1}{15} \).

**Final Answer for (ii):** \( 8 \frac{1}{15} \).

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