Simplify: `1'1/11 + 2'2/11 + 3'3/11 + 4/11 + 5/11 + 6/11`

Explanation

To simplify the expression `1'1/11 + 2'2/11 + 3'3/11 + 4/11 + 5/11 + 6/11`, we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
We start by converting the mixed numbers into improper fractions.

1. For `1'1/11`:
- Multiply the whole number (1) by the denominator (11) and add the numerator (1).
- \(1 \times 11 + 1 = 11 + 1 = 12\)
- So, \(1'1/11 = \frac{12}{11}\)

2. For `2'2/11`:
- \(2 \times 11 + 2 = 22 + 2 = 24\)
- So, \(2'2/11 = \frac{24}{11}\)

3. For `3'3/11`:
- \(3 \times 11 + 3 = 33 + 3 = 36\)
- So, \(3'3/11 = \frac{36}{11}\)

Now, we have:
\[ \frac{12}{11} + \frac{24}{11} + \frac{36}{11} + \frac{4}{11} + \frac{5}{11} + \frac{6}{11} \]

### Step 2: Combine All Fractions
Since all fractions have the same denominator (11), we can add the numerators directly.

\[ \frac{12 + 24 + 36 + 4 + 5 + 6}{11} \]

### Step 3: Calculate the Numerator
Now, we will add the numerators:
1. \(12 + 24 = 36\)
2. \(36 + 36 = 72\)
3. \(72 + 4 = 76\)
4. \(76 + 5 = 81\)
5. \(81 + 6 = 87\)

So, the total numerator is 87.

### Step 4: Write the Resulting Fraction
Now we can write the resulting fraction:
\[ \frac{87}{11} \]

### Step 5: Convert Improper Fraction to Mixed Number
To convert \(\frac{87}{11}\) into a mixed number:
1. Divide 87 by 11.
- \(87 \div 11 = 7\) remainder \(10\)
- So, we can write it as \(7\) with a remainder of \(10\).

Thus, \(\frac{87}{11} = 7 \frac{10}{11}\).

### Final Answer
The simplified expression is:
\[ 7 \frac{10}{11} \]

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