1 5/11:8/17 7 5/7:4 2/7 5 1/3:6 2/3 27 6/7:3 56:4 4/5

To solve the problems, we will convert each mixed number into an improper fraction and then simplify the ratios.

1. \(1 \frac{5}{11} : \frac{8}{17}\)

Step 1: Convert mixed number to improper fraction \[ 1 \frac{5}{11} = \frac{11 \cdot 1 + 5}{11} = \frac{16}{11} \]

Step 2: Set up the ratio \[ \frac{16/11}{8/17} \]

Step 3: Multiply by the reciprocal \[ \frac{16}{11} \div \frac{8}{17} = \frac{16}{11} \cdot \frac{17}{8} = \frac{16 \cdot 17}{11 \cdot 8} = \frac{272}{88} \]

Step 4: Simplify \[ \frac{272 \div 88}{88 \div 88} = \frac{34}{11} = 3 \frac{1}{11} \text{ (4:1)}\]

Thus, the simplified ratio is 34:11.

2. \(7 \frac{5}{7} : 4 \frac{2}{7}\)

Step 1: Convert to improper fractions \[ 7 \frac{5}{7} = \frac{7 \cdot 7 + 5}{7} = \frac{54}{7} \] \[ 4 \frac{2}{7} = \frac{4 \cdot 7 + 2}{7} = \frac{30}{7} \]

Step 2: Set up the ratio \[ \frac{54/7}{30/7} = \frac{54}{30} = \frac{27}{15} \]

Step 3: Simplify \[ \frac{27 \div 3}{15 \div 3} = \frac{9}{5} \text{ (9:5)}\]

Thus, the simplified ratio is 27:15.

3. \(5 \frac{1}{3} : 6 \frac{2}{3}\)

Step 1: Convert to improper fractions \[ 5 \frac{1}{3} = \frac{5 \cdot 3 + 1}{3} = \frac{16}{3} \] \[ 6 \frac{2}{3} = \frac{6 \cdot 3 + 2}{3} = \frac{20}{3} \]

Step 2: Set up the ratio \[ \frac{16/3}{20/3} = \frac{16}{20} \]

Step 3: Simplify \[ \frac{16 \div 4}{20 \div 4} = \frac{4}{5} \text{ (4:5)}\]

Thus, the simplified ratio is 16:20.

4. \(27 \frac{6}{7} : 3\)

Step 1: Convert to improper fraction \[ 27 \frac{6}{7} = \frac{27 \cdot 7 + 6}{7} = \frac{195}{7} \]

Step 2: Set up the ratio \[ \frac{195/7}{3} = \frac{195}{21} \]

Step 3: Simplify \[ \frac{195 \div 3}{21 \div 3} = \frac{65}{7} \text{ (65:7)}\]

Thus, the simplified ratio is 65:7.

5. \(56 : 4 \frac{4}{5}\)

Step 1: Convert to improper fraction \[ 4 \frac{4}{5} = \frac{4 \cdot 5 + 4}{5} = \frac{24}{5} \]

Step 2: Set up the ratio \[ \frac{56}{24/5} = 56 \cdot \frac{5}{24} = \frac{280}{24} \]

Step 3: Simplify \[ \frac{280 \div 8}{24 \div 8} = \frac{35}{3} \text{ (35:3)}\]

Thus, the simplified ratio is 280:24.

Summary of Simplified Ratios:

1. 34:11
2. 27:15
3. 4:5
4. 65:7
5. 35:3