Express the given ratios in simplest form:
a. 49:77 b. 55:99

Answer

To express the given ratios in simplest form, we will follow these steps:

### a. Simplifying the ratio 49:77

1. **Write the ratio as a fraction**:
\[ \frac{49}{77} \]

2. **Find the greatest common divisor (GCD)** of 49 and 77.
- The factors of 49 are: 1, 7, 49
- The factors of 77 are: 1, 7, 11, 77
- The common factor is 7.

3. **Divide both the numerator and the denominator by their GCD (7)**:
\[ \frac{49 \div 7}{77 \div 7} = \frac{7}{11} \]

4. **Final result**:
The simplest form of the ratio 49:77 is:
\[ 7:11 \]

### b. Simplifying the ratio 55:99

1. **Write the ratio as a fraction**:
\[ \frac{55}{99} \]

2. **Find the greatest common divisor (GCD)** of 55 and 99.
- The factors of 55 are: 1, 5, 11, 55
- The factors of 99 are: 1, 3, 9, 11, 33, 99
- The common factor is 11.

3. **Divide both the numerator and the denominator by their GCD (11)**:
\[ \frac{55 \div 11}{99 \div 11} = \frac{5}{9} \]

4. **Final result**:
The simplest form of the ratio 55:99 is:
\[ 5:9 \]

### Summary of Results:
- a. 49:77 = 7:11
- b. 55:99 = 5:9