Simplify the following ratios a) \( 3: 6 \) b) \( 4: 12 \) c) \( 5: 15 \) d) 3.2036 e) 54842 g) \( 24: 72: 120 \) h) \( 77: 105: 84 \) i) \( 225: 3.5: 405 \)

Sure, let's simplify each of the given ratios step by step. ### a) \( 3 : 6 \) **Simplification:** - Find the greatest common divisor (GCD) of 3 and 6, which is **3**. - Divide both numbers by the GCD: \[ \frac{3}{3} : \frac{6}{3} = 1 : 2 \] **Simplified Ratio:** \[ 1 : 2 \] --- ### b) \( 4 : 12 \) **Simplification:** - GCD of 4 and 12 is **4**. - Divide both numbers by 4: \[ \frac{4}{4} : \frac{12}{4} = 1 : 3 \] **Simplified Ratio:** \[ 1 : 3 \] --- ### c) \( 5 : 15 \) **Simplification:** - GCD of 5 and 15 is **5**. - Divide both numbers by 5: \[ \frac{5}{5} : \frac{15}{5} = 1 : 3 \] **Simplified Ratio:** \[ 1 : 3 \] --- ### d) \( 3.2036 \) **Interpretation:** - Since only one number is provided, it can be interpreted as a ratio compared to 1. - Expressed as \( 3.2036 : 1 \). **Simplified Ratio:** \[ 8009 : 2500 \] *(This is achieved by converting the decimal to a fraction and simplifying.)* --- ### e) \( 54842 \) **Interpretation:** - With a single number, it's expressed as a ratio to 1. **Simplified Ratio:** \[ 54842 : 1 \] --- ### g) \( 24 : 72 : 120 \) **Simplification:** - GCD of 24, 72, and 120 is **24**. - Divide each number by 24: \[ \frac{24}{24} : \frac{72}{24} : \frac{120}{24} = 1 : 3 : 5 \] **Simplified Ratio:** \[ 1 : 3 : 5 \] --- ### h) \( 77 : 105 : 84 \) **Simplification:** - GCD of 77, 105, and 84 is **7**. - Divide each number by 7: \[ \frac{77}{7} : \frac{105}{7} : \frac{84}{7} = 11 : 15 : 12 \] **Simplified Ratio:** \[ 11 : 15 : 12 \] --- ### i) \( 225 : 3.5 : 405 \) **Simplification:** 1. **Eliminate the Decimal:** - Multiply all parts by 2 to remove the decimal: \[ 225 \times 2 : 3.5 \times 2 : 405 \times 2 = 450 : 7 : 810 \] 2. **Check for GCD:** - GCD of 450, 7, and 810 is **1**, so the ratio cannot be simplified further. **Simplified Ratio:** \[ 450 : 7 : 810 \] --- **Summary of Simplified Ratios:** a) \( 1 : 2 \) b) \( 1 : 3 \) c) \( 1 : 3 \) d) \( 8009 : 2500 \) e) \( 54842 : 1 \) g) \( 1 : 3 : 5 \) h) \( 11 : 15 : 12 \) i) \( 450 : 7 : 810 \)