1. **Problem statement:** Three fifths of those present agreed, and the remaining 12 disagreed. Find: a. The fraction of those who disagreed. b. The total number of people present. c. The number of people who agreed. d. The ratio of those who agreed to those who disagreed. e. If 50 people were present and 30 agreed, would the ratio change? Why or why not? 2. **Formula and rules:** - Total fraction of people present = 1. - Fraction agreed = $\frac{3}{5}$. - Fraction disagreed = $1 - \frac{3}{5} = \frac{2}{5}$. - Number disagreed = 12. - Use the relation: fraction disagreed $\times$ total number = number disagreed. 3. **Step-by-step solution:** a. Fraction disagreed: $$1 - \frac{3}{5} = \frac{5}{5} - \frac{3}{5} = \frac{2}{5}$$ So, fraction disagreed is $\frac{2}{5}$. b. Total number present: Let total number be $x$. Number disagreed = fraction disagreed $\times x$: $$\frac{2}{5} x = 12$$ Solve for $x$: $$x = \frac{12}{\frac{2}{5}} = 12 \times \frac{5}{2} = 30$$ c. Number agreed: Fraction agreed $\times$ total number: $$\frac{3}{5} \times 30 = 18$$ d. Ratio agreed to disagreed: $$18 : 12 = \frac{18}{12} = \frac{3}{2}$$ e. If 50 people present and 30 agreed: Number disagreed = $50 - 30 = 20$. Ratio agreed to disagreed: $$30 : 20 = \frac{30}{20} = \frac{3}{2}$$ The ratio remains $\frac{3}{2}$, so it does not change. **Final answers:** - a. $\frac{2}{5}$ - b. 30 people - c. 18 people - d. Ratio $3:2$ - e. No change in ratio; it remains $3:2$ because the proportion of agreed to disagreed is the same.