1. Locate the rational numbers $3$, $4$, $\frac{4}{5}$, $0.4\overline{4}$, and $21\frac{3}{1}$ on the number line. - $3$ and $4$ are integers and can be located directly at points 3 and 4. - $\frac{4}{5} = 0.8$, so locate it between 0 and 1, closer to 1. - $0.4\overline{4}$ means $0.4444...$, which is $\frac{4}{9}$. - $21\frac{3}{1} = 21 + 3 = 24$, so locate at 24. 2. Represent each decimal as simplest fraction form. a. $2.6\overline{6}$ means $2.6666...$ - Let $x = 2.6\overline{6}$. - Multiply by 10: $10x = 26.6\overline{6}$. - Subtract original: $10x - x = 26.6\overline{6} - 2.6\overline{6} = 24$. - So, $9x = 24 \Rightarrow x = \frac{24}{9} = \frac{8}{3}$. b. $0.14\overline{14}$ means $0.141414...$ - Let $x = 0.14\overline{14}$. - Multiply by 100: $100x = 14.14\overline{14}$. - Subtract original: $100x - x = 14.14\overline{14} - 0.14\overline{14} = 14$. - So, $99x = 14 \Rightarrow x = \frac{14}{99}$. c. $0.716\overline{6}$ means $0.716666...$ - Let $x = 0.716\overline{6}$. - Multiply by 10: $10x = 7.1666...$ - Multiply by 1000: $1000x = 716.666...$ - Subtract: $1000x - 10x = 716.666... - 7.1666... = 709.5$ - So, $990x = 709.5 \Rightarrow x = \frac{709.5}{990} = \frac{1419}{1980}$. - Simplify by dividing numerator and denominator by 9: $\frac{1419/9}{1980/9} = \frac{157.666...}{220}$, but better to use a simpler approach: Alternate method: - Let $x = 0.716\overline{6}$. - Multiply by 10: $10x = 7.1666...$ - Multiply by 100: $100x = 71.6666...$ - Subtract: $100x - 10x = 71.6666... - 7.1666... = 64.5$ - So, $90x = 64.5 \Rightarrow x = \frac{64.5}{90} = \frac{129}{180} = \frac{43}{60}$. Therefore, $x = \frac{43}{60}$. d. $1.3212\overline{12}$ means $1.32121212...$ - Let $x = 1.3212\overline{12}$. - Multiply by 100: $100x = 132.121212...$ - Subtract original: $100x - x = 132.121212... - 1.321212... = 130.8$ - So, $99x = 130.8 \Rightarrow x = \frac{130.8}{99} = \frac{1308}{990}$. - Simplify numerator and denominator by 6: $\frac{218}{165}$. - Convert to mixed number: $1 + \frac{53}{165}$. e. $-0.53213\overline{13}$ means $-0.532131313...$ - Let $x = 0.53213\overline{13}$ (ignore negative for now). - Multiply by 1000: $1000x = 532.131313...$ - Multiply by 10: $10x = 5.32131313...$ - Subtract: $1000x - 10x = 532.131313... - 5.321313... = 526.81$ - So, $990x = 526.81 \Rightarrow x = \frac{526.81}{990} = \frac{52681}{99000}$. - Simplify fraction by dividing numerator and denominator by 11: $\frac{4799}{9000}$. - Add negative sign back: $x = -\frac{4799}{9000}$. Final answers: - a. $2.6\overline{6} = \frac{8}{3}$ - b. $0.14\overline{14} = \frac{14}{99}$ - c. $0.716\overline{6} = \frac{43}{60}$ - d. $1.3212\overline{12} = \frac{218}{165}$ - e. $-0.53213\overline{13} = -\frac{4799}{9000}$