Let's solve these problems step by step.
a. Given the fraction sequence [tex]\(\frac{2}{3}\)[/tex]:
1. [tex]\(\frac{2}{3} = \frac{4}{6}\)[/tex]
- Here, we can multiply both the numerator and the denominator by 2 to obtain 4 and 6, respectively.
2. [tex]\(\frac{2}{3} = \frac{4}{6} = \frac{8}{24}\)[/tex]
- For this fraction, we can multiply the numerator by 8 and the denominator by 8 to obtain 8 and 24, respectively.
3. [tex]\(\frac{2}{3} = \frac{4}{6} = \frac{8}{24} = \frac{20}{30}\)[/tex]
- For the last fraction part, to find the value in the denominator, we can set up the proportion: [tex]\(\frac{2}{3} = \frac{20}{x}\)[/tex]. Cross-multiplying, we get [tex]\(2x = 60\)[/tex]. Solving for [tex]\(x\)[/tex], we get [tex]\(x = 30\)[/tex].
So, the sequence for part a is:
[tex]\[ \frac{2}{3} = \frac{4}{6} = \frac{8}{24} = \frac{20}{30} \][/tex]
b. Given the fraction sequence [tex]\(\frac{3}{4}\)[/tex]:
1. [tex]\(\frac{3}{4} = \frac{6}{8}\)[/tex]
- Here, we can multiply both the numerator and the denominator by 2 to obtain 6 and 8, respectively.
2. [tex]\(\frac{3}{4} = \frac{6}{8} = \frac{21}{28}\)[/tex]
- For this fraction, we can set up the proportion: [tex]\(\frac{3}{4} = \frac{21}{x}\)[/tex]. Cross-multiplying, we get [tex]\(3x = 84\)[/tex]. Solving for [tex]\(x\)[/tex], we get [tex]\(x = 28\)[/tex].
3. [tex]\(\frac{3}{4} = \frac{6}{8} = \frac{21}{28} = \frac{24}{32}\)[/tex]
- For this last fraction part, we multiply the numerator by 8 to obtain 24 and the denominator by 8 to obtain 32, respectively.
So, the sequence for part b is:
[tex]\[ \frac{3}{4} = \frac{6}{8} = \frac{21}{28} = \frac{24}{32} \][/tex]
Thus, the completed sequences for both parts are:
a. [tex]\(\frac{2}{3} = \frac{\mathbf{4}}{6} = \frac{\mathbf{8}}{24} = \frac{20}{\mathbf{30}}\)[/tex]
b. [tex]\(\frac{3}{4} = \frac{\mathbf{6}}{8} = \frac{21}{\mathbf{28}} = \frac{\mathbf{24}}{32}\)[/tex]
