Sure, let's solve the given problem step-by-step.
Given:
[tex]\[ a = 0.\overline{6} \][/tex]
[tex]\[ b = 0.75 \][/tex]
### Step-by-Step Solution:
1. Convert the repeating decimal [tex]\( a \)[/tex] into a fraction:
To convert [tex]\( 0.\overline{6} \)[/tex] to a fraction. Let [tex]\( x = 0.\overline{6} \)[/tex].
[tex]\[ x = 0.666\ldots \][/tex]
Multiplying both sides by 10,
[tex]\[ 10x = 6.666\ldots \][/tex]
Subtracting the original equation from this one,
[tex]\[ 10x - x = 6.666\ldots - 0.666\ldots \][/tex]
[tex]\[ 9x = 6 \][/tex]
[tex]\[ x = \frac{6}{9} = \frac{2}{3} \][/tex]
So,
[tex]\[ a = \frac{2}{3} \][/tex]
2. Convert [tex]\( b \)[/tex] into a fraction:
Since [tex]\( b = 0.75 \)[/tex],
[tex]\[ b = \frac{75}{100} = \frac{3}{4} \][/tex]
3. Calculate [tex]\( a \cdot b \)[/tex]:
[tex]\[ a \cdot b = \frac{2}{3} \cdot \frac{3}{4} \][/tex]
Multiplying the fractions,
[tex]\[ a \cdot b = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2} \][/tex]
Hence,
[tex]\[ a \cdot b = 0.5 \][/tex]
4. Calculate [tex]\( \frac{a}{b} \)[/tex]:
[tex]\[ \frac{a}{b} = \frac{\frac{2}{3}}{\frac{3}{4}} \][/tex]
When dividing by a fraction, we multiply by its reciprocal,
[tex]\[ \frac{a}{b} = \frac{2}{3} \cdot \frac{4}{3} = \frac{2 \times 4}{3 \times 3} = \frac{8}{9} \][/tex]
Hence,
[tex]\[ \frac{a}{b} = 0.8888888888888888 \][/tex]
Putting these results in the boxes, we get:
[tex]\[ a \cdot b = 0.5 \][/tex]
[tex]\[ \frac{a}{b} = 0.8888888888888888 \][/tex]
