Sure! Let's solve each of these fractional problems step-by-step.
### First Problem: Multiplication of Fractions
We need to calculate [tex]\(\frac{5}{6} \times \frac{2}{9}\)[/tex].
Step 1: Multiply the numerators.
[tex]\[ 5 \times 2 = 10 \][/tex]
Step 2: Multiply the denominators.
[tex]\[ 6 \times 9 = 54 \][/tex]
Step 3: Combine the results.
[tex]\[ \frac{10}{54} \][/tex]
Step 4: Simplify the fraction if possible. The greatest common divisor (GCD) of 10 and 54 is 2.
[tex]\[ \frac{10 \div 2}{54 \div 2} = \frac{5}{27} \][/tex]
So, [tex]\(\frac{5}{6} \times \frac{2}{9} = \frac{5}{27}\)[/tex].
When we convert [tex]\(\frac{5}{27}\)[/tex] to a decimal, we get:
[tex]\[ \frac{5}{27} \approx 0.18518518518518517 \][/tex]
### Second Problem: Division of Fractions
We need to calculate [tex]\(\frac{9}{10} \div \frac{3}{6}\)[/tex].
Step 1: Find the reciprocal of the second fraction (since division by a fraction is the same as multiplying by its reciprocal).
[tex]\[ \frac{3}{6} \rightarrow \frac{6}{3} \][/tex]
Step 2: Multiply the first fraction by the reciprocal of the second fraction.
[tex]\[ \frac{9}{10} \times \frac{6}{3} \][/tex]
Step 3: Multiply the numerators.
[tex]\[ 9 \times 6 = 54 \][/tex]
Step 4: Multiply the denominators.
[tex]\[ 10 \times 3 = 30 \][/tex]
Step 5: Combine the results.
[tex]\[ \frac{54}{30} \][/tex]
Step 6: Simplify the fraction if possible. The GCD of 54 and 30 is 6.
[tex]\[ \frac{54 \div 6}{30 \div 6} = \frac{9}{5} \][/tex]
So, [tex]\(\frac{9}{10} \div \frac{3}{6} = \frac{9}{5}\)[/tex].
When we convert [tex]\(\frac{9}{5}\)[/tex] to a decimal, we get:
[tex]\[ \frac{9}{5} = 1.8 \][/tex]
### Summary
1. [tex]\(\frac{5}{6} \times \frac{2}{9} = \frac{5}{27} \approx 0.18518518518518517\)[/tex]
2. [tex]\(\frac{9}{10} \div \frac{3}{6} = \frac{9}{5} = 1.8\)[/tex]
Thus, the results are:
[tex]\[
\left(0.18518518518518517, 1.8\right)
\][/tex]
