Sure, let's work through these problems step-by-step.
### 12) [tex]\(\frac{2}{7} + \frac{2}{3}\)[/tex]
1. Find a common denominator: The denominators here are 7 and 3. The common denominator would be their product (since they are co-prime), which is [tex]\(21\)[/tex].
2. Convert each fraction to this common denominator:
- For [tex]\(\frac{2}{7}\)[/tex]:
[tex]\[
\frac{2}{7} = \frac{2 \times 3}{7 \times 3} = \frac{6}{21}
\][/tex]
- For [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
\frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21}
\][/tex]
3. Add the fractions:
[tex]\[
\frac{6}{21} + \frac{14}{21} = \frac{6 + 14}{21} = \frac{20}{21}
\][/tex]
4. Simplified fraction: [tex]\(\frac{20}{21}\)[/tex] is already in its simplest form.
Therefore,
[tex]\[
\frac{2}{7} + \frac{2}{3} = \frac{20}{21} \approx 0.9524
\][/tex]
### 14) [tex]\(\frac{1}{10} \cdot \frac{5}{6}\)[/tex]
1. Multiply the numerators together:
[tex]\[
1 \times 5 = 5
\][/tex]
2. Multiply the denominators together:
[tex]\[
10 \times 6 = 60
\][/tex]
3. Form the new fraction:
[tex]\[
\frac{5}{60}
\][/tex]
4. Simplify the fraction: [tex]\(\frac{5}{60}\)[/tex]
- Both numerator (5) and denominator (60) can be divided by their greatest common divisor, which is 5:
[tex]\[
\frac{5}{60} = \frac{5 \div 5}{60 \div 5} = \frac{1}{12}
\][/tex]
Therefore,
[tex]\[
\frac{1}{10} \cdot \frac{5}{6} = \frac{1}{12} \approx 0.0833
\][/tex]
So, the solutions to the given questions are:
12) [tex]\(\frac{2}{7} + \frac{2}{3} = \frac{20}{21} \approx 0.9524\)[/tex]
14) [tex]\(\frac{1}{10} \cdot \frac{5}{6} = \frac{1}{12} \approx 0.0833\)[/tex]
