Answer :

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]