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Answer :
Sure! Let's go through the problem and solve it step-by-step.
### Simplifying the Fractions:
1. [tex]\( \frac{20}{50} \)[/tex]:
- To simplify this fraction, find the greatest common divisor (GCD) of 20 and 50, which is 10.
- Divide both the numerator and the denominator by 10:
[tex]\(\frac{20}{50} = \frac{20 \div 10}{50 \div 10} = \frac{2}{5}\)[/tex].
2. [tex]\( \frac{42}{27} \)[/tex]:
- Find the GCD of 42 and 27, which is 3.
- Divide both the numerator and the denominator by 3:
[tex]\(\frac{42}{27} = \frac{42 \div 3}{27 \div 3} = \frac{14}{9}\)[/tex].
3. [tex]\( \frac{14}{21} \)[/tex]:
- The GCD of 14 and 21 is 7.
- Divide both the numerator and the denominator by 7:
[tex]\(\frac{14}{21} = \frac{14 \div 7}{21 \div 7} = \frac{2}{3}\)[/tex].
4. [tex]\( \frac{12}{16} \)[/tex]:
- The GCD of 12 and 16 is 4.
- Divide both the numerator and the denominator by 4:
[tex]\(\frac{12}{16} = \frac{12 \div 4}{16 \div 4} = \frac{3}{4}\)[/tex].
5. [tex]\( \frac{16}{18} \)[/tex]:
- The GCD of 16 and 18 is 2.
- Divide both the numerator and the denominator by 2:
[tex]\(\frac{16}{18} = \frac{16 \div 2}{18 \div 2} = \frac{8}{9}\)[/tex].
### Amplifying the Fractions:
1. [tex]\( \frac{2}{9} \)[/tex]:
- To amplify this fraction, we multiply both the numerator and the denominator by 2:
[tex]\(\frac{2}{9} \times 2 = \frac{4}{18} = \frac{4}{9}\)[/tex].
2. [tex]\( \frac{4}{5} \)[/tex]:
- Multiply both the numerator and the denominator by 2:
[tex]\(\frac{4}{5} \times 2 = \frac{8}{10} = \frac{8}{5}\)[/tex].
3. [tex]\( \frac{1}{12} \)[/tex]:
- Multiply both the numerator and the denominator by 2:
[tex]\(\frac{1}{12} \times 2 = \frac{2}{24} = \frac{1}{6}\)[/tex].
4. [tex]\( \frac{7}{8} \)[/tex]:
- Multiply both the numerator and the denominator by 2:
[tex]\(\frac{7}{8} \times 2 = \frac{14}{16} = \frac{7}{4}\)[/tex].
5. [tex]\( \frac{2}{3} \)[/tex]:
- Multiply both the numerator and the denominator by 2:
[tex]\(\frac{2}{3} \times 2 = \frac{4}{6} = \frac{4}{3}\)[/tex].
Therefore, the simplified and amplified fractions are as follows:
- Simplified Results:
[tex]\(\frac{20}{50} = \frac{2}{5}\)[/tex],
[tex]\(\frac{42}{27} = \frac{14}{9}\)[/tex],
[tex]\(\frac{14}{21} = \frac{2}{3}\)[/tex],
[tex]\(\frac{12}{16} = \frac{3}{4}\)[/tex],
[tex]\(\frac{16}{18} = \frac{8}{9}\)[/tex].
- Amplified Results:
[tex]\(\frac{2}{9} = \frac{4}{9}\)[/tex],
[tex]\(\frac{4}{5} = \frac{8}{5}\)[/tex],
[tex]\(\frac{1}{12} = \frac{1}{6}\)[/tex],
[tex]\(\frac{7}{8} = \frac{7}{4}\)[/tex],
[tex]\(\frac{2}{3} = \frac{4}{3}\)[/tex].
### Simplifying the Fractions:
1. [tex]\( \frac{20}{50} \)[/tex]:
- To simplify this fraction, find the greatest common divisor (GCD) of 20 and 50, which is 10.
- Divide both the numerator and the denominator by 10:
[tex]\(\frac{20}{50} = \frac{20 \div 10}{50 \div 10} = \frac{2}{5}\)[/tex].
2. [tex]\( \frac{42}{27} \)[/tex]:
- Find the GCD of 42 and 27, which is 3.
- Divide both the numerator and the denominator by 3:
[tex]\(\frac{42}{27} = \frac{42 \div 3}{27 \div 3} = \frac{14}{9}\)[/tex].
3. [tex]\( \frac{14}{21} \)[/tex]:
- The GCD of 14 and 21 is 7.
- Divide both the numerator and the denominator by 7:
[tex]\(\frac{14}{21} = \frac{14 \div 7}{21 \div 7} = \frac{2}{3}\)[/tex].
4. [tex]\( \frac{12}{16} \)[/tex]:
- The GCD of 12 and 16 is 4.
- Divide both the numerator and the denominator by 4:
[tex]\(\frac{12}{16} = \frac{12 \div 4}{16 \div 4} = \frac{3}{4}\)[/tex].
5. [tex]\( \frac{16}{18} \)[/tex]:
- The GCD of 16 and 18 is 2.
- Divide both the numerator and the denominator by 2:
[tex]\(\frac{16}{18} = \frac{16 \div 2}{18 \div 2} = \frac{8}{9}\)[/tex].
### Amplifying the Fractions:
1. [tex]\( \frac{2}{9} \)[/tex]:
- To amplify this fraction, we multiply both the numerator and the denominator by 2:
[tex]\(\frac{2}{9} \times 2 = \frac{4}{18} = \frac{4}{9}\)[/tex].
2. [tex]\( \frac{4}{5} \)[/tex]:
- Multiply both the numerator and the denominator by 2:
[tex]\(\frac{4}{5} \times 2 = \frac{8}{10} = \frac{8}{5}\)[/tex].
3. [tex]\( \frac{1}{12} \)[/tex]:
- Multiply both the numerator and the denominator by 2:
[tex]\(\frac{1}{12} \times 2 = \frac{2}{24} = \frac{1}{6}\)[/tex].
4. [tex]\( \frac{7}{8} \)[/tex]:
- Multiply both the numerator and the denominator by 2:
[tex]\(\frac{7}{8} \times 2 = \frac{14}{16} = \frac{7}{4}\)[/tex].
5. [tex]\( \frac{2}{3} \)[/tex]:
- Multiply both the numerator and the denominator by 2:
[tex]\(\frac{2}{3} \times 2 = \frac{4}{6} = \frac{4}{3}\)[/tex].
Therefore, the simplified and amplified fractions are as follows:
- Simplified Results:
[tex]\(\frac{20}{50} = \frac{2}{5}\)[/tex],
[tex]\(\frac{42}{27} = \frac{14}{9}\)[/tex],
[tex]\(\frac{14}{21} = \frac{2}{3}\)[/tex],
[tex]\(\frac{12}{16} = \frac{3}{4}\)[/tex],
[tex]\(\frac{16}{18} = \frac{8}{9}\)[/tex].
- Amplified Results:
[tex]\(\frac{2}{9} = \frac{4}{9}\)[/tex],
[tex]\(\frac{4}{5} = \frac{8}{5}\)[/tex],
[tex]\(\frac{1}{12} = \frac{1}{6}\)[/tex],
[tex]\(\frac{7}{8} = \frac{7}{4}\)[/tex],
[tex]\(\frac{2}{3} = \frac{4}{3}\)[/tex].
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