Let's simplify each of the given fractions step-by-step by finding the greatest common divisor (GCD) and reducing the fraction accordingly.
1. [tex]\(\frac{10}{30}\)[/tex]:
- The GCD of 10 and 30 is 10.
- [tex]\(\frac{10 \div 10}{30 \div 10} = \frac{1}{3}\)[/tex]
2. [tex]\(\frac{13}{15}\)[/tex]:
- The GCD of 13 and 15 is 1.
- [tex]\(\frac{13 \div 1}{15 \div 1} = \frac{13}{15}\)[/tex] (already in simplest form)
3. [tex]\(\frac{25}{50}\)[/tex]:
- The GCD of 25 and 50 is 25.
- [tex]\(\frac{25 \div 25}{50 \div 25} = \frac{1}{2}\)[/tex]
4. [tex]\(\frac{4}{20}\)[/tex]:
- The GCD of 4 and 20 is 4.
- [tex]\(\frac{4 \div 4}{20 \div 4} = \frac{1}{5}\)[/tex]
5. [tex]\(\frac{4}{16}\)[/tex]:
- The GCD of 4 and 16 is 4.
- [tex]\(\frac{4 \div 4}{16 \div 4} = \frac{1}{4}\)[/tex]
6. [tex]\(\frac{8}{24}\)[/tex]:
- The GCD of 8 and 24 is 8.
- [tex]\(\frac{8 \div 8}{24 \div 8} = \frac{1}{3}\)[/tex]
7. [tex]\(3 \cdot \frac{6}{8}\)[/tex]:
Start by simplifying [tex]\(\frac{6}{8}\)[/tex]:
- The GCD of 6 and 8 is 2.
- [tex]\(\frac{6 \div 2}{8 \div 2} = \frac{3}{4}\)[/tex]
Now, multiply by 3:
- [tex]\(3 \cdot \frac{3}{4} = \frac{9}{4}\)[/tex] (improper fraction, which is already in simplest form)
8. [tex]\(\frac{16}{32}\)[/tex]:
- The GCD of 16 and 32 is 16.
- [tex]\(\frac{16 \div 16}{32 \div 16} = \frac{1}{2}\)[/tex]
Summarizing all the fractions in their simplest forms:
1. [tex]\(\frac{10}{30} = \frac{1}{3}\)[/tex]
2. [tex]\(\frac{13}{15} = \frac{13}{15}\)[/tex]
3. [tex]\(\frac{25}{50} = \frac{1}{2}\)[/tex]
4. [tex]\(\frac{4}{20} = \frac{1}{5}\)[/tex]
5. [tex]\(\frac{4}{16} = \frac{1}{4}\)[/tex]
6. [tex]\(\frac{8}{24} = \frac{1}{3}\)[/tex]
7. [tex]\(3 \cdot \frac{6}{8} = \frac{9}{4}\)[/tex]
8. [tex]\(\frac{16}{32} = \frac{1}{2}\)[/tex]
Therefore, the simplified fractions are:
1. [tex]\(\frac{1}{3}\)[/tex]
2. [tex]\(\frac{13}{15}\)[/tex]
3. [tex]\(\frac{1}{2}\)[/tex]
4. [tex]\(\frac{1}{5}\)[/tex]
5. [tex]\(\frac{1}{4}\)[/tex]
6. [tex]\(\frac{1}{3}\)[/tex]
7. [tex]\(\frac{9}{4}\)[/tex]
8. [tex]\(\frac{1}{2}\)[/tex]
