Discover a world of knowledge and get your questions answered at IDNLearn.com. Get timely and accurate answers to your questions from our dedicated community of experts who are here to help you.
Sagot :
We need to determine which of the given fractions is in the simplest form. A fraction is in simplest form if the greatest common divisor (GCD) of its numerator and denominator is 1.
Let's check each fraction:
1. Fraction: [tex]\(\frac{39}{65}\)[/tex]
- Find the GCD of 39 and 65.
- Factors of 39: 1, 3, 13, 39
- Factors of 65: 1, 5, 13, 65
- The common factors are 1 and 13.
- The GCD is 13.
- Therefore, [tex]\(\frac{39}{65}\)[/tex] can be simplified by dividing both the numerator and the denominator by 13:
[tex]\[ \frac{39 \div 13}{65 \div 13} = \frac{3}{5} \][/tex]
- Hence, [tex]\(\frac{39}{65}\)[/tex] is not in simplest form.
2. Fraction: [tex]\(\frac{45}{65}\)[/tex]
- Find the GCD of 45 and 65.
- Factors of 45: 1, 3, 5, 9, 15, 45
- Factors of 65: 1, 5, 13, 65
- The common factors are 1 and 5.
- The GCD is 5.
- Therefore, [tex]\(\frac{45}{65}\)[/tex] can be simplified by dividing both the numerator and the denominator by 5:
[tex]\[ \frac{45 \div 5}{65 \div 5} = \frac{9}{13} \][/tex]
- Hence, [tex]\(\frac{45}{65}\)[/tex] is not in simplest form.
3. Fraction: [tex]\(\frac{46}{65}\)[/tex]
- Find the GCD of 46 and 65.
- Factors of 46: 1, 2, 23, 46
- Factors of 65: 1, 5, 13, 65
- The common factor is 1.
- The GCD is 1.
- Therefore, [tex]\(\frac{46}{65}\)[/tex] is already in simplest form since we cannot divide it any further. This fraction doesn't need any simplification.
4. Fraction: [tex]\(\frac{52}{65}\)[/tex]
- Find the GCD of 52 and 65.
- Factors of 52: 1, 2, 4, 13, 26, 52
- Factors of 65: 1, 5, 13, 65
- The common factors are 1 and 13.
- The GCD is 13.
- Therefore, [tex]\(\frac{52}{65}\)[/tex] can be simplified by dividing both the numerator and the denominator by 13:
[tex]\[ \frac{52 \div 13}{65 \div 13} = \frac{4}{5} \][/tex]
- Hence, [tex]\(\frac{52}{65}\)[/tex] is not in simplest form.
5. Fraction: [tex]\(\frac{55}{65}\)[/tex]
- Find the GCD of 55 and 65.
- Factors of 55: 1, 5, 11, 55
- Factors of 65: 1, 5, 13, 65
- The common factors are 1 and 5.
- The GCD is 5.
- Therefore, [tex]\(\frac{55}{65}\)[/tex] can be simplified by dividing both the numerator and the denominator by 5:
[tex]\[ \frac{55 \div 5}{65 \div 5} = \frac{11}{13} \][/tex]
- Hence, [tex]\(\frac{55}{65}\)[/tex] is not in simplest form.
Among the given fractions, the fraction [tex]\(\frac{46}{65}\)[/tex] is already in simplest form. Thus, the fraction in simplest form is:
[tex]\[ \boxed{\frac{46}{65}} \][/tex]
Let's check each fraction:
1. Fraction: [tex]\(\frac{39}{65}\)[/tex]
- Find the GCD of 39 and 65.
- Factors of 39: 1, 3, 13, 39
- Factors of 65: 1, 5, 13, 65
- The common factors are 1 and 13.
- The GCD is 13.
- Therefore, [tex]\(\frac{39}{65}\)[/tex] can be simplified by dividing both the numerator and the denominator by 13:
[tex]\[ \frac{39 \div 13}{65 \div 13} = \frac{3}{5} \][/tex]
- Hence, [tex]\(\frac{39}{65}\)[/tex] is not in simplest form.
2. Fraction: [tex]\(\frac{45}{65}\)[/tex]
- Find the GCD of 45 and 65.
- Factors of 45: 1, 3, 5, 9, 15, 45
- Factors of 65: 1, 5, 13, 65
- The common factors are 1 and 5.
- The GCD is 5.
- Therefore, [tex]\(\frac{45}{65}\)[/tex] can be simplified by dividing both the numerator and the denominator by 5:
[tex]\[ \frac{45 \div 5}{65 \div 5} = \frac{9}{13} \][/tex]
- Hence, [tex]\(\frac{45}{65}\)[/tex] is not in simplest form.
3. Fraction: [tex]\(\frac{46}{65}\)[/tex]
- Find the GCD of 46 and 65.
- Factors of 46: 1, 2, 23, 46
- Factors of 65: 1, 5, 13, 65
- The common factor is 1.
- The GCD is 1.
- Therefore, [tex]\(\frac{46}{65}\)[/tex] is already in simplest form since we cannot divide it any further. This fraction doesn't need any simplification.
4. Fraction: [tex]\(\frac{52}{65}\)[/tex]
- Find the GCD of 52 and 65.
- Factors of 52: 1, 2, 4, 13, 26, 52
- Factors of 65: 1, 5, 13, 65
- The common factors are 1 and 13.
- The GCD is 13.
- Therefore, [tex]\(\frac{52}{65}\)[/tex] can be simplified by dividing both the numerator and the denominator by 13:
[tex]\[ \frac{52 \div 13}{65 \div 13} = \frac{4}{5} \][/tex]
- Hence, [tex]\(\frac{52}{65}\)[/tex] is not in simplest form.
5. Fraction: [tex]\(\frac{55}{65}\)[/tex]
- Find the GCD of 55 and 65.
- Factors of 55: 1, 5, 11, 55
- Factors of 65: 1, 5, 13, 65
- The common factors are 1 and 5.
- The GCD is 5.
- Therefore, [tex]\(\frac{55}{65}\)[/tex] can be simplified by dividing both the numerator and the denominator by 5:
[tex]\[ \frac{55 \div 5}{65 \div 5} = \frac{11}{13} \][/tex]
- Hence, [tex]\(\frac{55}{65}\)[/tex] is not in simplest form.
Among the given fractions, the fraction [tex]\(\frac{46}{65}\)[/tex] is already in simplest form. Thus, the fraction in simplest form is:
[tex]\[ \boxed{\frac{46}{65}} \][/tex]
We are delighted to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. Thank you for choosing IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more solutions.
