Answer :
Let's reduce these rational numbers to their standard form. The standard form of a rational number is when the numerator and denominator have no common factors other than 1.
To achieve this, we will find the greatest common divisor (GCD) for the numerator and the denominator and then divide both by this GCD.
A. [tex]\(\frac{-27}{15}\)[/tex]
1. Find the GCD of 27 and 15. The GCD is 3.
2. Divide both the numerator and the denominator by 3:
[tex]\[
\frac{-27 \div 3}{15 \div 3} = \frac{-9}{5}
\][/tex]
B. [tex]\(\frac{100}{18}\)[/tex]
1. Find the GCD of 100 and 18. The GCD is 2.
2. Divide both the numerator and the denominator by 2:
[tex]\[
\frac{100 \div 2}{18 \div 2} = \frac{50}{9}
\][/tex]
C. [tex]\(\frac{-36}{124}\)[/tex]
1. Find the GCD of 36 and 124. The GCD is 4.
2. Divide both the numerator and the denominator by 4:
[tex]\[
\frac{-36 \div 4}{124 \div 4} = \frac{-9}{31}
\][/tex]
D. [tex]\(\frac{35}{15}\)[/tex]
1. Find the GCD of 35 and 15. The GCD is 5.
2. Divide both the numerator and the denominator by 5:
[tex]\[
\frac{35 \div 5}{15 \div 5} = \frac{7}{3}
\][/tex]
E. [tex]\(\frac{-14}{-49}\)[/tex]
1. Find the GCD of 14 and 49. The GCD is 7.
2. Divide both the numerator and the denominator by 7. Since both signs are negative, the result is positive:
[tex]\[
\frac{-14 \div 7}{-49 \div 7} = \frac{2}{7}
\][/tex]
Here are the rational numbers reduced to their standard forms:
- [tex]\(\frac{-27}{15} = \frac{-9}{5}\)[/tex]
- [tex]\(\frac{100}{18} = \frac{50}{9}\)[/tex]
- [tex]\(\frac{-36}{124} = \frac{-9}{31}\)[/tex]
- [tex]\(\frac{35}{15} = \frac{7}{3}\)[/tex]
- [tex]\(\frac{-14}{-49} = \frac{2}{7}\)[/tex]
To achieve this, we will find the greatest common divisor (GCD) for the numerator and the denominator and then divide both by this GCD.
A. [tex]\(\frac{-27}{15}\)[/tex]
1. Find the GCD of 27 and 15. The GCD is 3.
2. Divide both the numerator and the denominator by 3:
[tex]\[
\frac{-27 \div 3}{15 \div 3} = \frac{-9}{5}
\][/tex]
B. [tex]\(\frac{100}{18}\)[/tex]
1. Find the GCD of 100 and 18. The GCD is 2.
2. Divide both the numerator and the denominator by 2:
[tex]\[
\frac{100 \div 2}{18 \div 2} = \frac{50}{9}
\][/tex]
C. [tex]\(\frac{-36}{124}\)[/tex]
1. Find the GCD of 36 and 124. The GCD is 4.
2. Divide both the numerator and the denominator by 4:
[tex]\[
\frac{-36 \div 4}{124 \div 4} = \frac{-9}{31}
\][/tex]
D. [tex]\(\frac{35}{15}\)[/tex]
1. Find the GCD of 35 and 15. The GCD is 5.
2. Divide both the numerator and the denominator by 5:
[tex]\[
\frac{35 \div 5}{15 \div 5} = \frac{7}{3}
\][/tex]
E. [tex]\(\frac{-14}{-49}\)[/tex]
1. Find the GCD of 14 and 49. The GCD is 7.
2. Divide both the numerator and the denominator by 7. Since both signs are negative, the result is positive:
[tex]\[
\frac{-14 \div 7}{-49 \div 7} = \frac{2}{7}
\][/tex]
Here are the rational numbers reduced to their standard forms:
- [tex]\(\frac{-27}{15} = \frac{-9}{5}\)[/tex]
- [tex]\(\frac{100}{18} = \frac{50}{9}\)[/tex]
- [tex]\(\frac{-36}{124} = \frac{-9}{31}\)[/tex]
- [tex]\(\frac{35}{15} = \frac{7}{3}\)[/tex]
- [tex]\(\frac{-14}{-49} = \frac{2}{7}\)[/tex]
