Answer :
To write the given rational numbers in standard form, we need to simplify each fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD) and ensuring the denominator is positive.
### (i) [tex]\(\frac{6}{30}\)[/tex]
1. Find the GCD of 6 and 30:
- The factors of 6 are: 1, 2, 3, 6.
- The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30.
- The common factors are 1, 2, 3, 6.
- The GCD is 6.
2. Simplify [tex]\(\frac{6}{30}\)[/tex]:
- Divide both the numerator and the denominator by their GCD, 6:
[tex]\[ \frac{6 \div 6}{30 \div 6} = \frac{1}{5} \][/tex]
So, [tex]\(\frac{6}{30}\)[/tex] in standard form is [tex]\(\frac{1}{5}\)[/tex].
### (ii) [tex]\(\frac{12}{-17}\)[/tex]
1. Find the GCD of 12 and 17:
- The factors of 12 are: 1, 2, 3, 4, 6, 12.
- The factors of 17 are: 1, 17.
- The common factor is 1.
- The GCD is 1.
2. Simplify [tex]\(\frac{12}{-17}\)[/tex]:
- Since the GCD is 1, the fraction is already in its simplest form.
- However, we need to ensure the denominator is positive:
[tex]\[ \frac{12}{-17} \times \frac{-1}{-1} = \frac{-12}{17} \][/tex]
So, [tex]\(\frac{12}{-17}\)[/tex] in standard form is [tex]\(\frac{-12}{17}\)[/tex].
### (iii) [tex]\(\frac{-24}{32}\)[/tex]
1. Find the GCD of -24 and 32:
- The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.
- The factors of 32 are: 1, 2, 4, 8, 16, 32.
- The common factors are 1, 2, 4, 8.
- The GCD is 8.
2. Simplify [tex]\(\frac{-24}{32}\)[/tex]:
- Divide both the numerator and the denominator by their GCD, 8:
[tex]\[ \frac{-24 \div 8}{32 \div 8} = \frac{-3}{4} \][/tex]
So, [tex]\(\frac{-24}{32}\)[/tex] in standard form is [tex]\(\frac{-3}{4}\)[/tex].
Therefore, the simplified forms of the given fractions are:
(i) [tex]\(\frac{1}{5}\)[/tex]
(ii) [tex]\(\frac{-12}{17}\)[/tex]
(iii) [tex]\(\frac{-3}{4}\)[/tex]
### (i) [tex]\(\frac{6}{30}\)[/tex]
1. Find the GCD of 6 and 30:
- The factors of 6 are: 1, 2, 3, 6.
- The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30.
- The common factors are 1, 2, 3, 6.
- The GCD is 6.
2. Simplify [tex]\(\frac{6}{30}\)[/tex]:
- Divide both the numerator and the denominator by their GCD, 6:
[tex]\[ \frac{6 \div 6}{30 \div 6} = \frac{1}{5} \][/tex]
So, [tex]\(\frac{6}{30}\)[/tex] in standard form is [tex]\(\frac{1}{5}\)[/tex].
### (ii) [tex]\(\frac{12}{-17}\)[/tex]
1. Find the GCD of 12 and 17:
- The factors of 12 are: 1, 2, 3, 4, 6, 12.
- The factors of 17 are: 1, 17.
- The common factor is 1.
- The GCD is 1.
2. Simplify [tex]\(\frac{12}{-17}\)[/tex]:
- Since the GCD is 1, the fraction is already in its simplest form.
- However, we need to ensure the denominator is positive:
[tex]\[ \frac{12}{-17} \times \frac{-1}{-1} = \frac{-12}{17} \][/tex]
So, [tex]\(\frac{12}{-17}\)[/tex] in standard form is [tex]\(\frac{-12}{17}\)[/tex].
### (iii) [tex]\(\frac{-24}{32}\)[/tex]
1. Find the GCD of -24 and 32:
- The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.
- The factors of 32 are: 1, 2, 4, 8, 16, 32.
- The common factors are 1, 2, 4, 8.
- The GCD is 8.
2. Simplify [tex]\(\frac{-24}{32}\)[/tex]:
- Divide both the numerator and the denominator by their GCD, 8:
[tex]\[ \frac{-24 \div 8}{32 \div 8} = \frac{-3}{4} \][/tex]
So, [tex]\(\frac{-24}{32}\)[/tex] in standard form is [tex]\(\frac{-3}{4}\)[/tex].
Therefore, the simplified forms of the given fractions are:
(i) [tex]\(\frac{1}{5}\)[/tex]
(ii) [tex]\(\frac{-12}{17}\)[/tex]
(iii) [tex]\(\frac{-3}{4}\)[/tex]
