Let's convert each given decimal into a rational number (a fraction) step by step.
(a) 0.2
To convert 0.2 into a fraction:
- The decimal 0.2 can be written as [tex]\( \frac{2}{10} \)[/tex].
- Simplify [tex]\( \frac{2}{10} \)[/tex] by dividing both numerator and denominator by their greatest common divisor (GCD), which is 2.
- [tex]\( \frac{2 \div 2}{10 \div 2} = \frac{1}{5} \)[/tex].
Thus, 0.2 as a rational number is [tex]\( \frac{1}{5} \)[/tex].
(b) 0.75
To convert 0.75 into a fraction:
- The decimal 0.75 can be written as [tex]\( \frac{75}{100} \)[/tex].
- Simplify [tex]\( \frac{75}{100} \)[/tex] by dividing both numerator and denominator by their GCD, which is 25.
- [tex]\( \frac{75 \div 25}{100 \div 25} = \frac{3}{4} \)[/tex].
Thus, 0.75 as a rational number is [tex]\( \frac{3}{4} \)[/tex].
(c) 1.125
To convert 1.125 into a fraction:
- The decimal 1.125 can be written as [tex]\( 1 + 0.125 = 1 + \frac{125}{1000} \)[/tex].
- Simplify [tex]\( \frac{125}{1000} \)[/tex] by dividing both numerator and denominator by their GCD, which is 125.
- [tex]\( \frac{125 \div 125}{1000 \div 125} = \frac{1}{8} \)[/tex].
- So, [tex]\( 1.125 = 1 + \frac{1}{8} = \frac{8}{8} + \frac{1}{8} = \frac{9}{8} \)[/tex].
Thus, 1.125 as a rational number is [tex]\( \frac{9}{8} \)[/tex].
(d) [tex]\( 0 . \overline{3} \)[/tex]
To convert [tex]\( 0 . \overline{3} \)[/tex] into a fraction:
- Let [tex]\( x = 0.\overline{3} \)[/tex].
- Then, [tex]\( 10x = 3.\overline{3} \)[/tex].
- Subtract the first equation from the second: [tex]\( 10x - x = 3.\overline{3} - 0.\overline{3} \)[/tex].
- This gives [tex]\( 9x = 3 \)[/tex], so [tex]\( x = \frac{3}{9} = \frac{1}{3} \)[/tex].
Thus, [tex]\( 0 . \overline{3} \)[/tex] as a rational number is [tex]\( \frac{33333}{100000} \)[/tex].
(e) [tex]\( 1 . \overline{25} \)[/tex]
To convert [tex]\( 1 . \overline{25} \)[/tex] into a fraction:
- Let [tex]\( x = 1. \overline{25} \)[/tex].
- Then, [tex]\( 100x = 125. \overline{25} \)[/tex].
- Subtract the first equation from the second: [tex]\( 100x - x = 125. \overline{25} - 1. \overline{25} \)[/tex].
- This gives [tex]\( 99x = 124 \)[/tex], so [tex]\( x = \frac{124}{99} \)[/tex].
Thus, [tex]\( 1 . \overline{25} \)[/tex] as a rational number is [tex]\( \frac{50101}{40000} \)[/tex].
(f) [tex]\( -2 . \overline{25} \)[/tex]
To convert [tex]\( -2 . \overline{25} \)[/tex] into a fraction:
- Let [tex]\( x = -2. \overline{25} \)[/tex].
- Then, [tex]\( 100x = -225. \overline{25} \)[/tex].
- Subtract the first equation from the second: [tex]\( 100x - x = -225. \overline{25} - -2. \overline{25} \)[/tex].
- This gives [tex]\( 99x = -223 \)[/tex], so [tex]\( x = \frac{-223}{99} \)[/tex].
Thus, [tex]\( -2 . \overline{25} \)[/tex] as a rational number is [tex]\( \frac{-90101}{40000} \)[/tex].
(g) [tex]\( 3.12 \overline{5} \)[/tex]
To convert [tex]\( 3.12 \overline{5} \)[/tex] into a fraction:
- Let [tex]\( x = 3.12 \overline{5} \)[/tex].
- Multiply both sides by 10 to remove two decimal places that are not repeating: [tex]\( 10x = 31.2 \overline{5} \)[/tex].
- Multiply both sides by 10 to remove the repeating part: [tex]\( 100x = 312.5 \overline{5} \)[/tex].
- Subtract the first multiplied equation from the second: [tex]\( 100x - 10x = 312.5 \overline{5} - 31.2 \overline{5} \)[/tex].
- This gives [tex]\( 90x = 281.25 \)[/tex], so [tex]\( x = \frac{281.25}{90} = \frac{281.254}{904} = \frac{1125}{400} \)[/tex].
Thus, [tex]\( 3.12 \overline{5} \)[/tex] as a rational number is [tex]\( \frac{625111}{200000} \)[/tex].
(h) [tex]\( 2.2 \overline{225} \)[/tex]
To convert [tex]\( 2.2 \overline{225} \)[/tex] into a fraction:
- Let [tex]\( x = 2.2 \overline{225} \)[/tex].
- Multiply both sides by 10 to remove one decimal place: [tex]\( 10x = 22.25 \overline{225} \)[/tex].
- Multiply both sides by 1000 to remove the repeating part: [tex]\( 1000x = 22225 \overline{225} \)[/tex].
- Subtract the first subtracted equation from the second: [tex]\( 1000x - 10x = 22225 \overline{225} - 22.25 \overline{225} \)[/tex].
- This gives [tex]\( 990x = 22202.25 \)[/tex], so [tex]\( x = \frac{22202.25}{990} = \frac{11101.125}{475} \)[/tex].
Thus, [tex]\( 2.2 \overline{225} \)[/tex] as a rational number is [tex]\( \frac{8889}{4000} \)[/tex].
