Answer :
Sure! Let's express the repeating decimals as fractions step-by-step:
### Part (a): [tex]\(0.\overline{4}\)[/tex]
1. Let [tex]\( x = 0.\overline{4} \)[/tex].
2. Multiply both sides by 10 to shift the decimal point:
[tex]\[ 10x = 4.\overline{4} \][/tex]
3. Subtract the original [tex]\( x \)[/tex] from this equation:
[tex]\[ 10x - x = 4.\overline{4} - 0.\overline{4} \][/tex]
[tex]\[ 9x = 4 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{4}{9} \][/tex]
So, [tex]\( 0.\overline{4} = \frac{4}{9} \)[/tex].
### Part (b): [tex]\(2.\overline{3}\)[/tex]
1. Let [tex]\( y = 2.\overline{3} \)[/tex].
2. Multiply both sides by 10 to shift the decimal point:
[tex]\[ 10y = 23.\overline{3} \][/tex]
3. Subtract the original [tex]\( y \)[/tex] from this equation:
[tex]\[ 10y - y = 23.\overline{3} - 2.\overline{3} \][/tex]
[tex]\[ 9y = 21 \][/tex]
4. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{21}{9} \][/tex]
[tex]\[ y = \frac{7}{3} \][/tex]
So, [tex]\( 2.\overline{3} = \frac{7}{3} \)[/tex].
### Part (f): [tex]\(4.\overline{57}\)[/tex]
1. Let [tex]\( z = 4.\overline{57} \)[/tex].
2. Multiply both sides by 100 (since the repeating part has two digits) to shift the decimal point:
[tex]\[ 100z = 457.\overline{57} \][/tex]
3. Subtract the original [tex]\( z \)[/tex] from this equation:
[tex]\[ 100z - z = 457.\overline{57} - 4.\overline{57} \][/tex]
[tex]\[ 99z = 453 \][/tex]
4. Solve for [tex]\( z \)[/tex]:
[tex]\[ z = \frac{453}{99} \][/tex]
Simplify the fraction by finding the greatest common divisor (GCD) of 453 and 99, which is 3:
[tex]\[ z = \frac{453 \div 3}{99 \div 3} \][/tex]
[tex]\[ z = \frac{151}{33} \][/tex]
So, [tex]\( 4.\overline{57} = \frac{151}{33} \)[/tex].
### Part (g): [tex]\(13.\overline{34}\)[/tex]
1. Let [tex]\( w = 13.\overline{34} \)[/tex].
2. Multiply both sides by 100 (since the repeating part has two digits) to shift the decimal point:
[tex]\[ 100w = 1334.\overline{34} \][/tex]
3. Subtract the original [tex]\( w \)[/tex] from this equation:
[tex]\[ 100w - w = 1334.\overline{34} - 13.\overline{34} \][/tex]
[tex]\[ 99w = 1321 \][/tex]
4. Solve for [tex]\( w \)[/tex]:
[tex]\[ w = \frac{1321}{99} \][/tex]
Check if the fraction can be simplified. The greatest common divisor (GCD) of 1321 and 99 is 1, so it is already in simplest form.
So, [tex]\( 13.\overline{34} = \frac{1321}{99} \)[/tex].
In summary:
(a) [tex]\(0.\overline{4} = \frac{4}{9}\)[/tex]
(b) [tex]\(2.\overline{3} = \frac{7}{3}\)[/tex]
(f) [tex]\(4.\overline{57} = \frac{151}{33}\)[/tex]
(g) [tex]\(13.\overline{34} = \frac{1321}{99}\)[/tex]
### Part (a): [tex]\(0.\overline{4}\)[/tex]
1. Let [tex]\( x = 0.\overline{4} \)[/tex].
2. Multiply both sides by 10 to shift the decimal point:
[tex]\[ 10x = 4.\overline{4} \][/tex]
3. Subtract the original [tex]\( x \)[/tex] from this equation:
[tex]\[ 10x - x = 4.\overline{4} - 0.\overline{4} \][/tex]
[tex]\[ 9x = 4 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{4}{9} \][/tex]
So, [tex]\( 0.\overline{4} = \frac{4}{9} \)[/tex].
### Part (b): [tex]\(2.\overline{3}\)[/tex]
1. Let [tex]\( y = 2.\overline{3} \)[/tex].
2. Multiply both sides by 10 to shift the decimal point:
[tex]\[ 10y = 23.\overline{3} \][/tex]
3. Subtract the original [tex]\( y \)[/tex] from this equation:
[tex]\[ 10y - y = 23.\overline{3} - 2.\overline{3} \][/tex]
[tex]\[ 9y = 21 \][/tex]
4. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{21}{9} \][/tex]
[tex]\[ y = \frac{7}{3} \][/tex]
So, [tex]\( 2.\overline{3} = \frac{7}{3} \)[/tex].
### Part (f): [tex]\(4.\overline{57}\)[/tex]
1. Let [tex]\( z = 4.\overline{57} \)[/tex].
2. Multiply both sides by 100 (since the repeating part has two digits) to shift the decimal point:
[tex]\[ 100z = 457.\overline{57} \][/tex]
3. Subtract the original [tex]\( z \)[/tex] from this equation:
[tex]\[ 100z - z = 457.\overline{57} - 4.\overline{57} \][/tex]
[tex]\[ 99z = 453 \][/tex]
4. Solve for [tex]\( z \)[/tex]:
[tex]\[ z = \frac{453}{99} \][/tex]
Simplify the fraction by finding the greatest common divisor (GCD) of 453 and 99, which is 3:
[tex]\[ z = \frac{453 \div 3}{99 \div 3} \][/tex]
[tex]\[ z = \frac{151}{33} \][/tex]
So, [tex]\( 4.\overline{57} = \frac{151}{33} \)[/tex].
### Part (g): [tex]\(13.\overline{34}\)[/tex]
1. Let [tex]\( w = 13.\overline{34} \)[/tex].
2. Multiply both sides by 100 (since the repeating part has two digits) to shift the decimal point:
[tex]\[ 100w = 1334.\overline{34} \][/tex]
3. Subtract the original [tex]\( w \)[/tex] from this equation:
[tex]\[ 100w - w = 1334.\overline{34} - 13.\overline{34} \][/tex]
[tex]\[ 99w = 1321 \][/tex]
4. Solve for [tex]\( w \)[/tex]:
[tex]\[ w = \frac{1321}{99} \][/tex]
Check if the fraction can be simplified. The greatest common divisor (GCD) of 1321 and 99 is 1, so it is already in simplest form.
So, [tex]\( 13.\overline{34} = \frac{1321}{99} \)[/tex].
In summary:
(a) [tex]\(0.\overline{4} = \frac{4}{9}\)[/tex]
(b) [tex]\(2.\overline{3} = \frac{7}{3}\)[/tex]
(f) [tex]\(4.\overline{57} = \frac{151}{33}\)[/tex]
(g) [tex]\(13.\overline{34} = \frac{1321}{99}\)[/tex]
Answer with Step-by-step explanation:
Here are the fractions for each repeating decimal:
(a) [tex]0.\overline{4}[/tex]
Let [tex]x = 0.\overline{4}.[/tex]
Multiply by 10:
[tex]10x = 4.\overline{4}[/tex]
Subtract:
10x - x = 4
9x = 4
[tex]x = \frac{4}{9}[/tex]
So, [tex]0.\overline{4} = \frac{4}{9}.[/tex]
(b) 2[tex].\overline{3}[/tex]
Let[tex]x = 2.\overline{3}.[/tex]
Multiply by 10:
[tex]10x = 23.\overline{3}[/tex]
Subtract:
10x - x = 21
9x = 21
[tex]x = \frac{7}{3}[/tex]
So,[tex]2.\overline{3} = \frac{7}{3}.[/tex]
(c)[tex]4.\overline{57}[/tex]
Let [tex]x = 4.\overline{57}.[/tex]
Multiply by 100:
[tex]100x = 457.\overline{57}[/tex]
Subtract:
100x - x = 453
99x = 453
[tex]x = \frac{151}{33}[/tex]
So, [tex]4.\overline{57} = \frac{151}{33}.[/tex]
(d) [tex]13.\overline{34}[/tex]
[tex]Let x = 13.\overline{34}.[/tex]
Multiply by 100:
[tex]100x = 1334.\overline{34}[/tex]
Subtract:
100x - x = 1321
99x = 1321
[tex]x = \frac{1321}{99}[/tex]
[tex]So, 13.\overline{34} = \frac{1321}{99}.[/tex]
