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Prove algebraically that [tex]0.\dot{5}=\frac{5}{9}[/tex].

Write your proof using [tex]x[/tex] in the working. The concluding line has been written for you.

Therefore, [tex]x=\frac{5}{9}[/tex].

Sagot :

Let's prove algebraically that [tex]\(0.\dot{5} = \frac{5}{9}\)[/tex].

1. Assign the repeating decimal to a variable:
Let [tex]\( x = 0.\dot{5} \)[/tex]

2. Since [tex]\( 0.\dot{5} \)[/tex] is a repeating decimal, we know that:
[tex]\( x = 0.555555\ldots \)[/tex]

3. To eliminate the repeating part, multiply both sides of the equation by 10 to shift the decimal point one place to the right:
[tex]\( 10x = 5.555555\ldots \)[/tex]

4. Now, we have two equations:
[tex]\[ x = 0.555555\ldots \][/tex]
[tex]\[ 10x = 5.555555\ldots \][/tex]

5. Subtract the first equation from the second equation:
[tex]\[ 10x - x = 5.555555\ldots - 0.555555\ldots \][/tex]

6. Simplify both sides:
[tex]\[ 9x = 5 \][/tex]

7. Solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 9:
[tex]\[ x = \frac{5}{9} \][/tex]

Therefore, [tex]\( x = \frac{5}{9} \)[/tex].