Let's consider the repeating decimal [tex]\( x = 0.\overline{83} \)[/tex]. This means that [tex]\( x = 0.83838383\ldots \)[/tex].
We want to find the value of [tex]\( 100x \)[/tex].
Step-by-Step Solution:
1. Step 1: Define the variable
Let [tex]\( x \)[/tex] be the repeating decimal:
[tex]\[ x = 0.\overline{83} = 0.83838383\ldots \][/tex]
2. Step 2: Multiply by 100
To shift the repeating part two decimal places to the left, multiply [tex]\( x \)[/tex] by 100:
[tex]\[ 100x = 100 \times 0.83838383\ldots \][/tex]
Since multiplying by 100 moves the decimal point two places to the right, we get:
[tex]\[ 100x = 83.83838383\ldots \][/tex]
3. Step 3: Express the new equation
Now, we have two expressions:
[tex]\[ x = 0.83838383\ldots \][/tex]
[tex]\[ 100x = 83.83838383\ldots \][/tex]
4. Step 4: Subtract the original equation from the new equation
If we subtract the first equation from the second, we eliminate the repeating decimal part:
[tex]\[ 100x - x = 83.83838383\ldots - 0.83838383\ldots \][/tex]
[tex]\[ 99x = 83 \][/tex]
5. Step 5: Solve for [tex]\( x \)[/tex]
To solve for [tex]\( x \)[/tex], divide both sides of the equation by 99:
[tex]\[ x = \frac{83}{99} \][/tex]
So, [tex]\( x = \frac{83}{99} \)[/tex] represents the repeating decimal [tex]\( 0.\overline{83} \)[/tex].
Now, we multiply by 100 to find [tex]\( 100x \)[/tex]:
[tex]\[ 100x = 100 \times \frac{83}{99} \][/tex]
[tex]\[ 100x = \frac{8300}{99} \][/tex]
Thus, the value of [tex]\( 100x \)[/tex] is [tex]\( \frac{8300}{99} \)[/tex] which simplifies to approximately 83.83838383... which is exactly what we obtained in our calculation.
Therefore, [tex]\( 100x = 83.\overline{83} \)[/tex].
