Certainly! Let's consider a repeating decimal [tex]\( x = 0.\overline{32} \)[/tex]. This notation means that 32 is the repeating sequence, and it continues indefinitely.
To understand why the subtraction of [tex]\( 100x \)[/tex] and [tex]\( x \)[/tex] eliminates the repeating portion, let's go through a step-by-step explanation:
1. Initial Representation:
- Let [tex]\( x = 0.32323232\ldots \)[/tex]. This represents the repeating decimal.
2. Shifting the Decimal Point:
- To manage the repeating part, multiply [tex]\( x \)[/tex] by 100 (since the repeating block "32" has 2 digits).
- [tex]\( 100x = 32.32323232\ldots \)[/tex]. Notice that this representation also has the repeating sequence "32" after the decimal point, just shifted 2 places to the right.
3. Eliminating the Repeating Part:
- Now, subtract the original [tex]\( x \)[/tex] from [tex]\( 100x \)[/tex].
- Write down the equations explicitly:
[tex]\[
100x = 32.32323232\ldots
\][/tex]
[tex]\[
x = 0.32323232\ldots
\][/tex]
- When you subtract [tex]\( x \)[/tex] from [tex]\( 100x \)[/tex], you get:
[tex]\[
100x - x = 32.32323232\ldots - 0.32323232\ldots
\][/tex]
Notice what happens:
[tex]\[
100x - x = (32.32323232\ldots) - (0.32323232\ldots).
\][/tex]
- The repeating decimal portion [tex]\( 0.32323232\ldots \)[/tex] on the right-hand side cancels out the identical repeating portion in [tex]\( 32.32323232\ldots \)[/tex].
4. Simplified Equation:
- After cancellation, you're left with:
[tex]\[
99x = 32
\][/tex]
5. Solving for [tex]\( x \)[/tex]:
- To solve for [tex]\( x \)[/tex], divide both sides of the equation by 99:
[tex]\[
x = \frac{32}{99}
\][/tex]
Thus, the multiplication by 100 shifts the repeating sequence, and the subtraction effectively cancels out the repeating part, leaving a simple equation that can be solved for [tex]\( x \)[/tex]. This explains why [tex]\( 100x - x \)[/tex] eliminates the repeating portion of the decimal, making it easier to solve for the repeating decimal as a fraction.
