To express [tex]\( 1.\overline{7} \)[/tex] as a fraction in simplest form, follow these steps:
1. Identify the repeating decimal: The repeating part of the decimal is [tex]\( 0.\overline{7} \)[/tex].
2. Express the repeating decimal as a fraction:
- Let [tex]\( x = 0.\overline{7} \)[/tex].
- This means [tex]\( x = 0.7777\ldots \)[/tex].
3. Eliminate the repeating part by multiplying:
- Multiply both sides of the equation by 10 (since the repeating block is one digit long).
- [tex]\( 10x = 7.7777\ldots \)[/tex].
4. Subtract the original equation from this new equation:
- Subtract [tex]\( x = 0.7777\ldots \)[/tex] from [tex]\( 10x = 7.7777\ldots \)[/tex].
- This gives [tex]\( 10x - x = 7.7777\ldots - 0.7777\ldots \)[/tex] or [tex]\( 9x = 7 \)[/tex].
5. Solve for [tex]\( x \)[/tex]:
- [tex]\( x = \frac{7}{9} \)[/tex].
6. Combine the non-repeating and repeating parts:
- We initially started with [tex]\( 1.\overline{7} \)[/tex].
- This can be written as: [tex]\( 1 + 0.\overline{7} \)[/tex].
7. Add together:
- Replace [tex]\( 0.\overline{7} \)[/tex] with the fraction [tex]\(\frac{7}{9}\)[/tex].
- Therefore, it becomes: [tex]\( 1 + \frac{7}{9} \)[/tex].
8. Convert to a single fraction:
- Express 1 as a fraction with the same denominator: [tex]\( 1 = \frac{9}{9} \)[/tex].
- So, [tex]\( 1 + \frac{7}{9} = \frac{9}{9} + \frac{7}{9} = \frac{16}{9} \)[/tex].
Thus, the fraction equivalent to [tex]\( 1.\overline{7} \)[/tex] in simplest form is [tex]\( \frac{16}{9} \)[/tex].
Therefore, the correct answer is:
[tex]\[
\frac{16}{9}
\][/tex]
