To convert the repeating decimal [tex]\(7.\overline{6}\)[/tex] to a fraction in its lowest terms, follow these steps:
1. Assign a variable to the repeating decimal:
Let [tex]\( x = 7.\overline{6} \)[/tex].
2. Express the repeating decimal as an equation:
Notice that the repeating part is [tex]\(6\)[/tex], so we can write:
[tex]\[ x = 7.6666\ldots \][/tex]
3. Eliminate the repeating part using a multiplier:
To eliminate the repeating decimal, multiply [tex]\( x \)[/tex] by 10 (to shift the decimal point one place to the right):
[tex]\[ 10x = 76.6666\ldots \][/tex]
4. Set up an equation to find [tex]\( x \)[/tex]:
Now, you have:
[tex]\[ x = 7.6666\ldots \][/tex]
[tex]\[ 10x = 76.6666\ldots \][/tex]
5. Subtract the first equation from the second:
[tex]\[ 10x - x = 76.6666\ldots - 7.6666\ldots \][/tex]
This simplifies to:
[tex]\[ 9x = 69 \][/tex]
6. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{69}{9} \][/tex]
7. Simplify the fraction:
To simplify [tex]\(\frac{69}{9}\)[/tex], find the greatest common divisor (GCD) of 69 and 9. The GCD of 69 and 9 is 3.
Divide the numerator and the denominator by their GCD:
[tex]\[
\frac{69 \div 3}{9 \div 3} = \frac{23}{3}
\][/tex]
Thus, the fraction representation of [tex]\(7.\overline{6}\)[/tex] in its lowest terms is:
[tex]\[
\boxed{\frac{23}{3}}
\][/tex]
