First, let's convert [tex]\(76\%\)[/tex] to a decimal.
[tex]\[ 76\% = \frac{76}{100} = 0.76 \][/tex]
Our goal is to determine which of the given pairs of fractions this decimal value falls between. For comparison, we also need to convert the fractions to their decimal equivalents.
1. [tex]\( \frac{1}{5} \)[/tex] and [tex]\( \frac{1}{4} \)[/tex]:
[tex]\[ \frac{1}{5} = 0.2 \][/tex]
[tex]\[ \frac{1}{4} = 0.25 \][/tex]
[tex]\(0.76\)[/tex] does not lie between [tex]\(0.2\)[/tex] and [tex]\(0.25\)[/tex].
2. [tex]\( \frac{1}{3} \)[/tex] and [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[ \frac{1}{3} \approx 0.3333 \][/tex]
[tex]\[ \frac{1}{2} = 0.5 \][/tex]
[tex]\(0.76\)[/tex] does not lie between [tex]\(0.3333\)[/tex] and [tex]\(0.5\)[/tex].
3. [tex]\( \frac{2}{3} \)[/tex] and [tex]\( \frac{7}{10} \)[/tex]:
[tex]\[ \frac{2}{3} \approx 0.6666 \][/tex]
[tex]\[ \frac{7}{10} = 0.7 \][/tex]
[tex]\(0.76\)[/tex] does not lie between [tex]\(0.6666\)[/tex] and [tex]\(0.7\)[/tex].
4. [tex]\( \frac{3}{4} \)[/tex] and [tex]\( \frac{5}{6} \)[/tex]:
[tex]\[ \frac{3}{4} = 0.75 \][/tex]
[tex]\[ \frac{5}{6} \approx 0.8333 \][/tex]
[tex]\(0.76\)[/tex] lies between [tex]\(0.75\)[/tex] and [tex]\(0.8333\)[/tex].
Thus, the decimal 0.76 is between the fractions [tex]\( \frac{3}{4} \)[/tex] and [tex]\( \frac{5}{6} \)[/tex].
