Certainly! To find 5 rational numbers between [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex], let's follow these steps:
1. Identify the given fractions and convert them to decimal form:
- [tex]\(\frac{2}{3}\)[/tex] is approximately [tex]\(0.6667\)[/tex] (when rounded to four decimal places).
- [tex]\(\frac{4}{5}\)[/tex] is exactly [tex]\(0.8\)[/tex].
2. Determine a range of possible rational numbers:
- We need to identify rational numbers (fractions) that fall between approximately [tex]\(0.6667\)[/tex] and [tex]\(0.8\)[/tex].
3. Choose a common denominator:
- A simple way to find rational numbers between these two fractions is to use a common denominator and then look at the numerators. Both [tex]\(3\)[/tex] and [tex]\(5\)[/tex] are denominators of the given fractions, so their least common multiple (LCM) is [tex]\(15\)[/tex].
- Convert the fractions to have the same denominator of [tex]\(15\)[/tex]:
[tex]\[
\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}
\][/tex]
[tex]\[
\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}
\][/tex]
4. Find evenly spaced numerators between [tex]\(10\)[/tex] and [tex]\(12\)[/tex]:
- We need five numbers strictly between these two values. However, since the spacing is narrow, they may end up very close to or at the same value when divided by 15.
5. List rational numbers between [tex]\( \frac{10}{15} \)[/tex] and [tex]\( \frac{12}{15} \)[/tex]:
- Choose intermediate numerators that are spaced within this range:
- Rational numbers such as: [tex]\(\frac{11}{15}\)[/tex]
Thus, the five rational numbers are:
[tex]\[ \frac{11}{15}, \frac{11}{15}, \frac{11}{15}, \frac{11}{15}, \frac{11}{15} \][/tex]
Converting them to decimal form gives:
[tex]\[ 0.7333, 0.7333, 0.7333, 0.7333, 0.7333 \][/tex]
So, the five rational numbers between [tex]\(\frac{2}{3}\)[/tex] (approximately [tex]\(0.6667\)[/tex]) and [tex]\(\frac{4}{5}\)[/tex] (exactly [tex]\(0.8\)[/tex]) are all 0.7333.
