Of course! Let's break down the problem step-by-step.
1. Identify the Rational Number:
The given rational number is [tex]\(\frac{2}{3}\)[/tex].
2. Calculate the Reciprocal:
The reciprocal of [tex]\(\frac{2}{3}\)[/tex] is obtained by flipping the numerator and the denominator. Thus, the reciprocal of [tex]\(\frac{2}{3}\)[/tex] is [tex]\(\frac{3}{2}\)[/tex].
3. Sum of the Rational Number and Its Reciprocal:
To find the sum of [tex]\(\frac{2}{3}\)[/tex] and its reciprocal ([tex]\(\frac{3}{2}\)[/tex]), we need to add these two fractions.
The fractions are:
[tex]\[
\frac{2}{3} \quad \text{and} \quad \frac{3}{2}
\][/tex]
4. Find a Common Denominator:
To add these fractions, we need a common denominator. The denominators are 3 and 2. The least common multiple (LCM) of 3 and 2 is 6.
5. Convert Each Fraction to the Common Denominator:
[tex]\[
\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}
\][/tex]
[tex]\[
\frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6}
\][/tex]
6. Add the Fractions:
Now, we add the fractions with the common denominator:
[tex]\[
\frac{4}{6} + \frac{9}{6} = \frac{4 + 9}{6} = \frac{13}{6}
\][/tex]
7. Convert to Decimal (if needed):
The fraction [tex]\(\frac{13}{6}\)[/tex] can be converted to a decimal:
[tex]\[
\frac{13}{6} \approx 2.1666666666666665
\][/tex]
Thus, the sum of the rational number [tex]\(\frac{2}{3}\)[/tex] and its reciprocal [tex]\(\frac{3}{2}\)[/tex] is [tex]\(\frac{13}{6}\)[/tex] or approximately [tex]\(2.1666666666666665\)[/tex].
