To solve the given problem, follow these steps:
1. Convert Mixed Numbers to Improper Fractions:
- The fraction [tex]\(4 \frac{1}{3}\)[/tex] can be converted to an improper fraction. To do this, multiply the whole number (4) by the denominator (3) and add the numerator (1):
[tex]\[
4 \frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3}
\][/tex]
- Similarly, the fraction [tex]\(5 \frac{1}{6}\)[/tex] can be converted to an improper fraction. Multiply the whole number (5) by the denominator (6) and add the numerator (1):
[tex]\[
5 \frac{1}{6} = \frac{5 \times 6 + 1}{6} = \frac{30 + 1}{6} = \frac{31}{6}
\][/tex]
2. Identify the Fraction Needed for the Reciprocal:
- We need to find the reciprocal of one of these fractions. In this case, we are supposed to find the reciprocal of the fraction [tex]\(\frac{31}{6}\)[/tex].
3. Find the Reciprocal:
- The reciprocal of a fraction simply means to invert the fraction (swap the numerator and the denominator). Therefore, the reciprocal of [tex]\(\frac{31}{6}\)[/tex] is:
[tex]\[
\frac{6}{31}
\][/tex]
Thus, the reciprocal fraction that is required is [tex]\(\frac{6}{31}\)[/tex].
