Problem 76 - Algebra 1: Concepts and Skills

Question

Find the reciprocal of the mixed number. Write your answer in lowest terms. $$ 4 \frac{2}{5} $$

Step-by-Step Solution

Answer

The reciprocal of the mixed number \(4 \frac{2}{5}\) is \(\frac{5}{22}\).

Step 1: Convert the Mixed Number to an Improper Fraction

A mixed number \(a \frac{b}{c}\) can be converted into an improper fraction by the formula: \(\frac{(a*c) + b}{c}\). Applying this to the mixed number \(4 \frac{2}{5}\), we get the improper fraction: \(\frac{(4*5) + 2}{5} = \frac{22}{5}\).

Step 2: Find the Reciprocal of the Improper Fraction

The reciprocal of a fraction \(\frac{a}{b}\) is given by \(\frac{b}{a}\). Therefore, the reciprocal of \(\frac{22}{5}\) is \(\frac{5}{22}\).

Step 3: Simplify the Fraction into Lowest Terms

In this case, the reciprocal fraction \(\frac{5}{22}\) is already in its lowest terms as 5 and 22 do not share any common factors other than 1.

Key Concepts

Mixed Numbers

Mixed numbers are those numbers that consist of a whole number part and a fractional part. For instance, if you have the number \(4\frac{2}{5}\), 4 is the whole number and \(\frac{2}{5}\) is the fractional part. Mixed numbers are often used in everyday measurements and can make it easier to understand quantities that are more than a whole but less than two whole units.
An essential step when working with mixed numbers, especially if you're finding reciprocals, is converting them to improper fractions. This way, you can perform arithmetic operations more straightforwardly.
To convert a mixed number to an improper fraction, you follow this simple method:

• Multiply the whole number by the denominator of the fractional part.
• Add this result to the numerator of the fractional part.
• Write this sum as the new numerator, keeping the original denominator.

This process transforms the mixed number into a single fraction, making it much easier to handle in calculations.

Improper Fractions

An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). For example, the improper fraction \(\frac{22}{5}\) indicates that there are more parts than a whole that can be made from the denominator.
Improper fractions can be seen as just another form of expressing mixed numbers. They are very useful in mathematical calculations because they allow for seamless operations without needing to deal with separate whole number and fractional parts.
To convert a mixed number like \(4\frac{2}{5}\) into an improper fraction, you multiply the whole number by the denominator and add it to the numerator, resulting in \(\frac{22}{5}\).
Using improper fractions can simplify arithmetic processes, especially when dealing with division or when finding reciprocals, as it reduces potential errors associated with mixed numbers.

Fraction Simplification

Fraction simplification means reducing a fraction to its simplest or smallest possible form so that both the numerator and denominator are as low as possible while still being integers. A simplified fraction has no common factors other than 1 in its numerator and denominator.
Take, for example, the fraction \(\frac{5}{22}\). To determine if this fraction is in its simplest form, you need to check if there is any number other than 1 that can divide both the numerator and the denominator.

• List the factors of each number.
• Find the greatest common factor (GCF).
• If the GCF is 1, your fraction is already simplified.

In the case of \(\frac{5}{22}\), since there are no common factors between 5 (whose factors are 1 and 5) and 22 (whose factors are 1, 2, 11, and 22) other than 1, the fraction is already in its lowest terms.
Simplification can make fractions easier to work with and compare, and often, simple fractions are more intuitive to understand.