Problem 28 - Fundamentals of Mathematics

Question

For the following problems, find the reciprocal of each number. $$5 \frac{3}{4}$$

Step-by-Step Solution

Answer

The reciprocal is \(\frac{4}{23}\).

Step 1: Convert Mixed Number to Improper Fraction

The first step is to convert the mixed number into an improper fraction. A mixed number like \(5 \frac{3}{4}\) consists of a whole number and a fraction. Multiply the whole number by the denominator of the fraction, and then add the numerator. The denominator remains the same. For this example: \(5 \times 4 + 3 = 20 + 3 = 23\). Therefore, the mixed number \(5 \frac{3}{4}\) converts to the improper fraction \(\frac{23}{4}\).

Step 2: Find the Reciprocal

To find the reciprocal of a fraction, we swap the numerator and the denominator. For the fraction \(\frac{23}{4}\), the reciprocal is \(\frac{4}{23}\) because we interchange 23 and 4.

Key Concepts

Understanding Mixed Numbers

A mixed number is a way to express numbers that combines a whole number and a proper fraction. It provides a clearer picture of quantities that are greater than a whole but less than the next whole number. For example, the mixed number \(5 \frac{3}{4}\) denotes 5 whole units and an additional \(\frac{3}{4}\) of another unit.
Working with mixed numbers often requires conversions, especially in operations like finding reciprocals. This involves turning the mixed number into an improper fraction, which is a vital step in making calculations simpler. Remember:

• The whole number is multiplied by the denominator of the fractional part.
• The result is added to the numerator of the fraction.
• The original denominator stays the same.

After following these steps, you'll have an improper fraction that's ready for further operations.

Converting to Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. They are useful because they can simplify arithmetic operations, such as multiplication and division.
Let's see how the conversion from a mixed number works with our example, \(5 \frac{3}{4}\):

• Multiply 5 (the whole number) by 4 (the denominator): \(5 \times 4 = 20\).
• Add the result to 3 (the numerator): \(20 + 3 = 23\).
• Your improper fraction is \(\frac{23}{4}\).

This transformation allows for easier application of mathematical procedures, like finding reciprocals or performing arithmetic operations. Improper fractions are often intermediate forms in math processes that need to be converted back to mixed numbers for final answers or interpretations.

The Role of Fractions and Reciprocals

Fractions represent parts of a whole and are crucial for dealing with quantities that are not complete. While proper fractions have numerators smaller than the denominators, improper fractions, like those from mixed numbers, have larger numerators.
The reciprocal of a fraction is another fraction obtained by swapping its numerator and denominator. This is particularly useful in division and solving equations. For example, the reciprocal of \(\frac{23}{4}\) is \(\frac{4}{23}\). Finding reciprocals:

• Interchanging the numerator and the denominator gives the reciprocal.
• They help in dividing fractions—multiplicative inverses.
• Reciprocals pair up to equal 1 when multiplied together (\(\frac{a}{b} \times \frac{b}{a} = 1\)).

Overall, understanding fractions and their reciprocals is fundamental to mastering many areas of mathematics.