Problem 21 - Prealgebra

Question

Find the multiplicative inverse of each number. $$5 \frac{1}{4}$$

Step-by-Step Solution

Answer

The multiplicative inverse of \(5 \frac{1}{4}\) is \(\frac{4}{21}\).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \(5 \frac{1}{4}\) into an improper fraction. Multiply the whole number part, 5, by the denominator of the fractional part, 4, and add the numerator of the fractional part, 1. The calculation is: \(5 \times 4 + 1 = 20 + 1 = 21\). Thus, \(5 \frac{1}{4}\) becomes \(\frac{21}{4}\).

Step 2: Find the Multiplicative Inverse

The multiplicative inverse (or reciprocal) of a fraction \(\frac{a}{b}\) is the fraction \(\frac{b}{a}\). For the fraction \(\frac{21}{4}\), the multiplicative inverse is \(\frac{4}{21}\).

Step 3: Verify the Inverse

To verify, multiply the original fraction by its multiplicative inverse. The result should be 1. So, multiply \(\frac{21}{4}\) by \(\frac{4}{21}\): \(\frac{21}{4} \times \frac{4}{21} = \frac{21 \times 4}{4 \times 21} = \frac{84}{84} = 1\). The calculation confirms that \(\frac{4}{21}\) is indeed the multiplicative inverse of \(5 \frac{1}{4}\).

Key Concepts

Understanding Mixed Numbers

Mixed numbers combine a whole number with a fraction. They are often used in everyday language when dealing with measurements and quantities. For example, if you have 5 whole cakes and a quarter of another, you would express this as \(5 \frac{1}{4}\). This representation can make certain addition and subtraction problems more intuitive. However, in mathematical operations like multiplication or division, it's generally easier to work with improper fractions. To convert a mixed number into an improper fraction, use this simple process:

• Multiply the whole number by the denominator of the fractional part.
• Add the numerator to this product.
• The result becomes the numerator of your improper fraction, with the original denominator staying the same.

By following these steps, the mixed number \(5 \frac{1}{4}\) transforms into the improper fraction \(\frac{21}{4}\).

Improper Fractions Explained

Improper fractions are fractions where the numerator is equal to or greater than the denominator. They might look a bit unusual compared to proper fractions, but they play a vital role in mathematics.

• The numerator represents a count of parts, each part being 1 denominator of the whole.
• This means an improper fraction, like \(\frac{21}{4}\), represents more than a single whole.
• Such fractions are particularly useful for multiplication and division as they eliminate the need for separate whole and fractional parts.

By understanding an improper fraction, you normalize its computation, thus making it straightforward to find things like their multiplicative inverses or to compare and order them in various mathematical contexts.

The Reciprocal and Multiplicative Inverse

The reciprocal is another term for the multiplicative inverse. Using fractions, finding the reciprocal is quite simple:

• Just swap the numerator and the denominator.
• If you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).
• This operation transforms the fraction so that if it's multiplied by its reciprocal, it equals 1.

For example, the improper fraction \(\frac{21}{4}\) has a reciprocal of \(\frac{4}{21}\). Multiplying these two together confirms the concept: \(\frac{21}{4} \times \frac{4}{21} = \frac{84}{84} = 1\). This demonstrates that the reciprocal is, in fact, the multiplicative inverse. Recognizing and utilizing reciprocals is fundamental in solving equations and simplifying expressions across various branches of mathematics.