Problem 17 - Prealgebra

Question

Find the multiplicative inverse of each number. $$\frac{6}{11}$$

Step-by-Step Solution

Answer

The multiplicative inverse of \( \frac{6}{11} \) is \( \frac{11}{6} \).

Step 1: Understanding the Problem

The multiplicative inverse of a number is another number that, when multiplied with the original number, gives the result of 1. For a fraction \( \frac{a}{b} \), the multiplicative inverse is \( \frac{b}{a} \).

Step 2: Identify the Fraction Components

The given fraction is \( \frac{6}{11} \). Here, the numerator \( a = 6 \) and the denominator \( b = 11 \).

Step 3: Find the Inverse

To find the multiplicative inverse, simply swap the numerator and the denominator of the fraction. Thus, the inverse of \( \frac{6}{11} \) is \( \frac{11}{6} \).

Step 4: Verify the Inverse

Multiplying the original fraction with its inverse should yield 1: \( \frac{6}{11} \times \frac{11}{6} = \frac{6 \times 11}{11 \times 6} = \frac{66}{66} = 1 \). This confirms that \( \frac{11}{6} \) is indeed the multiplicative inverse.

Key Concepts

Understanding Fractions

Fractions are an essential part of mathematics. They express a portion of a whole and are written in the form \( \frac{a}{b} \). Here, \( a \) and \( b \) are integers. The top number, \( a \), is called the numerator, and it tells us how many parts of the whole there are. On the other hand, the bottom number, \( b \), is the denominator, and it indicates the total number of equal parts the whole is divided into.

Fractions can be greater than 1 if the numerator is larger than the denominator (this is called an improper fraction), or they can be less than 1 if the numerator is smaller. For example, \( \frac{3}{4} \) is less than 1, and \( \frac{7}{4} \) is more than 1. Understanding these distinctions is crucial for manipulating and identifying equivalents or inverses.

Role of the Numerator and Denominator

In any fraction, both the numerator and denominator play crucial roles in determining its value. When we talk about swapping them to find a multiplicative inverse, these roles get reversed. In our original fraction, \( \frac{6}{11} \), the numerator is 6, and the denominator is 11. By exchanging them, you create a new fraction \( \frac{11}{6} \).

This switching of positions essentially transforms the fraction's function, as making the numerator the larger number signifies an improper fraction. It's important because this new fraction, when multiplied with the original, must produce a whole (more specifically, number 1) to verify the multiplicative inverse.

Verifying Results for Accuracy

Verification is a vital step to complete when dealing with inverses. It's how you confirm that the calculations were successful. For fractions, this means checking that the product of a fraction and its inverse equals 1.

Let's take the exercise example again: multiply the original \( \frac{6}{11} \) by its inverse \( \frac{11}{6} \). When you perform the multiplication:

• Multiply the numerators: \( 6 \times 11 = 66 \)
• Multiply the denominators: \( 11 \times 6 = 66 \)

This multiplication results in \( \frac{66}{66} \), which simplifies to 1. This confirms that our inverse calculation is correct. Such verification ensures accuracy and understanding of how multiplicative inverses work with fractions, solidifying concepts that are universally applicable in mathematics.