Problem 53 - Beginning and Intermediate Algebra

Question

Find each reciprocal or multiplicative inverse. $$ \frac{2}{3} $$

Step-by-Step Solution

Answer

The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).

Step 1: Understanding the Reciprocal

The reciprocal (or multiplicative inverse) of a fraction is obtained by swapping its numerator and denominator. For a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).

Step 2: Applying the Rule

Given the fraction \( \frac{2}{3} \), we swap the numerator and the denominator. This means that the reciprocal of \( \frac{2}{3} \) is found by flipping the fraction to become \( \frac{3}{2} \).

Step 3: Verifying the Solution

To verify, multiply the original fraction \( \frac{2}{3} \) by its reciprocal \( \frac{3}{2} \). The product should be 1: \( \frac{2}{3} \times \frac{3}{2} = \frac{2 \times 3}{3 \times 2} = \frac{6}{6} = 1 \). This confirms that the found reciprocal is correct.

Key Concepts

Multiplicative Inverse

A multiplicative inverse is a key concept in mathematics, especially when dealing with fractions. The multiplicative inverse of a number is another number which, when multiplied with the original, results in 1. For fractions, this idea is straightforward: you simply swap the numerator and the denominator. For example, the multiplicative inverse of \( \frac{2}{3} \) is \( \frac{3}{2} \).
This operation is crucial because it helps simplify equations and solve for unknowns. Understanding the multiplicative inverse can make complex problems seem much more manageable. Remember, every non-zero fraction has a multiplicative inverse.

Verifying Reciprocal

After finding what you believe is the multiplicative inverse, it's important to verify that the reciprocal is correct. Verification strengthens your understanding and assures accuracy.
To verify, multiply the original fraction by its reciprocal. For example, if you take the original fraction \( \frac{2}{3} \), its reciprocal is \( \frac{3}{2} \). When these two are multiplied, the result should be 1:

• \( \frac{2}{3} \times \frac{3}{2} = \frac{6}{6} = 1 \)

This product confirms that you have the correct reciprocal. This simple check ensures that the calculations are accurate and the reciprocal is truly a multiplicative inverse.

Fractions Operations

Working with fractions involves understanding their operations, such as addition, subtraction, multiplication, and division. Each operation has rules that make it unique.
Adding or subtracting fractions involves finding a common denominator. However, when it comes to multiplication, you simply multiply the numerators and the denominators across:

• \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \)

For division, remember "invert and multiply." This is where the concept of reciprocal becomes crucial. You take the reciprocal of the divisor and multiply:

• \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \)

Understanding these operations is essential for moving confidently through fraction-based math problems, ensuring accuracy and efficiency in your calculations.