Problem 61 - Prealgebra

Question

Find the reciprocal of each number. $$\frac{2}{3}$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Concept of Reciprocity

To find the reciprocal of a number, you need to switch its numerator (top part) and its denominator (bottom part). The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \).

Step 2: Apply the Reciprocal Process

Given fraction is \( \frac{2}{3} \). To find the reciprocal, swap the numerator and the denominator. This means the reciprocal is \( \frac{3}{2} \).

Step 3: Verifying the Reciprocal

Check that \( \frac{2}{3} \times \frac{3}{2} = 1 \), which confirms that they are indeed reciprocals of each other. Multiply the numerators: \( 2 \times 3 = 6 \) and the denominators: \( 3 \times 2 = 6 \). Therefore, \( \frac{6}{6} = 1 \).

Key Concepts

Numerator

In a fraction, the numerator is the top number. It represents how many parts you have out of a whole.
For example, in the fraction \( \frac{2}{3} \), the number 2 is the numerator.
This means there are 2 parts being considered out of 3 total parts. Understanding the numerator is crucial when dealing with fractions, as it allows you to know exactly what portion of the entire unit is being discussed. If you imagine a pizza cut into 3 equal slices, the numerator tells you that you have 2 of those slices.
Whenever you reverse the numerator and denominator to find a reciprocal, you're essentially swapping these parts around. This changes the fraction's relationship, but the numerator's role as 'the count of parts' remains essential.

Denominator

The denominator is the bottom part of a fraction. It tells us the total number of equal parts the whole is divided into.
In the fraction \( \frac{2}{3} \), the denominator is 3.
This indicates that whatever is being measured is split into 3 equal sections. In visual terms, think of a pie divided into 3 equal slices. The denominator 3 tells us the total number of slices—the entire thing is accounted for by these sections.
When you find the reciprocal of a fraction, the denominator's position is swapped with that of the numerator. This inversion helps form a new fraction that maintains a unique balance with its original, often leading to a product of 1 when multiplied together. Understanding the denominator enables you to grasp how parts are totalized within the context of a whole number or larger entity.

Fraction Multiplication

Fraction multiplication is the process of multiplying two fractions together. To do this, you multiply the numerators together and the denominators together.
Let's see an example with the reciprocals, \( \frac{2}{3} \) and its reciprocal \( \frac{3}{2} \):* Multiply the numerators: \( 2 \times 3 = 6 \)
* Multiply the denominators: \( 3 \times 2 = 6 \)
The result is \( \frac{6}{6} \), which equals 1. This shows that a fraction and its reciprocal always multiply to 1, confirming the concept of reciprocal relationships. This balance holds because you're effectively multiplying a number by its 'inversion,' canceling all out to one whole unit.
Understanding fraction multiplication is key in confirming reciprocal identities and performing many mathematical operations confidently.