Problem 27 - Fundamentals of Mathematics

Question

For the following problems, find the reciprocal of each number. $$3 \frac{2}{7}$$

Step-by-Step Solution

Answer

The reciprocal of \(3 \frac{2}{7}\) is \(\frac{7}{23}\).

Step 1: Convert the Mixed Number to an Improper Fraction

To find the reciprocal of a mixed number like \(3 \frac{2}{7}\), first, we need to convert it into an improper fraction. Multiply the whole number part (3) by the denominator (7), then add the numerator (2): \(3 \times 7 + 2 = 21 + 2 = 23\). So, \(3 \frac{2}{7} = \frac{23}{7}\).

Step 2: Find the Reciprocal of the Improper Fraction

The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). So, the reciprocal of \(\frac{23}{7}\) is \(\frac{7}{23}\).

Step 3: Verify the Reciprocal

Multiply the original improper fraction by its reciprocal to ensure they yield 1: \(\frac{23}{7} \times \frac{7}{23} = \frac{161}{161} = 1\). This confirms that \(\frac{7}{23}\) is the correct reciprocal.

Key Concepts

Mixed Number

A mixed number combines a whole number with a fraction. It looks like this: 3\( \frac{2}{7} \). Mixed numbers are helpful because they portray quantities greater than 1 in a more intuitive way.
To convert a mixed number to an improper fraction, follow these steps:

• Multiply the whole number by the denominator of the fraction. For example, in 3\( \frac{2}{7} \), multiply 3 by 7 to get 21.
• Add the numerator to this result. So, 21 plus 2 equals 23.
• Use this sum as the new numerator over the original denominator. This gives \( \frac{23}{7} \).

These steps are crucial for operations like finding reciprocals or multiplying fractions, providing a consistent format for calculations.

Improper Fraction

Improper fractions have numerators larger than or equal to their denominators, such as \( \frac{23}{7} \). They represent the same value as mixed numbers, but in a different format.
Improper fractions are easy to work with since they can be directly used in mathematical operations, like multiplication, without additional conversion.
To convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator.
• Add the numerator from the fraction part.
• Place this result over the original denominator.

This conversion is the first step when dealing with mixed numbers in operations.

Fraction Multiplication

Fraction multiplication is straightforward yet essential in mathematics. It involves multiplying the numerators and the denominators separately.
Here's how it works: if you multiply \( \frac{a}{b} \) by \( \frac{c}{d} \), you do \( \frac{a \times c}{b \times d} \). This principle is useful for verifying reciprocals.
When you multiply a fraction by its reciprocal, the result will always be 1, as shown in the exercise: \( \frac{23}{7} \times \frac{7}{23} = \frac{161}{161} = 1 \).
Steps for multiplying fractions:

• Multiply the numerators to get the new numerator.
• Multiply the denominators to get the new denominator.
• Simplify the fraction if possible.

Understanding fraction multiplication helps in numerous mathematical problems.

Mathematics Education

In mathematics education, concepts like mixed numbers, improper fractions, and reciprocals are foundational. They are core ideas introduced in early learning and built upon as students progress.
The ability to convert between different forms of numbers, such as mixed numbers to improper fractions, is essential for succeeding in more advanced topics.
Educational strategies often include:

• Visual aids to help students grasp abstract concepts like fractions.
• Practical examples showing how these concepts apply in real-world contexts.
• Interactive activities that engage students in learning by doing.

Developing these skills forms a critical part of mathematics education, empowering students with the tools necessary for more complex problem solving.