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Chapter 3: Problem 109

For each number below, find a number to multiply it by to obtain 1. \(\frac{1}{3}\)

Short Answer

Expert verified
Multiply \(\frac{1}{3}\) by 3 to obtain 1.

Step by step solution

01

Understand the Problem

We need to find a number that, when multiplied by \(\frac{1}{3}\), results in 1.
02

Define the Multiplicative Inverse

The number we are looking for is called the *multiplicative inverse* or *reciprocal*. For any non-zero number \(a\), its multiplicative inverse is \(\frac{1}{a}\).
03

Apply the Rule to \(\frac{1}{3}\)

Since we are dealing with the fraction \(\frac{1}{3}\), we find its reciprocal by swapping the numerator and the denominator, which gives us 3.
04

Verify the Solution

Multiply \(\frac{1}{3}\) by its reciprocal 3: \(\frac{1}{3} \times 3 = 1\). The product is indeed 1, confirming that 3 is the correct multiplicative inverse.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Reciprocals
A reciprocal of a number is simply another number which, when multiplied with the original number, yields the result of 1. In more technical terms, the reciprocal is known as the multiplicative inverse. This concept applies to any non-zero number. For example, if you have a fraction like \(\frac{1}{3}\), its reciprocal would be 3.

To find the reciprocal of a fraction:
  • Swap the positions of the numerator and the denominator.
  • For a number such as \(\frac{a}{b}\), the reciprocal would be \(\frac{b}{a}\).
These principles are fundamental, especially when dealing with multiplication and division of fractions in mathematics.
Delving into Fractions
Fractions are expressions that represent a part of a whole, and they consist of two main elements: the numerator and the denominator. The numerator (the top number) indicates how many parts we have, while the denominator (the bottom number) shows how many equal parts make up a whole.

Understanding fractions is crucial because they are commonly used in problem-solving and real-life situations. For example, in the fraction \(\frac{1}{3}\), the numerator is 1 and the denominator is 3, indicating one part of something that is divided into three equal parts.
  • Dealing with fractions involves operations such as addition, subtraction, multiplication, and division.
  • Moreover, the concept of reciprocals directly links to fractions as swapping the numerator and denominator provides the reciprocal.
Grasping these ideas makes working with fractions much simpler.
Mastering Multiplication with Reciprocals
Multiplication involving reciprocals is an interesting concept, as it always results in the number 1. When you multiply a number by its reciprocal, the objective is to reach this unity value, demonstrating that these numbers effectively "cancel each other out."

Here's how this works with a fraction like \(\frac{1}{3}\):
  • Find the reciprocal of the fraction, which in this case is 3.
  • Multiply the original fraction by its reciprocal: \(\frac{1}{3} \times 3\).
  • Simplifying, you are left with 1 because the process negates the fraction with its inverse. \[ \frac{1}{3} \times \frac{3}{1} = \frac{3}{3} = 1 \]
This principle not only helps in simplifying complex calculations but also plays a key role in algebra and other higher mathematical computations.

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Most popular questions from this chapter

These problems review the four basic operations with fractions from this chapter. Perform the indicated operations. $$\frac{2}{3} \div 4$$

Divide. $$10 \div 3$$

These problems review the four basic operations with fractions from this chapter. Perform the indicated operations. $$8 \cdot \frac{5}{6}$$

Write 2 as a fraction with denominator 8 .

Multiply or divide as indicated. $$\frac{11}{12} \cdot \frac{10}{11} \cdot \frac{9}{10} \cdot \frac{8}{9} \cdot \frac{7}{8}$$
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