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Chapter 6: Problem 117

To divide fractions, you must first know how to multiply fractions. Explain.

Short Answer

Expert verified
To divide fractions, you need to understand multiplication of fractions because dividing a fraction by another fraction involves flipping the second fraction (taking its reciprocal) and multiplying it with the first fraction.

Step by step solution

01

Understanding the Question

You need to understand that dividing is the opposite of multiplying. Similarly, in fractions, dividing one fraction by another is the same as multiplying that fraction by the reciprocal of the other fraction.
02

Illustrating the Process

Assume two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \). To divide the first fraction by the second \(\frac{a}{b} \div \frac{c}{d}\), you would actually have to multiply the first fraction by the reciprocal of the second fraction and that would be \(\frac{a}{b} \times \frac{d}{c}\) . This is why you need to understand multiplication of fractions.
03

Solving Further

Multiplying any two fractions involves multiplying the numerators with each other, and the denominators with each other. So for the example given, the result would be \(\frac{a \times d}{b \times c}\) .

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Key Concepts

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

Multiplying Fractions
Understanding the process of muliplying fractions is essential in many areas of mathematics, including when it comes to division. To multiply two fractions, simply multiply their numerators (the top numbers) to get the new numerator, and their denominators (the bottom numbers) to get the new denominator.

For example, if you multiply \( \frac{2}{3} \) by \( \frac{4}{5} \), you would calculate it as follows: \[ \frac{2 \times 4}{3 \times 5} = \frac{8}{15} \.\] This resulting fraction is your answer, and if possible, you should always simplify the fraction to its lowest terms.
Reciprocal of a Fraction
The reciprocal of a fraction is simply a way to flip the fraction upside down. In other words, you make the numerator the denominator, and the denominator the numerator.

It's crucial when dividing fractions because to divide one fraction by another, we multiply by the reciprocal of the divisor. For instance, if you have the fraction \( \frac{3}{4} \), its reciprocal would be \( \frac{4}{3} \). We use the concept of reciprocals because dividing by a number is the same as multiplying by its reciprocal.
Fraction Arithmetic
When it comes to fraction arithmetic, there are a few key operations to master: addition, subtraction, multiplication, and division. Each one has its own set of rules for dealing with the numerators and denominators.

Adding and Subtracting Fractions

For addition and subtraction, the denominators must be the same; if they are not, you'll need to find a common denominator.

Multiplying and Dividing Fractions

For multiplication, multiply the numerators together and the denominators together, as previously discussed. For division, multiply by the reciprocal of the fraction you are dividing by.
Numerators and Denominators
A fraction consists of two parts 鈥� a numerator and a denominator. The numerator, located above the fraction bar, indicates how many parts of the whole you have. The denominator, below the fraction bar, represents the total number of equal parts into which the whole is divided.

Understanding the roles of numerators and denominators is critical for performing all operations with fractions. For example, when adding fractions, if the denominators are not the same, you cannot simply add the numerators; you must first create a common denominator.

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