/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 122 Simplify. $$ -\frac{2}{3} \d... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 122

Simplify. $$ -\frac{2}{3} \div \frac{4}{9} $$

Short Answer

Expert verified
-\( \frac{3}{2} \)

Step by step solution

01

- Rewrite the Division as Multiplication

To simplify the division of fractions, rewrite the division as a multiplication by the reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator. So, rewrite \( -\frac{2}{3} \div \frac{4}{9} \) as \( -\frac{2}{3} \times \frac{9}{4} \).
02

- Multiply the Fractions

Multiply the numerators together and the denominators together. This gives \( -\frac{2 \times 9}{3 \times 4} \).
03

- Simplify the Fraction

Calculate the product in both the numerator and the denominator. This results in \( -\frac{18}{12} \). Now, simplify this fraction by finding the greatest common divisor (GCD) of 18 and 12, which is 6. Thus, \( -\frac{18}{12} = -\frac{18 \div 6}{12 \div 6} = -\frac{3}{2} \).

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Key Concepts

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

Reciprocals
When we talk about reciprocals in mathematics, we refer to flipping a fraction upside down. The numerator becomes the denominator, and the denominator becomes the numerator. For example, the reciprocal of \(\frac{4}{9}\) is \(\frac{9}{4}\). Reciprocals are especially important in fraction division. When you divide by a fraction, you actually multiply by its reciprocal. This is a helpful way to think about division problems involving fractions.
Fraction Multiplication
Multiplying fractions might seem challenging at first, but it's straightforward if you break it down. To multiply fractions, follow these steps:
  • Multiply the numerators together.
  • Multiply the denominators together.
For instance, let鈥檚 multiply \(\frac{2}{3}\) by \(\frac{9}{4}\). You multiply 2 by 9 to get 18 and 3 by 4 to get 12. So, \(\frac{2}{3} \times \frac{9}{4} = \frac{18}{12}\). Remember to simplify the resulting fraction, which brings us to the next concept.
Simplifying Fractions
Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator have no common factors other than 1. For example, let's take \(\frac{18}{12}\). We need to find the greatest common divisor (GCD) of 18 and 12. Dividing both the numerator and the denominator by their GCD will give us our simplified fraction. Simplification makes fractions easier to understand and work with, particularly in more complex problems.
Greatest Common Divisor
The greatest common divisor (GCD) is the largest integer that divides both the numerator and the denominator evenly. Finding the GCD involves identifying the common factors of two numbers. For \(\frac{18}{12}\), the common factors are 1, 2, 3, and 6. The largest of these is 6. So, dividing both parts of the fraction by 6, we get \(\frac{18 \div 6}{12 \div 6} = \frac{3}{2}\). Using the GCD to simplify fractions provides a clear and accurate way to reduce them to their simplest form.

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Most popular questions from this chapter

Find the domain of the function \(f\) given by each of the following. $$f(x)=\frac{x-5}{9 x-18 x^{2}}$$

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