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Chapter 4: Problem 202

In the following exercises, simplify the complex fraction. $$\frac{-\frac{4}{5}}{2}$$

Short Answer

Expert verified
\(\frac{-2}{5}\)

Step by step solution

01

Understand the Problem

The complex fraction to simplify is \(\frac{-\frac{4}{5}}{2}\).
02

Rewrite the Fraction

Rewrite the complex fraction as the product of fractions. This means dividing by 2 is the same as multiplying by its reciprocal: \(\frac{1}{2}\). So, \(\frac{-\frac{4}{5}}{2} = -\frac{4}{5} \times \frac{1}{2}\).
03

Multiply the Fractions

Multiply the numerators together and the denominators together: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\). In this case, it becomes \(\frac{-4 \times 1}{5 \times 2} = \frac{-4}{10}\).
04

Simplify the Resulting Fraction

Simplify \(\frac{-4}{10}\) by finding the greatest common divisor (GCD) of 4 and 10, which is 2. Divide both the numerator and the denominator by 2: \(\frac{-4 \div 2}{10 \div 2} = \frac{-2}{5}\).

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Key Concepts

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

Fraction Multiplication
In simplifying fractions, multiplication plays a vital role. When you need to multiply two fractions, multiply their numerators together and their denominators together.
For instance, with fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), the product is \(\frac{a \times c}{b \times d}\).
For example:
  • \(\frac{3}{4} \times \frac{2}{5} = \frac{3 \times 2}{4 \times 5} = \frac{6}{20} = \frac{3}{10}\)
Always remember to simplify the final fraction if possible.
Simplifying involves dividing numerator and denominator by their greatest common divisor (GCD). This ensures that the fraction is in its simplest form and easier to understand.
Reciprocal of a Number
Understanding reciprocals is critical when dealing with complex fractions. The reciprocal of a number is simply one divided by that number.
For example, if you have a number \(x\), its reciprocal is \(1/x\).
Here are a few points to grasp this concept:
  • The reciprocal of \(2\) is \( \frac{1}{2} \).
  • For any fraction \( \frac{a}{b} \), its reciprocal is \(\frac{b}{a}\).
  • Multiplying a number by its reciprocal always results in \(1\), because \(x \times \frac{1}{x} = 1\).
In the original exercise, we simplified by multiplying by the reciprocal of \(2\), which is \( \frac{1}{2} \). This technique transformed the division problem into a multiplication problem, which is typically easier to handle.
Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) is crucial for simplifying fractions.
It allows us to reduce fractions to their simplest form by dividing both the numerator and the denominator by their GCD.
Here's a step-by-step guide to find the GCD:
  • List the factors of each number.
    For instance, for \(4\), the factors are \{1, 2, 4\}. For \(10\), the factors are \{1, 2, 5, 10\}.
  • Identify the common factors.
    In our example, the common factors are \(1\) and \(2\).
  • Select the largest common factor.
    The GCD of \(4\) and \(10\) is \(2\).
In the exercise, we took the fraction \( \frac{-4}{10} \) and simplified it by dividing both the numerator and the denominator by \(2\), leading to \( \frac{-2}{5}\), the fraction's simplest form.

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Most popular questions from this chapter

In the following exercises, perform the indicated operations and simplify. $$-\frac{11}{36}-\frac{11}{20}$$

In the following exercises, simplify. $$ \frac{9 a}{10} \div \frac{15 a}{8} $$

In the following exercises, subtract. $$\frac{8}{11}-\frac{6}{11}$$

In the following exercises, solve the equation. $$ a-\frac{3}{10}=-\frac{9}{10} $$

In the following exercises, perform the indicated operations and simplify. $$\frac{\left(\frac{2}{3}\right)^{2}}{\left(\frac{5}{8}\right)^{2}}$$
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