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Chapter 4: Problem 50

Perform each indicated operation. See Examples 1 through 15. $$ -100 \div \frac{1}{2} $$

Short Answer

Expert verified
-200

Step by step solution

01

Understand the Problem

We need to divide -100 by \( \frac{1}{2} \). When dividing by a fraction, we actually multiply by its reciprocal.
02

Identify the Reciprocal

The reciprocal of \( \frac{1}{2} \) is 2, since the reciprocal of any fraction \( \frac{a}{b} \) is \( \frac{b}{a} \).
03

Rewrite the Division as Multiplication

Convert the division into multiplication using the reciprocal. So, \( -100 \div \frac{1}{2} \) becomes \( -100 \times 2 \).
04

Multiply the Numbers

Now calculate \( -100 \times 2 \). Multiply the numbers: \(-100 \times 2 = -200\).
05

Verify the Sign

Since the multiplication involves a negative number and a positive number, the result will be negative. Thus, the final answer is -200.

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Key Concepts

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

Reciprocal of a Fraction
When you hear the term "reciprocal," think about flipping a fraction upside down. This operation is fundamental in division involving fractions. Every fraction, such as \( \frac{a}{b} \), has a reciprocal, which is \( \frac{b}{a} \). Reciprocals are useful because they turn division into multiplication, simplifying calculations.

For example, the reciprocal of \( \frac{1}{2} \) is 2. Why? Because:
  • You switch the numerator and the denominator of the fraction.
  • This means if you divide by \( \frac{1}{2} \), you effectively multiply by 2.
Using the reciprocal is a crucial step in dividing fractions, making it easier and more manageable to handle in arithmetic operations.
Multiplying Negative Numbers
Multiplying negative numbers might seem tricky at first, but here's a simple rule to remember: When you multiply a negative number by a positive number, the result is negative.

Why does this happen? Well, it's because you're essentially reversing the direction on the number line:
  • Think of multiplication as repeated addition. So, multiplying by a negative number reverses each step.
  • For example, \( -100 \times 2 \) means you add \(-100\) two times, resulting in \(-200\).
Keeping track of signs is important and helps ensure that the final answer reflects the true value.
Arithmetic Operations
Arithmetic operations form the building blocks of math, including addition, subtraction, multiplication, and division. When working with fractions and negative numbers, understanding these operations becomes vital.

Here's a quick guide to help you:
  • When dividing by a fraction, switch to multiplying by the reciprocal. This simplifies the process significantly.
  • Always pay attention to the signs. Multiplying or dividing a negative with a positive yields a negative.
  • Double-check your steps, especially with more complex equations, to ensure accuracy.
Mastering these operations paves the way for smoother mathematical problem-solving, ensuring you're well-equipped for any arithmetic challenge.

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Most popular questions from this chapter

Solve. Which of the following are equivalent to \(7 \frac{3}{4} ?\) a. \(6 \frac{7}{4}\) b. \(5 \frac{11}{4}\) c. \(7 \frac{12}{16}\) d. all of them

Solve. True or false? When the fraction \(\frac{7}{12}\) is rewritten with a denominator of \(48,\) the result is \(\frac{11}{48} .\) If false, give the correct fraction.

$$\text { Evaluate each expression if } x=\frac{3}{4} \text { and } y=-\frac{4}{7}$$ $$\frac{\frac{9}{14}}{x+y}$$

Find the prime factorization of each number. Two students work to prime factor \(120 .\) One student starts by writing 120 as \(12 \times 10 .\) The other student writes 120 as \(24 \times 5\). Finish each prime factorization. Are they the same? Why or why not?

Solve. A student asked you to check his work below. Is it correct? If not, where is the error? \(3 \frac{2}{3} \cdot 1 \frac{1}{7} \stackrel{?}{=} 3 \frac{2}{21}\)
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