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Chapter 5: Problem 4

Simplify to lowest terms. $$\frac{-16}{24}$$

Short Answer

Expert verified
Answer: $$\frac{-2}{3}$$.

Step by step solution

01

Find the Greatest Common Divisor

We will first find the greatest common divisor (GCD) of 16 and 24. The factors of 16 are {1, 2, 4, 8, 16}, and the factors of 24 are {1, 2, 3, 4, 6, 8, 12, 24}. The factors common to both numbers are {1, 2, 4, 8}, so the GCD is 8.
02

Simplify the fraction using the GCD

Now, we will divide both numerator and denominator by the GCD, which is 8: $$\frac{-16}{24} \div \frac{8}{8} = \frac{-16 \div 8}{24 \div 8} = \frac{-2}{3}$$ So the simplified fraction is: $$\frac{-2}{3}$$.

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Key Concepts

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

Greatest Common Divisor (GCD)
Understanding the Greatest Common Divisor (GCD) is crucial when it comes to simplifying fractions. GCD, also known as the greatest common factor, is the largest number that divides into two or more numbers without leaving a remainder. To find the GCD of two numbers, you list out the factors of each number and look for the largest number that appears in both lists. For instance, when we consider the numbers 16 and 24:
  • Factors of 16: \(1, 2, 4, 8, 16\)
  • Factors of 24: \(1, 2, 3, 4, 6, 8, 12, 24\)

Both lists share the numbers \(1, 2, 4, 8\), among which 8 is the largest, hence the GCD of 16 and 24 is 8. Identifying the GCD is the first and a very significant step in the process of fraction simplification as it helps in reducing the fraction to its lowest terms.
Reduce Fractions
To reduce a fraction, meaning to simplify it to its lowest terms, you divide the numerator and the denominator by their GCD. Reducing fractions is about making them easier to understand and work with. In the given example, we have the fraction \(\frac{-16}{24}\). Here's what reducing this fraction looks like in a clear, step-by-step process:
  • Start with the fraction \(\frac{-16}{24}\).
  • Find the GCD of the numerator and denominator, which is 8.
  • Divide both the numerator and denominator by the GCD: \(\frac{-16 \/ 8}{24 \/ 8}\).
  • You get the reduced fraction \(\frac{-2}{3}\).

Reduction makes fractions simpler and often easier to add, subtract, multiply, or divide in further calculations. It鈥檚 an essential skill in math and its applications.
Fraction Simplification Process
The fraction simplification process consists of several systematic steps that ensure any fraction is reduced to its simplest form. During this process, you'll be dividing both the numerator and the denominator by their GCD. Let's break it down even more:
  1. Identify the numerator (the top number) and the denominator (the bottom number) of the fraction.
  2. Calculate the GCD of these two numbers.
  3. Divide both the numerator and the denominator by their GCD to get the simplified fraction.

Using the example of \(\frac{-16}{24}\), we apply these steps. The GCD of 16 and 24 is 8. Upon dividing both -16 and 24 by 8, the fraction simplifies to \(\frac{-2}{3}\). This process is a fundamental element of mathematical problem-solving involving fractions, as it can greatly simplify the complexity of calculations and help students understand and compare fractions with ease.

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Most popular questions from this chapter

Add or subtract. $$-2 \frac{1}{6}-\left(-3 \frac{1}{2}\right)$$

Estimate and find the actual sum expressed as a mixed mumber in simplest form. $$10 \frac{5}{12}-6 \frac{5}{8}$$

Add or subtract. $$6 \frac{3}{4}+\left(-2 \frac{1}{3}\right)$$

Solve. Use \(\frac{22}{7}\) for \(\pi\). The radius of the bottom of the Liberty Bell is \(22 \frac{87}{88}\) inches. a. What is the diameter? b. What is the circumference?

Use the LCD to simplify the equation, then solve and check. $$3 b-\frac{1}{4}=\frac{1}{2}$$
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