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Chapter 4: Problem 57

Determine whether each pair of fractions is equivalent. \(\frac{3}{9}\) and \(\frac{6}{18}\)

Short Answer

Expert verified
Yes, the two fractions are equivalent.

Step by step solution

01

Simplify the First Fraction

To check if fractions are equivalent, we simplify each fraction to its lowest terms. Begin with \( \frac{3}{9} \). Find the greatest common divisor (GCD) of 3 and 9, which is 3. Divide both the numerator and the denominator by 3: \( \frac{3}{9} = \frac{1}{3} \).
02

Simplify the Second Fraction

Now simplify \( \frac{6}{18} \). Find the greatest common divisor (GCD) of 6 and 18, which is 6. Divide both the numerator and the denominator by 6: \( \frac{6}{18} = \frac{1}{3} \).
03

Compare the Simplified Fractions

Both fractions, \( \frac{3}{9} \) and \( \frac{6}{18} \), simplify to \( \frac{1}{3} \). Since their simplest forms are equal, the original fractions are equivalent.

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Key Concepts

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

Simplifying Fractions
Simplifying fractions means reducing the fraction to its smallest possible equivalent. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD). A fraction is fully simplified when no number other than 1 can divide both the numerator and the denominator evenly.
  • Start with identifying the GCD of the numbers in the fraction.
  • Divide both the numerator and the denominator by the GCD.
For example, in the fraction \( \frac{3}{9} \), the GCD of 3 and 9 is 3. Thus, we divide both by 3 to get \( \frac{1}{3} \). The same method applies to any fraction you want to simplify.
Greatest Common Divisor
The Greatest Common Divisor (GCD), also known as the greatest common factor, is the largest integer that divides two or more numbers without leaving a remainder. Understanding the GCD is crucial for simplifying fractions because it tells us precisely how much we can reduce both parts of a fraction.
Here's how to find the GCD:
  • List the factors of both numbers.
  • Identify the largest factor that appears in both lists.
For instance, the factors of 6 are 1, 2, 3, and 6. The factors of 18 are 1, 2, 3, 6, 9, and 18. The largest number they share is 6, so the GCD of 6 and 18 is 6.
Fraction Comparison
Fraction comparison often involves simplifying the fractions first. Once they are in their simplest form, you can directly compare them to see if they are equivalent or one is larger than the other. Equivalent fractions are two fractions that express the same portion of the whole, even if they look different.
To determine if two fractions are equivalent:
  • Simplify each fraction using the GCD.
  • Check if their simplest forms are identical.
If both simplified fractions are the same, like \( \frac{1}{3} \) from both \( \frac{3}{9} \) and \( \frac{6}{18} \), then the fractions are equivalent. This process ensures accurate comparison by removing the variations due to larger numerators and denominators.

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Most popular questions from this chapter

$$\text { Evaluate each expression if } x=\frac{3}{4} \text { and } y=-\frac{4}{7}$$ $$x^{2}+7 y$$

The Gauge Act of 1846 set the standard gauge for U.S. railroads at \(56 \frac{1}{2}\) inches. (See figure.) If the standard gauge in Spain is \(65 \frac{9}{10}\) inches, how much wider is Spain's standard gauge than the U.S. standard gauge? (Source: San Diego Railroad Museum)

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. See the Concept Checks in this section. Round the mixed number \(5 \frac{11}{12}\) to the nearest whole number

Write each improper fraction as a mixed number or a whole number. See Example 21. $$\frac{37}{8}$$
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