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Chapter 4: Problem 72

Fill in the blanks with \(<,>,\) or \(=\). $$-\frac{4}{20} \square-\frac{6}{30}$$

Short Answer

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Step by step solution

01

Simplify the fractions

Simplify both fractions to their lowest terms. First, simplify \(-\frac{4}{20}\). The greatest common divisor (GCD) of 4 and 20 is 4. Thus, \(-\frac{4}{20} = -\frac{1}{5}\). Next, find the GCD of 6 and 30, which is 6. Hence, \(-\frac{6}{30} = -\frac{1}{5}\).
02

Compare the simplified fractions

Now compare \(-\frac{1}{5}\) and \(-\frac{1}{5}\). Since both fractions are equal, the correct symbol to use is \(=\).

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Key Concepts

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

simplifying fractions
Simplifying fractions makes comparing them much easier. To simplify a fraction, divide both the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD).
For example, let's simplify \(\frac{4}{20}\).
- First, find the highest number that can evenly divide both 4 and 20. This number is 4.
- Divide both the numerator and denominator by 4: \(\frac{4 \div 4}{20 \div 4} = \frac{1}{5}\).
Now, repeat the process for \(\frac{6}{30}\). The GCD of 6 and 30 is 6:
- Divide both the numerator and denominator by 6: \(\frac{6 \div 6}{30 \div 6} = \frac{1}{5}\).
Always remember: simplifying fractions helps make comparisons straightforward!
greatest common divisor
The Greatest Common Divisor (GCD) is the largest number that can divide both the numerator and the denominator of a fraction without leaving a remainder.
To find the GCD of two numbers, you can use several methods, like listing out the factors or using the Euclidean algorithm. Let's see how this works with \(\frac{4}{20}\):
- Factors of 4 are \{1, 2, 4\}.
- Factors of 20 are \{1, 2, 4, 5, 10, 20\}.
The largest common factor is 4, which is the GCD.
Using the Euclidean algorithm:
- Divide 20 by 4 to get 5 (remainder is 0).
- Since the remainder is 0, the divisor (4) is the GCD.
This shows why identifying the GCD is essential for simplifying fractions.
fraction comparison
Comparing fractions is easier when they are simplified. Once you simplify a fraction, you can directly compare the values.
Here's an example of comparing two fractions: \(-\frac{4}{20}\) and \(-\frac{6}{30}\):
- Simplify \(-\frac{4}{20}\) to \(-\frac{1}{5}\).
- Simplify \(-\frac{6}{30}\) to \(-\frac{1}{5}\).
Now both fractions are simplified to \(-\frac{1}{5}\).
Since the two values are identical, we use the symbol \ = \ to show they are equal.
By simplifying first, you ensure that comparison becomes a straightforward task.
Whether fractions are negative (like in our example) or positive, the approach remains the same: simplify first, then compare.

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Most popular questions from this chapter

Simplify the fraction to lowest terms. Write the answer as a fraction or a whole number. $$\frac{69}{92}$$

Simplify to lowest terms by first reducing the powers of 10. $$\frac{120}{160}$$

Convert the mixed number to an improper fraction. $$12 \frac{1}{6}$$

Divide. Write the quotient as a mixed number. $$695 \div 34$$

For a recent year, the population of the United States was reported to be \(296,000,000\). During the same year, the population of California was \(36,458,000\). a. Round the U.S. population to the nearest hundred million. b. Round the population of California to the nearest million. c. Using the results from parts (a) and (b), write a simplified fraction showing the portion of the U.S. population represented by California.
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