Problem 71 Fill in the blanks with \(<,&... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

$Math Studyset 91影视 Explanations$ Math

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Chapter 4: Problem 71

Fill in the blanks with $<,>,$ or $=$. $$-\frac{3}{12} \square-\frac{2}{8}$$

Short Answer

Expert verified

-\frac{3}{12} = -\frac{2}{8}

Step by step solution

Simplify the Fractions

First, simplify both fractions to their lowest terms. For $-\frac{3}{12}$, divide the numerator and the denominator by their greatest common divisor, which is 3. $-\frac{3}{12} = -\frac{1}{4}$. For $-\frac{2}{8}$, divide the numerator and the denominator by their greatest common divisor, which is 2. $-\frac{2}{8} = -\frac{1}{4}$.

Compare Simplified Fractions

Now compare $-\frac{1}{4}$ to $-\frac{1}{4}$. Since both fractions are equal, the relationship 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 means reducing them to their lowest terms. This makes it easier to compare them.

Here's how to simplify any fraction:

1. Find the greatest common divisor (GCD) of the numerator and the denominator.
2. Divide both the numerator and the denominator by this GCD.

For example, if we start with $-\frac{3}{12}$, we can simplify it by finding the GCD of 3 and 12, which is 3. Dividing both terms by 3, we get $-\frac{1}{4}$.
Similarly, for $-\frac{2}{8}$, the GCD of 2 and 8 is 2. Dividing both terms by 2, we obtain $-\frac{1}{4}$.
Sometimes, finding the GCD can be tricky, but a simple way to approach it is to break down the numbers into their prime factors. The GCD will be the product of the lowest power of common prime factors.

Comparing Fractions

Comparing fractions involves more than just looking at the numerators or denominators. It's important to understand the value represented by each fraction.

If the fractions have the same denominators, compare the numerators directly.
If they have different denominators, you may need to bring both fractions to a common denominator before comparing.

In our exercise, after simplifying both fractions to $-\frac{1}{4}$, it becomes clear that \'-1' in both fractions signifies that they are equivalent. Therefore, $-\frac{1}{4} = -\frac{1}{4}$.
Generally speaking, for fractions with the same denominator, the fraction with the larger numerator is the larger fraction, and vice versa. With fractions having different denominators, finding a common denominator helps lay the groundwork for a fair comparison.

Negative Fractions

Negative fractions can sometimes be confusing, but they follow the same rules as positive fractions鈥攐nly the direction of inequality changes.

A fraction with a larger negative numerator (in absolute value) is smaller.
For example, \-\frac{1}{4}\ is larger than \-\frac{2}{4}\.

In our exercise, $-\frac{3}{12}$ and $-\frac{2}{8}$ simplified to \-\frac{1}{4}\ respectively, indicating they are equal.
Moreover, if we were comparing \-\frac{1}{3}\ and \-\frac{1}{4}\, \-\frac{1}{4}\ would be the larger fraction as $-\frac{1}{4}$ is closer to 0 on the number line than $-\frac{1}{3}$.
Remember this rule, and negative fractions will become much less intimidating to handle.

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91影视

Fill in the blanks with \(<,>,\) or \(=\). $$-\frac{3}{12} \square-\frac{2}{8}$$

Short Answer

Step by step solution

Simplify the Fractions

Compare Simplified Fractions

Key Concepts

Simplifying Fractions

Comparing Fractions

Negative Fractions

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