Problem 13 - Prealgebra

Question

Replace each with \(<,>,\) or \(=\) to make a true sentence. $$-\frac{3}{4} \circ-\frac{7}{8}$$

Step-by-Step Solution

Answer

\(-\frac{3}{4} > -\frac{7}{8}\)

Step 1: Understand the Signs of the Fractions

Both fractions given are negative: \(-\frac{3}{4}\) and \(-\frac{7}{8}\). When comparing negative fractions, the fraction closer to zero (less negative) is the greater number.

Step 2: Compare the Absolute Values

To compare \(-\frac{3}{4}\) and \(-\frac{7}{8}\), first compare their absolute values: \(\frac{3}{4}\) and \(\frac{7}{8}\). Since fractions have a common denominator, these can be compared directly.

Step 3: Convert to a Common Denominator

The denominators are 4 and 8. Convert \(\frac{3}{4}\) to have the same denominator as 8. Multiply the numerator and the denominator of \(\frac{3}{4}\) by 2 to get \(\frac{6}{8}\).

Step 4: Compare the New Fractions

Now compare \(\frac{6}{8}\) and \(\frac{7}{8}\). Since 6 is less than 7, \(\frac{6}{8}\) is less than \(\frac{7}{8}\).

Step 5: Translate to the Negative Fractions

Since \(\frac{6}{8} < \frac{7}{8}\), then \(-\frac{6}{8} > -\frac{7}{8}\). Hence, \(-\frac{3}{4} > -\frac{7}{8}\).

Key Concepts

Understanding Negative Numbers

Negative numbers are simply numbers that are less than zero. They are represented with a minus (-) sign. For example,

• -3 is a negative number.
• -1/2 is a negative fraction.

In the number line, negative numbers are found to the left of zero. When comparing negative numbers, the number closer to zero is always greater. For instance, -1 is greater than -2 because -1 is closer to zero on the number line. This rule also applies to fractions!

The Role of Absolute Values

The absolute value of a number is its "distance" from zero, without considering if it's negative or positive. To find the absolute value, you simply strip the negative sign.

• The absolute value of -4 is 4.
• The absolute value of \(-\frac{3}{4}\) is \(\frac{3}{4}\).

When we compare or work with negative numbers or fractions, like in our exercise, it's often helpful to first find their absolute values. This allows us to focus on the sizes of the numbers without getting confused by the negative signs.

How to Compare Fractions

Comparing fractions involves determining which fraction is larger or smaller. One way to compare fractions is by looking at their numerators and denominators. However, this is easier said than done!

• If fractions have the same denominator, simply compare their numerators.
• If they have different denominators, it's necessary to find a common baseline by converting them using a common denominator.

In our example, even with negative fractions, we first focused on their absolute values, converting fractions into a format where comparison is straightforward.

Finding Common Denominators

Having a common denominator means converting all fractions involved so they share the same denominator, which allows for straightforward comparison.

• This is crucial when comparing fractions that originally have different denominators, such as \(\frac{1}{3}\) and \(\frac{2}{5}\).
• In our specific problem, we made sure both \(\frac{3}{4}\) and \(\frac{7}{8}\) had the denominator 8 (transforming \(\frac{3}{4}\) into \(\frac{6}{8}\)).

Choosing a common denominator that is a multiple of both original denominators allows us to compare fractions easily by simply comparing their numerators.