Problem 43 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true sentence. $$-0.75 \circ-\frac{7}{9}$$

Step-by-Step Solution

Answer

-0.75 > -\frac{7}{9}.

Step 1: Convert Decimal to Fraction

The decimal \(-0.75\) can be converted to a fraction. The fraction equivalent is \(-\frac{3}{4}\) since 0.75 is \(\frac{75}{100}\), which simplifies to \(\frac{3}{4}\).

Step 2: Compare the Denominators

Let's compare the two fractions \(-\frac{3}{4}\) and \(-\frac{7}{9}\). First, convert them to have a common denominator. The least common denominator for 4 and 9 is 36.

Step 3: Convert Fractions to Common Denominator

Convert \(-\frac{3}{4}\) to having a denominator of 36: \[-\frac{3}{4} = -\frac{3 \times 9}{4 \times 9} = -\frac{27}{36}\]Convert \(-\frac{7}{9}\) to having a denominator of 36: \[-\frac{7}{9} = -\frac{7 \times 4}{9 \times 4} = -\frac{28}{36}\]

Step 4: Compare the Numerators

Now, compare the two fractions with the common denominator. We have: \(-\frac{27}{36}\) and \(-\frac{28}{36}\). Since \(-27\) is greater than \(-28\), thus \(-\frac{27}{36} > -\frac{28}{36}\).

Step 5: Conclude the Relationship

Based on the comparison, the correct relationship between \(-0.75\) and \(-\frac{7}{9}\) is \(-0.75 > -\frac{7}{9}\). Thus, the original expression becomes \(-0.75 > -\frac{7}{9}\).

Key Concepts

Fractions and Decimals

Fractions and decimals are different ways to represent numbers that are not whole numbers. Understanding how they relate is crucial in mathematical exercises, such as solving inequalities. When you see a number like \(-0.75\), it's in decimal form. This decimal can be converted into a fraction to simplify comparisons or calculations. For example, the fraction that corresponds to \(-0.75\) is \(-\frac{3}{4}\).

• Firstly, recognize that \(-0.75\) equals \(-\frac{75}{100}\) because decimals are always based on powers of 10.
• Then, reduce the fraction by finding the greatest common factor (GCF) of the numerator and denominator, which in this case is 25. Dividing both top and bottom by 25, we get \(-\frac{3}{4}\).

This conversion helps in better comparing with other fractions, which is useful when solving inequalities that involve both decimals and fractions.

Comparing Fractions

Comparing fractions might seem tricky at first, but with a systematic approach, it's quite manageable. The essential step is to find a common denominator, making it easy to compare the numerators directly.

• For instance, to compare \(-\frac{3}{4}\) and \(-\frac{7}{9}\), determine the Least Common Denominator (LCD). The LCD of 4 and 9 is 36 because it’s the smallest number that both denominators can divide into evenly.
• Next, convert each fraction to have this common denominator: \(-\frac{3}{4}\) becomes \(-\frac{27}{36}\) and \(-\frac{7}{9}\) becomes \(-\frac{28}{36}\).

Now you can easily compare their numerators: -27 is greater than -28, thus indicating \(-\frac{27}{36} > -\frac{28}{36}\). This is a neat strategy for handling fraction comparisons!

Mathematical Conversion

Mathematical conversion is a powerful skill that makes math easier by allowing you to switch between numbers in different forms. It involves several key processes:

• Turning decimals to fractions: As mentioned, \(-0.75\) becomes \(-\frac{3}{4}\), an easier form for comparison.
• Adjusting fractions to the same denominator: This is crucial for comparison, where changing \(-\frac{3}{4}\) and \(-\frac{7}{9}\) to have a denominator of 36 simplifies the task of direct comparison.

Mastering these conversions not only helps in problems involving fractions and decimals but also generally in algebra and beyond, reinforcing understanding and problem-solving efficiency.

Negative Numbers

Working with negative numbers requires careful consideration, especially when comparing their sizes. Unlike positive numbers, larger negative numbers are actually smaller because they lie further left on the number line.

• For example, \(-27\) is greater than \(-28\) since \(-27\) is closer to zero.
• When comparing fractions with negative numerators, as with \(-\frac{27}{36}\) and \(-\frac{28}{36}\), this distinction is essential as \(-27\) being closer to zero than \(-28\) makes the former greater. Keeping this in mind helps solve problems where negative numbers appear tricky.

Understanding the behavior of negatives ensures accuracy in calculations and comparisons, crucial in dealing with any math-related queries involving inequalities and negative values.