Problem 84 - Elementary Algebra Within Reach

Question

In Exercises 79-84, evaluate the expression. $$ -7 \frac{3}{5}-3 \frac{1}{2} $$

Step-by-Step Solution

Answer

-11.1

Step 1: Convert Mixed Numbers to Fractions

The first step is to convert each mixed number into an improper fraction. \( -7 \frac{3}{5} \) converts to \( -\frac{38}{5} \) and \( 3 \frac{1}{2} \) converts to \( \frac{7}{2} \).

Step 2: Convert into Like Fractions

The next step is to convert the fractions into like fractions so they can be subtracted. The lowest common denominator for \( 5 \) and \( 2 \) is \( 10 \). So, multiply the numerator and denominator of \( -\frac{38}{5} \) by \( 2 \) to get \( -\frac{76}{10} \) and do the same for \( \frac{7}{2} \) to get \( \frac{35}{10} \).

Step 3: Subtract the Fractions

The final step is to subtract the two fractions using the rule a/b - c/d = (ad - bc) / bd. Using this rule, \( -\frac{76}{10} \) - \( \frac{35}{10} \) = \(- \frac{111}{10}\). This provides the final answer.

Key Concepts

Mixed Numbers

Mixed numbers are numbers that combine a whole number and a proper fraction. Examples of mixed numbers include expressions like \(3 \frac{1}{2}\) and \(7 \frac{3}{5}\). Understanding how to convert mixed numbers into improper fractions can simplify calculations like addition, subtraction, multiplication, and division. To perform this conversion, here's a simple approach:

• Multiply the whole number by the denominator of the fraction.
• Add the resulting product to the numerator of the fraction.
• Place this sum as the new numerator in the fraction, keeping the original denominator the same.

For example, converting \(7 \frac{3}{5}\) into an improper fraction involves these steps:
Multiply \(7 \times 5 = 35\), then add \(3\), resulting in \(38\). The improper fraction is \(\frac{38}{5}\). Performing these steps allows mixed numbers to be easily manipulated and simplified.

Improper Fractions

An improper fraction has a numerator that is larger or equal to its denominator, meaning the fraction's value is one or more. Improper fractions are often derived from mixed numbers for ease of computation. Here's a simple breakdown on how to convert mixed numbers to improper fractions, as seen in clearing fraction expressions:

• For \(7 \frac{3}{5}\), multiply: \(7 \times 5 = 35\) then add the fraction's numerator: \(35 + 3 = 38\). So, \(\frac{38}{5}\) is your improper fraction.
• When comparing or combining fractions, improper fractions provide a straightforward format for operations like addition and subtraction.

Working with improper fractions can make complex expressions more manageable, especially when finding like denominators is necessary.

Lowest Common Denominator

To effectively subtract fractions, they must have the same denominator, known as the lowest common denominator (LCD). The LCD is the smallest number that both denominators can divide into without leaving a remainder. To achieve this commonality:

• List the multiples of each denominator until a common number is found. For this exercise, the denominators \(5\) and \(2\) have a lowest common multiple of \(10\).
• Adjust the fractions to the LCD by multiplying both the numerator and denominator of each fraction to create equivalent fractions with the LCD.
• For instance, \(-\frac{38}{5}\) becomes \(-\frac{76}{10}\) and \(\frac{7}{2}\) becomes \(\frac{35}{10}\).

Utilizing the LCD simplifies the fraction subtraction process, making calculations more fluid and reducing errors.

Fraction Subtraction Steps

Subtracting fractions involves a few essential steps. First, ensure that the fractions have a common denominator, which allows their numerators to be directly subtracted.

• Begin with like fractions by converting them using the lowest common denominator, as discussed in previous sections.
• Subtract the second fraction's numerator from the first, keeping the common denominator unchanged.
• In the example, subtract \(-\frac{76}{10}\) from \(\frac{35}{10}\). The numerators are calculated as \(-76 - 35 = -111\), giving \(-\frac{111}{10}\).

This result represents the difference, which can then be simplified if possible. Understanding these steps can greatly enhance your ability to manage and calculate similar problems efficiently.