Problem 60 - Intermediate Algebra

Question

Simplify each numerical expression. $$ -4 \frac{3}{5}-\left(1 \frac{1}{5}-2 \frac{3}{10}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{7}{2}\).

Step 1: Convert Mixed Numbers to Improper Fractions

The first step involves converting all mixed numbers to improper fractions. For \(-4 \frac{3}{5}\), calculate \(-\left(4 \times 5 + 3\right)/5 = -23/5\).For \(1 \frac{1}{5}\), calculate \(\left(1 \times 5 + 1\right)/5 = 6/5\).For \(2 \frac{3}{10}\), calculate \(\left(2 \times 10 + 3\right)/10 = 23/10\).

Step 2: Simplify the Expression Inside the Parentheses

Subtract the fractions inside the parentheses: \(1 \frac{1}{5} - 2 \frac{3}{10} = \frac{6}{5} - \frac{23}{10}\).First, find a common denominator, which is 10, so we convert half fractions: \(\frac{6}{5} = \frac{12}{10}\).Now, subtract: \(\frac{12}{10} - \frac{23}{10} = \frac{-11}{10}\).

Step 3: Subtract the Resulting Fractions

Subtract the result from Step 2 from \(-4 \frac{3}{5}\):The expression becomes \(-\frac{23}{5} - \left(-\frac{11}{10}\right)\).Convert \(-\frac{23}{5}\) to a common denominator of 10, \(-\frac{46}{10}\).Now, add: \(-\frac{46}{10} + \frac{11}{10} = -\frac{35}{10}\).

Step 4: Simplify the Final Fraction

Finally, simplify \(-\frac{35}{10}\). Divide both numerator and denominator by their greatest common divisor (5): \(-\frac{35}{10} = -\frac{7}{2}\).

Key Concepts

Mixed Numbers to Improper Fractions

Converting mixed numbers to improper fractions is an essential step in simplifying expressions that include fractions. A mixed number consists of a whole number and a fraction, such as \(-4 \frac{3}{5}\). When simplifying, it's easier to work with improper fractions, where the numerator is greater than or equal to the denominator.

Here's how you can convert a mixed number:

• Multiply the whole number part by the denominator of the fraction.
• Add the result to the numerator of the fraction.
• Place the sum above the original denominator to get your improper fraction.

For example, to convert \(1 \frac{1}{5}\), multiply \(1\) by \(5\) to get \(5\), then add \(1\), resulting in \(6\). So, \(1 \frac{1}{5}\) becomes \(\frac{6}{5}\).

Understanding this conversion is crucial as it sets the stage for further operations involving fractions.

Finding Common Denominators

Finding a common denominator is vital when you are dealing with the addition or subtraction of fractions. Since fractions represent parts of a whole, having a common size for these parts allows us to directly add or subtract the numerators.

Here's a simple way to find a common denominator:

• Identify the denominators of the fractions in question.
• Find the least common multiple (LCM) of these denominators; that will be your common denominator.
• Convert each fraction to an equivalent fraction with the found common denominator.

In our example, for fractions such as \(\frac{6}{5}\) and \(\frac{23}{10}\), the denominators are \(5\) and \(10\). The LCM is \(10\), so convert \(\frac{6}{5}\) to \(\frac{12}{10}\).

This step ensures fractions can be added or subtracted directly and accurately.

Subtracting Fractions

Once you have fractions with a common denominator, subtracting them is straightforward. You simply subtract the numerators, keeping the denominator the same.

Here’s a quick guide:

• Ensure both fractions have the same denominator.
• Subtract the second fraction's numerator from the first fraction's numerator.
• Write the result above the common denominator.

For instance, with \(\frac{12}{10}\) and \(\frac{23}{10}\), subtract to get \(\frac{-11}{10}\). It involves simple arithmetic with the numerators: \(12 - 23 = -11\).

Understanding and practicing this will greatly enhance your ability to work with fractions in various mathematical contexts.