Problem 17 - Algebra for College Students

Question

Perform the following operations with real numbers. $$-2 \frac{3}{8}+5 \frac{7}{8}$$

Step-by-Step Solution

Answer

The result is \(3 \frac{1}{2}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert both mixed numbers into improper fractions. \(-2 \frac{3}{8}\) can be written as \(-\frac{19}{8}\) because \(-2 \times 8 + 3 = -19\). Similarly, \(5 \frac{7}{8}\) can be written as \(\frac{47}{8}\) because \(5 \times 8 + 7 = 47\).

Step 2: Add the Improper Fractions

The two fractions have a common denominator, which is 8. Add the fractions by combining their numerators: \(-\frac{19}{8} + \frac{47}{8} = \frac{-19 + 47}{8} = \frac{28}{8}\).

Step 3: Simplify the Fraction

Simplify \(\frac{28}{8}\) by finding the greatest common divisor (GCD) of 28 and 8, which is 4. Divide both the numerator and the denominator by 4: \(\frac{28}{8} = \frac{28 \div 4}{8 \div 4} = \frac{7}{2}\).

Step 4: Convert to a Mixed Number (if needed)

Convert \(\frac{7}{2}\) to a mixed number by performing division. Divide 7 by 2 to get 3 with a remainder of 1, so \(\frac{7}{2}\) can be expressed as \(3 \frac{1}{2}\).

Key Concepts

Improper Fractions

When working with mixed numbers, a common task is to convert them into improper fractions. This makes mathematical operations like addition simpler. An improper fraction is where the numerator (the top number) is larger than the denominator (the bottom number). For example, in the expression \(-2 \frac{3}{8}\), you need to transform it to an improper fraction to make calculations easier.

• Multiply the whole number part by the denominator.
• Add the numerator to the result from the first step.
• The sum becomes the new numerator, keeping the original denominator.

Using the method above, for \(-2 \frac{3}{8}\) you get \(-\frac{19}{8}\) because \(-2 \times 8 + 3 = -19\). This transformation is essential for performing further operations.

Mixed Numbers

Mixed numbers, which include a whole number and a fraction, are useful in representing values greater than one. However, they aren't ideal for direct arithmetic calculations, especially addition or subtraction.
When dealing with improper fractions and mixed numbers, it's useful to convert mixed numbers to improper fractions first.

• It simplifies calculations because you work with a single fraction rather than a combination of a whole number and a fraction.
• At the end of a problem, converting back from improper fractions can make your answer clearer.

For example, the improper fraction \(\frac{7}{2}\) can be converted back to a mixed number by dividing 7 by 2, giving \(3 \frac{1}{2}\). This makes it easier to interpret the result in practical contexts.

Fraction Simplification

Simplifying a fraction involves reducing it to its simplest form. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD).
For instance, with the fraction \(\frac{28}{8}\), the GCD of 28 and 8 is 4. Divide both 28 and 8 by 4 to get the simplified fraction \(\frac{7}{2}\).
Here’s how you simplify a fraction:

• Identify the largest number that both the numerator and denominator can be divided by, which is the GCD.
• Divide both the numerator and the denominator by the GCD.

Simplification not only makes a fraction easier to work with, but also often reveals a clearer picture of what the fraction represents. This step is crucial before converting back to mixed numbers or before making practical use of the result.

Addition of Fractions

Adding fractions can seem daunting, but it becomes simple once you understand the basics. The most important rule to remember when adding fractions is that they must have a common denominator.
In the problem given, \(-\frac{19}{8} + \frac{47}{8}\), these fractions already share the denominator 8. This makes the addition straightforward:

• Keep the denominator the same.
• Add the numerators: \(-19 + 47 = 28\).

The result is \(\frac{28}{8}\), which can then be simplified as needed.
Always ensure you simplify the resulting fraction, as was done in the example, to have the cleanest final answer.