Problem 5 - Prealgebra

Question

Find each sum or difference. Write in simplest form. \(-2 \frac{4}{5}+\left(-\frac{2}{5}\right)\)

Step-by-Step Solution

Answer

The sum is \(-\frac{16}{5}\).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \(-2 \frac{4}{5}\) into an improper fraction. The whole number \(-2\) is converted by multiplying with the denominator \(5\): \(-2\times5 = -10\). Adding the numerator gives \(-10 + 4 = -14\). Thus, \(-2\frac{4}{5} = -\frac{14}{5}\).

Step 2: Add the Fractions

Now we add the two fractions: \(-\frac{14}{5}\) and \(-\frac{2}{5}\). Since the denominators are the same, we directly add the numerators: \(-14 + (-2) = -16\). This gives us the fraction \(-\frac{16}{5}\).

Step 3: Simplify the Fraction if Needed

Since \(-\frac{16}{5}\) is an improper fraction, we do not need to simplify it further because the numerator and denominator have no common factors other than 1, and the fraction is already in simplest form.

Key Concepts

Mixed Numbers

Mixed numbers are a combination of whole numbers and fractions. They are often used because they can be easier to visualize and work with than improper fractions. For instance, instead of saying "13 divided by 2," you might say "6 and a half," which is clearer and quicker to understand.

When dealing with arithmetic operations like addition or subtraction, mixed numbers need to be converted into improper fractions for ease of calculation. For example, a mixed number like \(2 \frac{4}{5}\) would need to be converted. This involves multiplying the whole number (2) by the denominator (5) and adding the numerator (4), resulting in an improper fraction. Learning to convert mixed numbers comfortably is an essential skill in managing fractions effectively.

Improper Fractions

Improper fractions have numerators that are larger than their denominators. While they may look a bit strange at first, they are rather useful, especially in calculations. They allow you to easily combine fractions since they maintain a single representation instead of splitting into whole numbers and fractions.

An improper fraction like \(-\frac{14}{5}\) can simply represent a larger division - meaning it portrays something that's more whole than a simple fraction but not complex like a mixed number. When performing operations with fractions, it makes the math more straightforward to have them in improper form because you only work with a single numerator and denominator throughout your calculations.

• Always convert mixed numbers to improper fractions before processing.
• They help manage subtraction, addition, and multiplication tasks more seamlessly.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. This means making the numerator and denominator as small as possible with whole numbers that still represent the same value as the original fraction.

To simplify a fraction correctly, identify any common factors between the numerator and the denominator. If they exist, divide both by their greatest common factor. Take the expression \(-\frac{16}{5}\); it's already in its simplest form because the greatest common factor between 16 and 5 is 1.

• Checking if a fraction can be simplified makes it easier to read and work with.
• If no common factors exist above 1, the fraction is already simplest.

Make your fractions easier for everyone to interpret by remembering these steps and always presenting them in their reduced form when possible.