Problem 63 - Elementary and Intermediate Algebra

Question

Add. $$ -\frac{9}{16}+\frac{7}{16} $$

Step-by-Step Solution

Answer

The sum is \(-\frac{1}{8}\).

Step 1: Identify the Problem

We are asked to add the fractions \(-\frac{9}{16}\) and \(\frac{7}{16}\). Both fractions have the same denominator.

Step 2: Add the Numerators

Since the denominators are the same, we can directly add the numerators of the fractions: \(-9 + 7\).

Step 3: Perform the Addition

Calculate the sum of the numerators: \(-9 + 7 = -2\).

Step 4: Write the Resulting Fraction

Use the sum from Step 3 over the common denominator: \(-\frac{2}{16}\).

Step 5: Simplify the Fraction

Simplify \(-\frac{2}{16}\) by dividing both the numerator and the denominator by 2, giving \(-\frac{1}{8}\).

Key Concepts

Same Denominator

When working with fractions, one key step in addition is ensuring that both fractions have the same denominator. A denominator is the bottom number in a fraction, representing how many equal parts the whole is divided into.

If the denominators are the same, it means that both fractions represent parts of the same whole. This makes the process of adding the fractions significantly easier because you can focus solely on adding the numerators.

Having a common denominator means you don’t need to find equivalent fractions or worry about cross-multiplying. For example, in \(-\frac{9}{16}\) and \(\frac{7}{16}\), the denominator is already the same: 16. To proceed, you simply keep this denominator consistent and focus on the numerators.

Adding Numerators

Once you have confirmed the same denominator, the next step is to add the numerators. The numerator is the top number in a fraction, indicating how many parts of the whole you have.

With equal denominators, simply add the numerators together while retaining the common denominator. In our problem, the numerators are \(-9\) and \(7\). So, you need to perform the calculation:

• \(-9 + 7 = -2\)

This result is placed over the common denominator of 16, creating a new fraction: \(-\frac{2}{16}\).

Remember, the operation only affects the numerators when dealing with like denominators. This step is as straightforward as standard integer addition—just be cautious of the signs! In this case, subtracting instead of adding, as you are dealing with a negative number.

Simplifying Fractions

After adding fractions, you might end up with a fraction that is not in its simplest form. Simplifying fractions involves reducing the fraction to its smallest possible equivalent by removing common factors from the numerator and the denominator.

Start by identifying the greatest common factor (GCF) that the numerator and denominator share.
In \(-\frac{2}{16}\), this GCF is 2.
To simplify, divide both the numerator and the denominator by 2:

• \(-2 \div 2 = -1\)
• \(16 \div 2 = 8\)

Hence, the simplified form of \(-\frac{2}{16}\) is \(-\frac{1}{8}\).

Simplifying makes the fraction easier to interpret and is a crucial skill when working with fractions. It ensures the fraction is in its most useful form, which is often necessary for solving mathematical problems efficiently.