Problem 84 - Elementary and Intermediate Algebra

Question

Add. $$ -\frac{3}{32}+\left(-\frac{1}{2}\right) $$

Step-by-Step Solution

Answer

The sum is \(-\frac{19}{32}\).

Step 1: Find a Common Denominator

The first step in adding fractions is to find a common denominator. The denominators of the fractions are 32 and 2. The least common denominator (LCD) for these two numbers is 32.

Step 2: Convert Fractions to Common Denominator

The first fraction, \(-\frac{3}{32}\), already has the denominator 32. For the fraction \(-\frac{1}{2}\), we need to convert it to have the denominator of 32. To do this, multiply the numerator and the denominator of \(-\frac{1}{2}\) by 16: \(-\frac{1}{2} \times \frac{16}{16} = -\frac{16}{32}\).

Step 3: Add the Fractions

Now that both fractions have the same denominator, add them by adding the numerators: \(-\frac{3}{32} + -\frac{16}{32} = \frac{-3 - 16}{32} = \frac{-19}{32}\).

Step 4: Simplify the Result

The fraction \(\frac{-19}{32}\) is already in its simplest form since 19 is a prime number and does not divide 32.

Key Concepts

Common Denominator

When adding fractions, it's crucial to view them in terms of a common language, so to speak. Just like communicating effectively requires a common language, adding fractions requires a common denominator. A common denominator is essentially a shared multiple between the denominators of the fractions you aim to add or subtract.

Here's a basic rundown on utilizing a common denominator:

• Identify the denominators of your fractions. In our exercise, these are 32 and 2.
• Calculate a number that both these denominators can easily divide into—a.k.a. the common denominator. This helps transform the fractions into comparable parts.

By turning different fractions into equivalent fractions with common denominators, you can add or subtract them just by working with their numerators. This clears the way for a straightforward addition or subtraction.

Least Common Denominator

While a common denominator allows you to add fractions, sometimes you may inadvertently choose a larger number than necessary. This is when the concept of the Least Common Denominator (LCD) comes in handy. The LCD is the smallest number that both denominators can evenly divide into. Finding it ensures we don’t work with large numerators unnecessarily.

To determine the LCD:

• List out multiples of each denominator until you find the smallest one they share.
• In our example, the denominators 32 and 2 share the least common denominator of 32, which is convenient as 32 is already a suitable denominator for one of the fractions.

Utilizing the LCD simplifies calculations and keeps numbers manageable. Using the smallest possible values often makes simplifying the result easier later on.

Simplifying Fractions

Once fractions are added, it’s essential to simplify the resulting fraction to its simplest form, which means representing it in the smallest possible whole numbers without changing its value. This ensures your answer is neat and can be easily interpreted.

To simplify a fraction:

• Check if the numerator and the denominator have any common factors other than 1.
• Divide both the numerator and the denominator by their greatest common factor (GCF) to simplify.

In our case of \(-\frac{19}{32}\), the numerator 19 is a prime number, meaning it only divides evenly by 1 and itself. Since 19 doesn’t divide 32, the fraction \(-\frac{19}{32}\) is already simplified as much as it can be. Sometimes, particularly with prime numbers, the initial fraction you calculate might already be the simplest form.