Problem 17 - Prealgebra

Question

Find each sum or difference. Write in simplest form. $$\frac{3}{4}+\left(-\frac{5}{8}\right)$$

Step-by-Step Solution

Answer

The simplest form of the expression \( \frac{3}{4} + \left(-\frac{5}{8}\right) \) is \( \frac{1}{8} \).

Step 1: Identify the Operation

The expression given is a combination of addition and negative signs: \( \frac{3}{4} + \left(-\frac{5}{8}\right) \). This means we are adding \( \frac{3}{4} \) and \(-\frac{5}{8} \).

Step 2: Find a Common Denominator

To add or subtract fractions, they must have the same denominator. The denominators here are 4 and 8. The least common denominator between the two is 8.

Step 3: Convert the Two Fractions

Convert \( \frac{3}{4} \) to its equivalent form with a denominator of 8. Multiply both numerator and denominator by 2: \( \frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8} \). The second fraction \( -\frac{5}{8} \) already has 8 as its denominator.

Step 4: Perform the Addition

Now add \( \frac{6}{8} \) and \(-\frac{5}{8} \): \[ \frac{6}{8} + \left(-\frac{5}{8}\right) = \frac{6 - 5}{8} = \frac{1}{8} \].

Step 5: Simplify the Result

The fraction \( \frac{1}{8} \) is already in its simplest form, as 1 and 8 have no common factors other than 1.

Key Concepts

Understanding the Least Common Denominator

When working with fractions, especially in addition and subtraction, having a shared or common denominator is crucial. This makes it easier to combine the fractions. The smallest shared multiple of the denominators involved is what we call the "Least Common Denominator" or LCD. This process sometimes requires finding multiples of each denominator until you find a common one. For example:

• Consider the fractions \( \frac{3}{4} \) and \( -\frac{5}{8} \).
• Here, the denominators are 4 and 8.
• The multiples of 4 are 4, 8, 12, 16,...
• The multiples of 8 are 8, 16, 24,...

The smallest number that is found in both lists is 8, so the LCD is 8.
By finding the LCD, the fractions can easily be converted to this new denominator, allowing simpler addition or subtraction of the fractions.

Fraction Simplification Made Easy

Once the fractions are added or subtracted, it’s important to check if the resulting fraction can be simplified. Simplification helps express the fraction in the simplest possible form without changing its value.
To simplify a fraction, you divide both the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD). Here’s how it works:

• Consider the fraction \( \frac{1}{8} \). The numbers in this fraction are 1 and 8.
• The only common factor is 1, so the GCD is 1.

Since there are no other common factors to divide by, \( \frac{1}{8} \) is already in its simplest form.
Always simplify your final answer when working with fractions; it keeps your answer neat and easier to understand.

Handling Negative Fractions

Negative fractions can sometimes be confusing, but they are simply fractions with a negative sign in front. This can indicate that the fraction represents a negative value.
For example, in \( -\frac{5}{8} \), the negative sign can be considered in one of three ways:

• Beside the whole fraction as \( -\frac{5}{8} \)
• With the numerator as \( \frac{-5}{8} \)
• With the denominator as \( \frac{5}{-8} \)

Any of these notations mean the same thing: the fraction represents a negative value. When you add or subtract a negative fraction:

• Think of it as subtracting its positive counterpart.
• So \( 3/4 + (-5/8) \) can also be understood as \( 3/4 - 5/8 \).

This will help you clearly handle negative fractions with confidence while performing fraction operations.