Problem 3 - Prealgebra

Question

Find the following sums and differences, and reduce to lowest terms. (Add or subtract as indicated.) $$\frac{3}{8}-\frac{5}{8}$$

Step-by-Step Solution

Answer

The simplified result is \(\frac{-1}{4}\).

Step 1: Align the Denominators

Both fractions have the same denominator of 8. This means we can directly perform the subtraction without needing to find a common denominator.

Step 2: Subtract the Numerators

Since the denominators are already aligned, subtract the numerators: \(3 - 5 = -2\).

Step 3: Write the Result as a Fraction

Combine the result of the subtraction over the common denominator: \(\frac{-2}{8}\).

Step 4: Simplify the Fraction

Simplify \(\frac{-2}{8}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \(\frac{-2 \div 2}{8 \div 2} = \frac{-1}{4}\).

Key Concepts

Subtraction of Fractions

Subtracting fractions involves working with the numerators and denominators. If the fractions have the same denominator, like \( \frac{3}{8} \) and \( \frac{5}{8} \) in our example, you can directly subtract the numerators. The denominator remains unchanged. Here’s how you do it:

• Keep the denominator: Because both fractions have the same denominator (8), there's no need to change it.
• Subtract the numerators: Simply subtract the top numbers (numerators) from each other. So, \( 3 - 5 = -2 \).
• Write the result: Put the numerator result over the denominator to create a new fraction: \( \frac{-2}{8} \).

Once you have subtracted, you should always check if the fraction can be simplified.

Simplifying Fractions

Simplifying fractions means reducing them to the smallest equivalent form. After performing subtraction, we got \( \frac{-2}{8} \). This fraction can be simplified, as both 2 and 8 share common factors. Here's how you simplify:

• Identify common factors: Check for the greatest number that can divide both the numerator and denominator. For \( -2 \) and 8, it's 2.
• Divide by the greatest common factor: Divide both the numerator and denominator by 2. \( \frac{-2 \div 2}{8 \div 2} = \frac{-1}{4} \).

Simplifying is an essential step because it makes the fraction easier to understand and work with.

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept in simplifying fractions. It is the largest number that can exactly divide both the numerator and the denominator of a fraction. Finding the GCD can be done in a few simple steps:

• List the factors: Enumerate the factors of each number. For instance, factors of 2 are 1 and 2, while factors of 8 are 1, 2, 4, and 8.
• Find the largest common factor: Compare the lists and find the biggest number in common. For \(-2\) and \(8\), the GCD is 2.

Using the GCD helps ensure that fractions are simplified to their lowest possible terms. This process transforms \( \frac{-2}{8} \) into \( \frac{-1}{4} \), making calculations and comparisons simpler.