Problem 79 - Algebra 2

Question

Evaluate each expression. \(\frac{1}{4}-\frac{2}{3}\)

Step-by-Step Solution

Answer

The expression evaluates to \(-\frac{5}{12}\).

Step 1: Find a Common Denominator

To subtract fractions, we need to have a common denominator. The denominators here are 4 and 3. The least common multiple of 4 and 3 is 12.

Step 2: Convert Fractions to Common Denominator

Convert each fraction to an equivalent fraction with a denominator of 12. The first fraction: \( \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \). The second fraction: \( \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \).

Step 3: Subtract the Numerators

Subtract the numerators of the fractions while keeping the common denominator: \( \frac{3}{12} - \frac{8}{12} = \frac{3 - 8}{12} = \frac{-5}{12} \).

Step 4: Simplify the Result

The resulting fraction \( \frac{-5}{12} \) is already in its simplest form, as 5 and 12 have no common factors other than 1.

Key Concepts

Understanding Common Denominator

When working with fraction subtraction, the first step is to find a common denominator. A common denominator is a shared multiple of the denominators of the fractions you are working with. This is crucial because you can only subtract (or add) fractions that have the same denominator.

To find a common denominator, you look for the least common multiple (LCM) of the given denominators. In our example, the denominators are 4 and 3. To find the LCM, list the multiples of each number:

• Multiples of 4: 4, 8, 12, 16, 20, ...
• Multiples of 3: 3, 6, 9, 12, 15, ...

The smallest multiple that both lists share is 12. This becomes the common denominator for both fractions.

Finding the Least Common Multiple

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both. In fraction subtraction, finding the LCM is a key step to determine the common denominator.

Here's a quick method to find the LCM:

• List the prime factors of each number.
• Multiply each factor the greatest number of times it occurs in either number.

In our context with 4 and 3:

• Prime factors of 4: 2 \( \times \) 2
• Prime factor of 3: 3

Multiply the factors, making sure each appears the maximum times it does in any number, which gives 2 \( \times \) 2 \( \times \) 3 = 12. And there you have the LCM, which is used as the common denominator.

Creating Equivalent Fractions

To subtract fractions successfully, once you have a common denominator, you need to convert each original fraction into an equivalent fraction with this common denominator.

Equivalent fractions have the same value but use different numerators and denominators. You achieve them by multiplying both the numerator and the denominator by the same number.

For instance, to convert \( \frac{1}{4} \) to have a denominator of 12, multiply both the numerator and denominator by 3, resulting in \( \frac{3}{12} \). Similarly, to convert \( \frac{2}{3} \), multiply both by 4, resulting in \( \frac{8}{12} \). These fractions, \( \frac{3}{12} \) and \( \frac{8}{12} \), now share a common denominator and are equivalent to the originals.

Simplifying Fractions

Simplifying fractions means reducing them to their smallest form where the numerator and denominator have no common factors other than 1. This process makes fractions easier to interpret and use.

To simplify a fraction, check the greatest common factor (GCF) of the numerator and the denominator, and divide both by this number. In cases where no common factor exists, the fraction is already simplest.

In our step-by-step problem, we started with \( \frac{-5}{12} \). The numerator and the denominator here, 5 and 12, do not share any factors other than 1. Hence, \( \frac{-5}{12} \) is already simplified. Simplification is an essential skill to present your answers neatly in the most reduced form possible.