Problem 59 - Algebra A Combined Function

Question

Perform each indicated operation. $$ \frac{2}{6}-\frac{3}{4} $$

Step-by-Step Solution

Answer

The result of the operation \(\frac{2}{6}-\frac{3}{4}\) is \(-\frac{5}{12}\).

Step 1: Find a Common Denominator

To subtract fractions, they must have the same denominator. The denominators here are 6 and 4. The least common multiple of 6 and 4 is 12. Thus, the common denominator will be 12.

Step 2: Convert Fractions to Have the Same Denominator

Convert \(\frac{2}{6}\) and \(\frac{3}{4}\) so that they both have the denominator of 12. Convert \(\frac{2}{6}\):\[\frac{2}{6} = \frac{2 \times 2}{6 \times 2} = \frac{4}{12}\]Convert \(\frac{3}{4}\):\[\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\]

Step 3: Subtract the Fractions

Now that both fractions have the same denominator, subtract them:\[\frac{4}{12} - \frac{9}{12} = \frac{4-9}{12} = -\frac{5}{12}\]

Step 4: Simplify the Result

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

Key Concepts

Least Common Multiple

When subtracting fractions, the first step is to find a common denominator. This requires determining the least common multiple (LCM) of the denominators. For the fractions \( \frac{2}{6} \) and \( \frac{3}{4} \), their denominators are 6 and 4, respectively. The LCM is the smallest number that both denominators divide into evenly.
The prime factorizations of these numbers can help. For 6, it is 2 and 3. For 4, it is 2 and 2. The LCM is 12, which combines the highest power of each prime (2 x 2 x 3 = 12). Once you have this, you can set both fractions over this common denominator.

Common Denominator

Using a common denominator helps align the fractions so they can be subtracted easily. Since we've determined that the LCM of 6 and 4 is 12, this number becomes our common denominator.
To convert, adjust the numerators accordingly. For \( \frac{2}{6} \), multiply the numerator and denominator by 2 to get \( \frac{4}{12} \). For \( \frac{3}{4} \), multiply by 3 to reach \( \frac{9}{12} \). Now both fractions are prepared for direct subtraction.

Simplifying Fractions

Once you subtract fractions, it's essential to simplify your result, which means reducing it to its simplest form. After subtracting, the result of \( \frac{4}{12} - \frac{9}{12} \) is \( -\frac{5}{12} \).
In this case, the fraction \( -\frac{5}{12} \) is already simplified because 5 is a prime number and has no common factors with 12 other than 1. Simplifying helps to ensure the fraction is easy to understand and use in further calculations.

Fraction Conversion

To work with fractions in operations like subtraction, conversion to the same denominator is crucial. You change each fraction to equivalent fractions with a shared denominator.
This involves scaling up both the numerator and denominator so they match without altering the value of the fraction. By converting \( \frac{2}{6} \) and \( \frac{3}{4} \) to have the denominator of 12, we created \( \frac{4}{12} \) and \( \frac{9}{12} \). It's a simple but essential technique that forms the basis for adding, subtracting, and comparing fractions.