Problem 30 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{3}{4}-\frac{7}{12} $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{1}{6} \).

Step 1: Find a common denominator

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

Step 2: Convert fractions to a common denominator

Convert the fraction 3/4 to an equivalent fraction with a denominator of 12. You do this by multiplying both the numerator and the denominator by 3: \( \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \).

Step 3: Subtract the fractions

Now that both fractions have a common denominator, subtract the fractions: \( \frac{9}{12} - \frac{7}{12} = \frac{9 - 7}{12} = \frac{2}{12} \).

Step 4: Simplify the fraction

Simplify \( \frac{2}{12} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \( \frac{2 \div 2}{12 \div 2} = \frac{1}{6} \).

Key Concepts

Common Denominator

When subtracting fractions, it's important for them to share the same denominator. This helps align the fractions, so you can easily perform operations like addition and subtraction. To begin, identify the denominators of your fractions. In our example, the two fractions are \( \frac{3}{4} \) and \( \frac{7}{12} \). The denominators are 4 and 12.
Finding a common denominator involves determining a number that both denominators can divide into evenly. You can often use the least common multiple (LCM) of the numbers as the common denominator. This ensures that transformations are minimized, and operations remain simple.

Least Common Multiple

The least common multiple is crucial because it is the smallest number that is divisible by two or more numbers. Here, we calculate the LCM of the denominators 4 and 12.

• For 4, the multiples are: 4, 8, 12, 16, ...
• For 12, the multiples are: 12, 24, 36, 48, ...

The first shared multiple is 12. Therefore, the LCM of 4 and 12 is 12, and we will use this number as the common denominator.
This allows for straightforward conversion of fractions to equivalent forms, having matching denominators.

Simplifying Fractions

After performing operations like subtraction, simplifying fractions reduces the results to their simplest form. It involves dividing both the numerator and denominator by their greatest common divisor (GCD). In our example, after subtraction, the fraction \( \frac{2}{12} \) was obtained.

• The GCD of 2 and 12 is 2. Therefore, divide both the numerator (2) and the denominator (12) by 2:
• \( \frac{2 \div 2}{12 \div 2} = \frac{1}{6} \)

Simplifying makes fractions easier to understand and work with. Also, a simpler fraction is often more visually satisfying and standard in mathematics.