Problem 29 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(\frac{1}{4} \cdot \frac{2}{3}-\frac{1}{6}\)

Step-by-Step Solution

Answer

The simplified expression is 0.

Step 1: Simplify the Product of Fractions

First, simplify \( \frac{1}{4} \cdot \frac{2}{3} \). Multiply the numerators together and do the same for the denominators: \( \frac{1 \times 2}{4 \times 3} = \frac{2}{12} \).

Step 2: Simplify the Resulting Fraction

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

Step 3: Subtract the Fractions

Now, subtract \( \frac{1}{6} \) from the simplified product \( \frac{1}{6} \). Since the fractions have the same denominator, you can directly subtract the numerators: \( \frac{1}{6} - \frac{1}{6} = \frac{0}{6} = 0 \).

Step 4: State the Simplified Expression

The simplified result of the given expression \( \frac{1}{4} \cdot \frac{2}{3} - \frac{1}{6} \) is \( 0 \).

Key Concepts

Multiplying Fractions

When we multiply fractions, we follow a straightforward process that involves multiplying the numerators (the numbers on the top) and the denominators (the numbers on the bottom) separately. For example, if we have two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \), to multiply them we do:

• Multiply the numerators: \( a \times c \)
• Multiply the denominators: \( b \times d \)

Therefore, the resulting product of the two fractions is \( \frac{a \times c}{b \times d} \). In the original exercise, this process helped transform \( \frac{1}{4} \times \frac{2}{3} \) into \( \frac{2}{12} \). Multiply fractions just the way you would multiply whole numbers, but keep them in fraction form. This will keep your calculations simpler and lead to an easier way to simplify.

Subtracting Fractions

Subtracting fractions can seem tricky, but it's mostly about ensuring the fractions have the same denominator. A common denominator is essential when dealing with addition or subtraction. This is because fractions represent parts of a whole, and having the same bottom number (denominator) makes sure we are dealing with the same sized parts. In our example, the fractions \( \frac{1}{6} \) and \( \frac{1}{6} \) already have the same denominator.

• Make sure both fractions share the same denominator
• Subtract the numerators while keeping the same denominator

So, \( \frac{1}{6} - \frac{1}{6} \) proceeds to \( \frac{0}{6} \), which simplifies to zero. Always simplify your answers to express them in their simplest form.

Greatest Common Divisor

Simplifying fractions is a breeze with an understanding of the Greatest Common Divisor (GCD). This is the largest number that divides exactly into both the numerator and denominator of a fraction without leaving a remainder. It helps reduce fractions to their simplest form. To find the GCD of two numbers, follow these steps:

• List the factors of each number
• Identify the greatest factor common to both

In the exercise, to simplify \( \frac{2}{12} \), the factors of 2 are 1 and 2, while the factors of 12 include 1, 2, 3, 4, 6, and 12. The greatest common factor here is 2. Dividing both the numerator and the denominator by 2 gives the simplest form \( \frac{1}{6} \). By identifying the GCD, you make the fraction as simple as possible, making calculations much more straightforward.