Problem 31 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

0

Step 1: Perform Multiplication

Start by multiplying the fractions at the beginning of the expression: \( \frac{1}{4} \cdot \frac{2}{3} \). Multiply the numerators and the denominators separately: \( \frac{1 \times 2}{4 \times 3} = \frac{2}{12} \). This simplifies to \( \frac{1}{6} \) by dividing the numerator and denominator by their greatest common divisor, which is 2.

Step 2: Subtract the Fractions

Subtract the resulting fraction from Step 1 (\( \frac{1}{6} \)) from the fraction in the expression:\( \frac{1}{6} - \frac{1}{6} = \frac{0}{6} \). Any fraction with 0 in the numerator is 0.

Key Concepts

Multiplying Fractions

When multiplying fractions, the process is straightforward. You simply multiply the numerators (the top numbers) and then multiply the denominators (the bottom numbers). For the fractions \( \frac{1}{4} \) and \( \frac{2}{3} \), you multiply the numerators: \( 1 \times 2 = 2 \). Then, you multiply the denominators: \( 4 \times 3 = 12 \).
This gives you a new fraction: \( \frac{2}{12} \).

• Always multiply the numerators with each other.
• Next, multiply the denominators with each other.
• Combine them into a new fraction.

Before proceeding, it is a good idea to simplify the fraction \( \frac{2}{12} \). This makes the subtraction step easier in the next parts of the problem.

Subtracting Fractions

Subtracting fractions require a common denominator. However, in this scenario, both fractions \( \frac{1}{6} - \frac{1}{6} \) already have the same denominator, making things simpler.

• Ensure both fractions have the same denominator.
• Subtract the numerators: \( 1 - 1 = 0 \).
• Keep the denominator as it is: 6.

So, the result is \( \frac{0}{6} \), which simplifies to 0. Remember, any fraction with 0 as the numerator is equal to just 0. This is a simple subtraction because the fractions are equivalent.

Greatest Common Divisor

Simplifying fractions often involves the greatest common divisor (GCD). The GCD of two numbers is the largest positive integer that divides both numbers completely without leaving a remainder. In the multiplication step, we received \( \frac{2}{12} \).
To simplify this, determine the GCD of 2 and 12, which is 2.

• Divide the numerator (2) by the GCD (2).
• Divide the denominator (12) by the GCD (2).
• Resulting in a simplified fraction: \( \frac{1}{6} \).

Using GCD makes calculations simpler and helps in presenting the final answer in its lowest form. Learning to find the GCD is a powerful skill for simplifying any fractions.