Problem 106 - Algebra 1: Concepts and Skills

Question

Divide. Write the answer as a fraction or as a mixed number in simplest form. $$ \frac{1}{2} \div 4 $$

Step-by-Step Solution

Answer

\(\frac{1}{8}\)

Step 1: Identify the Divisor's Reciprocal

Identify the reciprocal of the divisor, 4. The reciprocal of a number is obtained by switching the number's numerator and denominator. Since 4 is a whole number and can be written as \(\frac{4}{1}\), its reciprocal is \(\frac{1}{4}\).

Step 2: Apply Division as Multiplication by the Reciprocal

When dividing by a fraction, we actually multiply by its reciprocal. So convert the division into multiplication by swapping the divisor, 4, with its reciprocal. The equation becomes: \(\frac{1}{2} \times \(\frac{1}{4}\)

Step 3: Multiply Fractions

To multiply fractions, multiply the numerators together for the new numerator and the denominators together for the new denominator. So, \(\frac{1}{2} \times \frac{1}{4} = \frac{1 \times 1}{2 \times 4}\)

Step 4: Simplify the Result

Upon multiplication, we get \(\frac{1}{8}\) as the result which is already in its simplest form.

Key Concepts

Understanding Reciprocals

Before diving into fraction division, it's critical to grasp the idea of a reciprocal. A reciprocal flips the numerator and the denominator of a fraction. In simpler terms, if you have a whole number like 4, consider it as a fraction: \(\frac{4}{1}\). Flip it around, and you get \(\frac{1}{4}\), the reciprocal.
A helpful tip to remember: the product of a number and its reciprocal is always 1. For example:

• \(4 \times \frac{1}{4} = 1\)
• \(\frac{3}{5} \times \frac{5}{3} = 1\)

This concept is crucial when dividing fractions because dividing by a number is the same as multiplying by its reciprocal.

Simplifying Fractions Made Easy

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This step often follows multiplication or division of fractions.

• First, look at the numerator and denominator.
• See if there is a common factor you can divide them by.
• Keep simplifying until no larger common factor exists.

For example, if you end up with \(\frac{6}{8}\), divide by 2 to get \(\frac{3}{4}\).
Working with fractions becomes a lot easier when they're in their simplest form, making calculations more straightforward and comparison between fractions clearer.

Mastering Multiplication of Fractions

Multiplying fractions is a straightforward process. Unlike addition or subtraction, you don't need a common denominator.
Here's how to multiply two fractions:

• Multiply the numerators to get the new numerator.
• Multiply the denominators to get the new denominator.

For example, if you have \(\frac{1}{2} \times \frac{1}{4}\), multiply:

• Numerators: \(1 \times 1 = 1\)
• Denominators: \(2 \times 4 = 8\)

So, \(\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}\).
Multiplication helps significantly in division situations as well because dividing by a fraction can be rewritten as multiplying by its reciprocal.