Problem 64 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{1}{8} \div \frac{1}{4}$$

Step-by-Step Solution

Answer

The answer to \( \frac{1}{8} \div \frac{1}{4} \) is \( \frac{1}{2} \).

Step 1: Rewrite the Division as Multiplication

To divide fractions, rewrite the problem as \( \frac{1}{8} \) times the reciprocal of \( \frac{1}{4} \), or \( 4/1 \). So the equation is now \( \frac{1}{8} \times \frac{4}{1} \).

Step 2: Multiply Numerators and Denominators

Now, multiply the numerators (top number of the fraction) together and the denominators (bottom number of the fraction) together. It becomes \( \frac{1 \times 4}{8 \times 1} = \frac{4}{8} \).

Step 3: Simplify the Result

Now, simplify the fraction to its lowest terms. The numerator and denominator are both divisible by 4, so dividing them both by 4 gives \( \frac{4 \div 4}{8 \div 4} = \frac{1}{2} \).

Key Concepts

Understanding the Reciprocal

When diving into dividing fractions, one of the first terms you'll encounter is "reciprocal." But what does this mean? The reciprocal of a fraction is essentially flipping it upside down. For any fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). This concept is crucial since division of fractions turns into a multiplication problem by using reciprocals.
Let's consider an example: The reciprocal of \( \frac{1}{4} \) is \( \frac{4}{1} \). By using this method, dividing fractions becomes intuitive and simple, making it easy to solve problems with ease. Remember this handy approach whenever faced with a fraction division!

• Reciprocal flips the numerator and the denominator.
• The reciprocal helps transform division into multiplication.
• Understanding reciprocals simplifies dividing fractions significantly.

Mastering Multiplication of Fractions

Multiplying fractions might seem challenging at first, but it is quite straightforward once you get the hang of it. To multiply two fractions, simply multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator.
For example, if you multiply \( \frac{1}{8} \) by \( \frac{4}{1} \), you would do the following:
- Multiply the numerators: \(1 \times 4 = 4\)
- Multiply the denominators: \(8 \times 1 = 8\)
Thus, \( \frac{1}{8} \times \frac{4}{1} = \frac{4}{8} \). After multiplying, you might need to proceed to the final step: simplifying the result. Multiplying fractions is generally quick, and knowing this process turns daunting problems into solveable steps.

• Numerators and denominators are multiplied separately.
• The resulting fraction may need simplification.
• This method turns fraction multiplication into an easy job.

Easily Simplifying Fractions

Simplifying fractions brings them down to their simplest form. This means finding the greatest common divisor (GCD) of both the numerator and the denominator and dividing them by it.
Let's simplify \( \frac{4}{8} \): both 4 and 8 can be divided evenly by 4, their GCD. Thus, \( \frac{4}{8} \) simplifies to \( \frac{1}{2} \).
Simplifying is key for most fractional operations because it makes the fraction more understandable and ensures it's in the most concise form.

• Identify the greatest common divisor of both numbers.
• Divide both the numerator and the denominator by the GCD.
• Achieve a simplified, clearer fraction.

Knowing how to simplify fractions effectively allows you to solve problems neatly and correctly. It ensures that your final answers are both precise and easily interpretable.