Problem 25 - Basic College Mathematics

Question

Divide and simplify. \(\frac{5}{8} \div \frac{5}{8}\)

Step-by-Step Solution

Answer

1

Step 1: Understand Division of Fractions

Dividing fractions involves multiplying by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by flipping its numerator and denominator.

Step 2: Reciprocal of the Second Fraction

Find the reciprocal of \(\frac{5}{8}\), which is \(\frac{8}{5}\).

Step 3: Multiply by the Reciprocal

Transform the division into multiplication: \(\frac{5}{8} \times \frac{8}{5}\).

Step 4: Multiply the Fractions

Multiply the numerators and denominators: \(\frac{5 \times 8}{8 \times 5} = \frac{40}{40}\).

Step 5: Simplify the Fraction

Simplify \(\frac{40}{40}\) to \(1\).

Key Concepts

Understanding the Reciprocal

When dividing fractions, the first essential step involves understanding the concept of the reciprocal. The reciprocal of a fraction is obtained by flipping its numerator and denominator. For instance, the reciprocal of \( \frac{5}{8} \), the fraction from our exercise, is \( \frac{8}{5} \).

By flipping the fraction, we make it possible to change the division operation into multiplication, which simplifies the process. Remembering how to find the reciprocal is crucial because it transforms a complex division problem into a more manageable multiplication problem.

Multiplying Fractions

Now that we understand the reciprocal, the next step is to multiply the fractions. In our example, instead of directly dividing \( \frac{5}{8} \) by \( \frac{5}{8} \), we convert the operation into a multiplication problem using the reciprocal: \( \frac{5}{8} \times \frac{8}{5} \).

Multiplication of fractions is straightforward: multiply the numerators together and the denominators together.

• Numerators: \( 5 \times 8 = 40 \)
• Denominators: \( 8 \times 5 = 40 \)

This gives us the fraction \( \frac{40}{40} \). Multiplication reduces the problem to a fraction that simplifies more easily.

Simplifying Fractions

The final step is to simplify the fraction obtained from the multiplication. Simplifying means reducing the fraction to its lowest terms.

In our exercise, we ended up with \( \frac{40}{40} \). This fraction can be simplified by dividing the numerator and the denominator by their common factor. In this case, \( 40 \) divides evenly by \( 40 \), resulting in \( \frac{40 \/ \40}{40 \/ \40} = 1 \.

Therefore, \ ( \frac{5}{8} \div \frac{5}{8} = 1 \). Simplifying fractions helps us find more manageable and easily understandable solutions.