Problem 44 - Prealgebra

Question

Find each quotient. $$\frac{x}{5} \div \frac{x}{20}$$

Step-by-Step Solution

Answer

The quotient is 4.

Step 1: Understand Division of Fractions

To divide by a fraction, you multiply by its reciprocal. This means we need to rewrite the problem from \( \frac{x}{5} \div \frac{x}{20} \) to \( \frac{x}{5} \times \frac{20}{x} \).

Step 2: Set Up the Multiplication

Rewrite the division problem as a multiplication: \( \frac{x}{5} \times \frac{20}{x} \). This uses the reciprocal of the second fraction \( \frac{x}{20} \), which is \( \frac{20}{x} \).

Step 3: Simplify the Expression

Before multiplying, observe that \( x \) in the numerator of one fraction and \( x \) in the denominator of the other can be canceled out. Simplifying \( \frac{x}{5} \times \frac{20}{x} \), we remove \( x \) to get \( \frac{20}{5} \).

Step 4: Compute the Final Quotient

Now simplify \( \frac{20}{5} \) by dividing 20 by 5, which gives 4.

Key Concepts

Division of Fractions

Dividing fractions might sound confusing, but with practice, it becomes easier. To divide one fraction by another, the most important rule to remember is to multiply by the reciprocal of the second fraction. This is often expressed as "invert and multiply."
Here's a simple way to do it:

• Identify the numerator (top number) and the denominator (bottom number) of both fractions.
• Find the reciprocal of the second fraction (flip the numerator and denominator).
• Multiply the first fraction by this reciprocal.

For example, when you have \( \frac{x}{5} \div \frac{x}{20} \), rather than directly dividing these two, you change it to a multiplication problem: \( \frac{x}{5} \times \frac{20}{x} \). This turns division into a simpler, more familiar operation, helping you solve problems with greater ease.

Multiplication of Fractions

Multiplying fractions is simpler than dividing them because it doesn't require finding a reciprocal first. To multiply fractions, remember just two steps: multiply all the numerators together to get the new numerator, and then multiply all the denominators together to get the new denominator.
Here's a quick breakdown:

• Write down the fractions you are multiplying.
• Multiply the numerators together and place the product over the product of the denominators.
• Simplify the result if possible by dividing any common factors.

Taking the product of \( \frac{x}{5} \times \frac{20}{x} \), you multiply \( x \times 20 \) for the new numerator and \( 5 \times x \) for the new denominator. Before multiplying, be on the lookout for terms that cancel each other, like "\( x \)" in the numerator and "\( x \)" in the denominator, leading to a simpler expression \( \frac{20}{5} \), which simplifies further to 4.

Reciprocal of a Fraction

Understanding reciprocals is key to dividing fractions. The reciprocal of a fraction simply means flipping the numerator and the denominator. For instance, if the fraction is \( \frac{3}{4} \), its reciprocal is \( \frac{4}{3} \). This flipping turns a division problem into a multiplication one, which is easier to handle.
Knowing how to find the reciprocal is especially helpful when tackling complex fraction problems:

• The reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).
• For whole numbers, consider them as fractions with a denominator of 1. Thus, the reciprocal of 5 is \( \frac{1}{5} \).

In our example \( \frac{x}{5} \div \frac{x}{20} \), the reciprocal of \( \frac{x}{20} \) is \( \frac{20}{x} \), allowing us to change the operation to multiplication: \( \frac{x}{5} \times \frac{20}{x} \). This simple exchange brings clarity to dividing fractions, showcasing how mathematics is often about transforming complex ideas into simpler forms.