Problem 82 - Prealgebra

Question

Simplify. $$\frac{\frac{1}{2}-\frac{1}{4}}{\frac{1}{4}-\frac{1}{8}}$$

Step-by-Step Solution

Answer

The simplified expression is 2.

Step 1: Simplify the Numerator

The numerator of the expression is \( \frac{1}{2} - \frac{1}{4} \). To subtract these fractions, find a common denominator. The common denominator for 2 and 4 is 4. Convert \( \frac{1}{2} \) to have a denominator of 4: \( \frac{1}{2} = \frac{2}{4} \). Now subtract: \( \frac{2}{4} - \frac{1}{4} = \frac{1}{4} \).

Step 2: Simplify the Denominator

The denominator of the expression is \( \frac{1}{4} - \frac{1}{8} \). Find a common denominator for 4 and 8, which is 8. Convert \( \frac{1}{4} \) to \( \frac{2}{8} \). Now subtract: \( \frac{2}{8} - \frac{1}{8} = \frac{1}{8} \).

Step 3: Divide the Simplified Numerator by the Denominator

After simplifying, the expression becomes \( \frac{\frac{1}{4}}{\frac{1}{8}} \). Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, \( \frac{1}{4} \div \frac{1}{8} = \frac{1}{4} \times 8 = 2 \).

Step 4: Final Answer

The fully simplified expression is \( 2 \).

Key Concepts

Common Denominator

When working with fractions that you need to add or subtract, it's crucial to have a common denominator. The denominator is the bottom part of the fraction, and having a common one means both fractions are split into parts of the same size. This makes it easier to combine or contrast them.
For example, consider the fractions \( \frac{1}{2} \) and \( \frac{1}{4} \). To subtract these fractions, you need them to share the same denominator. Since 4 is a multiple of 2, you can convert \( \frac{1}{2} \) into \( \frac{2}{4} \). Now both fractions have the same denominator, and you can easily subtract \( \frac{1}{4} \) from \( \frac{2}{4} \) to get \( \frac{1}{4} \).

• Find a number that both denominators can divide into evenly.
• Convert each fraction to an equivalent one with this common denominator.

This step is fundamental for simplifying expressions involving multiple fractions.

Subtracting Fractions

Subtracting fractions is straightforward once you've found a common denominator. With the fractions properly aligned, you only need to subtract the numerators (the numbers on top).
For instance, with \( \frac{2}{4} \) and \( \frac{1}{4} \), you subtract the numerators 2 and 1, which gives you 1. The denominator remains the same: 4. So, \( \frac{2}{4} - \frac{1}{4} = \frac{1}{4} \).

• Ensure both fractions have the same denominator before subtracting.
• Subtract only the numerators, keeping the denominator unchanged.
• Simplify the resulting fraction if possible.

This method ensures the operation is simple and accurate when working with any fractional subtraction task.

Dividing Fractions

Dividing fractions can seem daunting at first, but there's a simple rule that makes the process much less complex. Instead of directly dividing, you multiply by the reciprocal of the fraction you're dividing by.
Consider you have the fraction \( \frac{1}{4} \) and you want to divide it by \( \frac{1}{8} \). Instead, you multiply \( \frac{1}{4} \) by the reciprocal of \( \frac{1}{8} \), which is 8 (since \( \frac{1}{8} \rightarrow 8 \)). This changes the division into multiplication: \( \frac{1}{4} \times 8 = 2 \).

• Invert the divisor (the second fraction you're dividing by).
• Change the operation to multiplication.
• Perform the multiplication as you would with regular fractions.

Using the reciprocal transforms a challenging division into a simple multiplication, making the solution more approachable.