Problem 169 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{-\frac{4}{5}}{2} $$

Step-by-Step Solution

Answer

The simplified form is \(-\frac{2}{5}\).

Step 1: Understand the Problem

First, look at the problem \( \frac{-\frac{4}{5}}{2} \) and understand that it involves simplifying a fraction within another fraction.

Step 2: Simplify the Inner Fraction

The inner fraction is already simplified: \( -\frac{4}{5} \). So we don't need to do anything further with \(-\frac{4}{5}\).

Step 3: Divide the Outer Fraction

Divide the inner fraction by 2. Dividing by 2 is the same as multiplying by \frac{1}{2}\, so the expression becomes \(-\frac{4}{5} \times \frac{1}{2}\).

Step 4: Multiply the Fractions

Multiply the numerators together and the denominators together: \(-\frac{4 \times 1}{5 \times 2} = -\frac{4}{10}\).

Step 5: Simplify the Result

Simplify \(-\frac{4}{10}\) by finding the greatest common divisor of 4 and 10, which is 2. So, \(-\frac{4 \div 2}{10 \div 2} = -\frac{2}{5} \).

Key Concepts

Fraction Division

The given exercise requires understanding how to handle complex fractions, specifically dividing fractions. When you see a fraction divided by another number or fraction, you can solve it by multiplying with the reciprocal of that number or fraction.
In the exercise, we have \(\frac{-\frac{4}{5}}{2}\). To divide \(-\frac{4}{5}\) by 2, you multiply \(-\frac{4}{5}\) by the reciprocal of 2, which is \(\frac{1}{2}\). This simplifies the problem and makes it easier to solve.
When dividing fractions, remember these key steps:

• Identify the inner fraction and simplify it if necessary.
• Divide by multiplying with the reciprocal of the divisor.
• Simplify the result, if possible.

This approach is essential for solving complex fraction problems correctly and efficiently.

Multiplying Fractions

Multiplying fractions is a critical skill for dealing with more complex mathematical problems involving fractions. The fundamental rule is to multiply the numerators together and the denominators together.
For the problem \(\frac{-\frac{4}{5} \times \frac{1}{2}}\), you multiply the numerators (\(-4 \times 1\)) and the denominators (\(5 \times 2\)). This results in \(\frac{-4 \times 1}{5 \times 2} = \frac{-4}{10}\).
Here are some general steps for multiplying fractions:

• Multiply the top numbers (numerators) together.
• Multiply the bottom numbers (denominators) together.
• Simplify the resulting fraction, if necessary.

This technique is straightforward and can be applied to any fraction multiplication problem, making it a foundational skill in fraction operations.

Simplifying Fractions

After performing operations with fractions, it's often necessary to simplify the result. Simplifying a fraction means reducing it to its lowest terms by dividing the numerator and the denominator by their greatest common divisor (GCD).
In our example with \(\frac{-4}{10}\), we simplify by finding the GCD of 4 and 10, which is 2. Dividing both the numerator and the denominator by 2, we get \(\frac{-4 \div 2}{10 \div 2} = \frac{-2}{5}\).
The steps to simplify fractions are:

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• The fraction you obtain is the simplified form.

Remember, simplifying fractions makes them easier to work with and is crucial for checking the final answer's correctness and neatness.