Problem 31 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{\frac{-3+y}{4}}{\frac{8+y}{28}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{28(y-3)}{4(y+8)} \).

Step 1: Understand the problem

We need to simplify the complex fraction \( \frac{\frac{-3+y}{4}}{\frac{8+y}{28}} \). This involves simplifying a fraction where both the numerator and the denominator are themselves fractions.

Step 2: Rewrite the complex fraction

Write the complex fraction as a division problem. It becomes: \( \left( \frac{-3+y}{4} \right) \div \left( \frac{8+y}{28} \right) \).

Step 3: Apply the reciprocal rule

To divide by a fraction, multiply by its reciprocal. Multiply \( \frac{-3+y}{4} \) by the reciprocal of \( \frac{8+y}{28} \), which is \( \frac{28}{8+y} \). The expression now is: \( \frac{-3+y}{4} \times \frac{28}{8+y} \).

Step 4: Multiply the fractions

Multiply the numerators and the denominators: \( \frac{(-3+y) \cdot 28}{4 \cdot (8+y)} \).

Step 5: Simplify the multiplication

Calculate the products: the numerator is \( 28(-3+y) \) and the denominator is \( 4(8+y) \).

Step 6: Distribute the multiplication

Distribute the multiplications in both numerator and denominator: Numerator becomes \( 28y - 84 \) and the denominator becomes \( 32 + 4y \).

Step 7: Simplify the fraction

The expression is now \( \frac{28y - 84}{4y + 32} \). Look for any further simplification by factoring or reducing common terms.

Key Concepts

Simplifying Fractions

When dealing with complex fractions, our goal is to make them simpler and more manageable. A complex fraction is essentially a fraction within a fraction, which might look daunting at first. However, we can break down the process using simple steps.

• The first thing to do is to rewrite the complex fraction as a simple division problem. For instance, in our problem: \[ \frac{\frac{-3+y}{4}}{\frac{8+y}{28}} \]This can be rewritten as \( \left( \frac{-3+y}{4} \right) \div \left( \frac{8+y}{28} \right) \).
• The division of fractions turns into multiplication by the reciprocal, making our calculations easier.

Understanding these initial simplifications paves the way for further steps, such as applying the reciprocal rule and distributive property.

Reciprocal Rule

The reciprocal rule is crucial when simplifying complex fractions. When you divide by a fraction, you can multiply by its reciprocal instead. This rule transforms a division problem into a multiplication one, which is often easier to manage.

• In the example given, we have the division \( \left( \frac{-3+y}{4} \right) \div \left( \frac{8+y}{28} \right) \).
• To apply the reciprocal rule, we flip the second fraction. So, \( \frac{8+y}{28} \) becomes \( \frac{28}{8+y} \).
• Now, we multiply: \( \frac{-3+y}{4} \times \frac{28}{8+y} \).

By using the reciprocal rule, the expression becomes easier to handle and prepares it for further simplification through multiplication and distribution.

Distributive Property

The distributive property allows us to simplify expressions by distributing a single term across terms inside parentheses. In our complex fraction problem, once we've set up the multiplication, we can distribute the numbers through both the numerator and the denominator.

• We start with \( \frac{(-3+y) \cdot 28}{4 \cdot (8+y)} \).
• Distribute in the numerator: Multiply each term inside the parentheses by 28, which gives us \( 28y - 84 \).
• In the denominator, distribute 4 over \( 8+y \), resulting in \( 32 + 4y \).

This application of the distributive property helps us in managing complex multiplications in both the numerator and denominator, ultimately simplifying the problem further for reduction or factoring.