Problem 12 - Elementary Algebra

Question

Perform the indicated multiplications and divisions and express your answers in simplest form. $$\frac{4 x}{11 y} \div \frac{12 x}{33}$$

Step-by-Step Solution

Answer

The simplest form is \(\frac{1}{y}\).

Step 1: Understand Division of Fractions

Begin by understanding that dividing by a fraction is equivalent to multiplying by its reciprocal. So, \(\frac{4x}{11y} \div \frac{12x}{33}\) can be rewritten as \(\frac{4x}{11y} \times \frac{33}{12x}\).

Step 2: Multiply the Fractions

Multiply the numerators together and the denominators together: \(\frac{4x \cdot 33}{11y \cdot 12x}\). This results in \(\frac{132x}{132yx}\).

Step 3: Simplify the Expression

Notice that the \(132x\) in the numerator and \(132x\) in the denominator are common factors. Cancel these common terms: \(\frac{132x}{132yx} = \frac{1}{y}\).

Key Concepts

Multiplying Fractions

When multiplying fractions, the process is straightforward. You simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. This rule applies whether you're dealing with simple numbers or algebraic terms. For example, in the original exercise, after finding the reciprocal of the second fraction, you utilize this rule to multiply:

• The numerators: \(4x\) and \(33\)
• The denominators: \(11y\) and \(12x\)

Then, you multiply them to get:\[\frac{4x \cdot 33}{11y \cdot 12x} = \frac{132x}{132yx}\]Remember:

• Always multiply across to avoid mistakes.
• If there are common factors between numerators and denominators, you can simplify them later.

This structured approach ensures organized calculations and minimizes errors.

Reciprocals in Fractions

Understanding reciprocals is key to solving algebraic fraction division problems. A reciprocal of a fraction simply means flipping the numerator and denominator. For instance, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). Using the reciprocal is necessary when dividing fractions, as division is converted into multiplication using this technique.

In our exercise, the expression \(\frac{4x}{11y} \div \frac{12x}{33}\) is transformed by finding the reciprocal of \(\frac{12x}{33}\), which becomes \(\frac{33}{12x}\). This turns the original division problem into:\[\frac{4x}{11y} \times \frac{33}{12x}\]This step is crucial because it simplifies the process, making it easier to move forward with multiplying fractions. Mastering reciprocals greatly simplifies division of fractions, both in numbers and algebra.

Simplifying Expressions

Simplifying expressions is a critical step in algebra to make calculations easier and results more meaningful. In the multiplication result we have:\[\frac{132x}{132yx}\]
Notice the expression in both the numerator and the denominator shares a common term of \(132x\). You can cancel this common factor in both the numerator and denominator, essentially dividing both by \(132x\):

• In the numerator: \(132x \div 132x = 1\)
• In the denominator: \(132yx \div 132x = y\)

So the simplified result is:\[\frac{1}{y}\]Simplifying not only involves dividing out common factors but also checking if there’s more reduction possible. This final step of simplification is essential for reaching the most reduced form of algebraic expressions, providing a cleaner and often more useful result.