Problem 35 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{5 x}{3} \div \frac{10 x}{6} $$

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Write the Division as Multiplication

To simplify the expression \( \frac{\frac{5x}{3}}{\frac{10x}{6}} \), first rewrite the division as a multiplication by the reciprocal. This step involves flipping the second fraction: \[ \frac{5x}{3} \times \frac{6}{10x} \]

Step 2: Cancel Common Factors Before Multiplication

Before multiplying, identify and cancel any common factors in the numerators and denominators of the fractions. Here, the common factor \( x \) can be cancelled from both the \(5x\) in the numerator and the \(10x\) in the denominator:\[ \frac{5}{3} \times \frac{6}{10} \]

Step 3: Multiply the Remaining Fractions

Now multiply the numerators together and the denominators together:Numerator: \( 5 \times 6 = 30 \) Denominator: \( 3 \times 10 = 30 \) So the expression becomes:\[ \frac{30}{30} \]

Step 4: Simplify to Obtain the Final Result

Finally, simplify the fraction \( \frac{30}{30} \) to its simplest form. Since both the numerator and the denominator are the same, the fraction simplifies to:\[ 1 \]

Key Concepts

Fraction Simplification

Simplifying fractions is a crucial skill in algebra that makes expressions easier to work with. It involves reducing the fraction to its simplest form without changing its value. Here are the key steps:

• Identify common factors in the numerator and the denominator.
• Divide both by these common factors.

Let's look at the fraction \( \frac{30}{30} \). Both numerator and denominator are divisible by 30. Simplifying it gives \( \frac{30 \div 30}{30 \div 30} = \frac{1}{1} = 1 \).
This shows that every non-zero value divided by itself equals one. In fraction simplification, finding the greatest common divisor makes the process more efficient, bringing fractions to their simplest form.If both parts of the fraction share no common factors other than 1, they are considered simplified.

Division of Fractions

Dividing fractions can be simplified by converting the operation into multiplication. Instead of dividing by a fraction, multiply by its reciprocal. The reciprocal of a fraction is achieved by swapping its numerator and denominator.For example, in the expression \( \frac{5x}{3} \div \frac{10x}{6} \), follow these steps:

• Convert \( \frac{10x}{6} \) into its reciprocal, \( \frac{6}{10x} \).
• Change the division to multiplication: \( \frac{5x}{3} \times \frac{6}{10x} \).

Now, perform the multiplication as normal, which brings us to the next core concept. Remembering this conversion allows you to simplify complex division problems into easier multiplication equations.

Canceling Common Factors

When you simplify fractions, canceling common factors is an essential technique that makes calculations straightforward. Before multiplying fractions, check for factors that appear in both the numerator and denominator.In the problem \( \frac{5x}{3} \times \frac{6}{10x} \), the variable \( x \) appears in both the numerator of the first fraction and the denominator of the second fraction. This allows it to be canceled out:

• Cancel \( x \) in \( 5x \) and \( 10x \) to get \( \frac{5}{3} \times \frac{6}{10} \).

Make sure every remaining number or term isn't common before the final multiplication. Canceling helps reduce the numbers you work with, leading directly to the simplest form of the fraction.