Problem 19 - Elementary and Intermediate Algebra

Question

Multiply, and then simplify, if possible. \(\frac{x+5}{5} \cdot \frac{x}{x+5}\)

Step-by-Step Solution

Answer

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

Step 1: Multiply the Numerators

To multiply the expressions \(\frac{x+5}{5} \cdot \frac{x}{x+5}\), start by multiplying the numerators: \((x+5) \times x = x(x+5) = x^2 + 5x\).

Step 2: Multiply the Denominators

Next, multiply the denominators: \(5 \times (x+5) = 5(x+5)\).

Step 3: Form the New Fraction

Combine the results from Step 1 and Step 2 into a single fraction: \(\frac{x^2 + 5x}{5(x + 5)}\).

Step 4: Simplify the Fraction

Since both the numerator \(x^2 + 5x\) and the denominator \(5(x + 5)\) involve the expression \((x+5)\), factor \(x\) out of the numerator: \(x(x + 5)\). This gives \(\frac{x(x+5)}{5(x+5)}\). Cancel \(x+5\) from the numerator and the denominator: \(\frac{x(x+5)}{5(x+5)} = \frac{x}{5}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is much like cleaning up a messy room; it makes everything clearer and easier to work with. When faced with a fraction, your goal is to make it as simple as possible. This means that the numerator and the denominator should have no common factors other than 1.

For instance, in the problem presented, simplifying \[ \frac{x(x+5)}{5(x+5)} \]meant canceling out the common factor of \((x+5)\) from both the numerator and the denominator.

• The first step is checking if there is a factor present in both parts of the fraction.
• Next, if possible, divide both by that common factor.

The fraction reduces to\( \frac{x}{5} \),since you have effectively divided out the \((x+5)\), leaving the simplest form behind.

Numerator and Denominator

A basic understanding of fractions involves knowing the numerator and the denominator, much like knowing the parts of a sandwich.

The numerator is the top part of the fraction, it shows how many parts you have. On the other hand, the denominator is at the bottom, indicating the total number of equal parts or units that make up a whole.
For example, in the fraction \(\frac{x(x+5)}{5(x+5)}\):

• The expression \(x(x+5)\) is the numerator which comes from multiplying the original numerators \(x(x+5)\).
• The expression \(5(x+5)\) is the denominator and results from multiplying the original denominators.

A firm grasp of these parts helps in simplifying fractions as you can easily identify common factors for cancellation.

Factoring Expressions

Factoring is the mathematical version of peeling layers off an onion; you're breaking down complex expressions into simpler parts. This is crucial in simplifying fractions and solving equations.

To factor an expression, look for common terms or patterns, such as the distributive property or recognizing a simple quadratic expression. For example, in our exercise:

• The numerator \(x(x+5)\) was obtained by factoring out \(x\) from \(x^2 + 5x\).
• This factoring is essential as it revealed the common factor \((x+5)\) shared with the denominator.

Successfully factoring expressions allows for simplification through cancellation, paving the way to reduce fractions to their simplest form.