Problem 9 - Algebra A Combined Function

Question

Find each product and simplify if possible. See Examples 1 through 3. $$ \frac{6 x+6}{5} \cdot \frac{10}{36 x+36} $$

Step-by-Step Solution

Answer

The simplified product is \( \frac{1}{3} \).

Step 1: Understand the Problem

We need to find the product of two fractions: \( \frac{6x+6}{5} \) and \( \frac{10}{36x+36} \). Then, we will simplify the resulting expression if possible.

Step 2: Multiply the Numerators

Multiply the numerators of the two fractions: \((6x+6) \times 10 = 60x + 60\).

Step 3: Multiply the Denominators

Multiply the denominators of the two fractions: \(5 \times (36x + 36) = 180x + 180\).

Step 4: Form the Resulting Fraction

Combine the results into a single fraction: \( \frac{60x + 60}{180x + 180} \).

Step 5: Factor the Numerator and Denominator

Factor the numerator: \(60x + 60 = 60(x + 1)\). Factor the denominator: \(180x + 180 = 180(x + 1)\).

Step 6: Simplify the Fraction

Cancel the common factor \((x + 1)\) from both the numerator and the denominator: \( \frac{60(x + 1)}{180(x + 1)} = \frac{60}{180} \).

Step 7: Simplify Further

Simplify \( \frac{60}{180} \) by reducing the fraction using the greatest common divisor, which is 60: \( \frac{60}{180} = \frac{1}{3} \).

Key Concepts

Factoring Algebraic Expressions

Factoring an algebraic expression is like finding out what numbers or expressions are multiplied together to create your original expression. Imagine turning a big, bulky expression into a neat package of its smaller parts.
To factor an expression, first, look for the greatest common factor (GCF) in all the terms. A GCF is the largest expression that can divide each term of your algebraic expression without leaving a remainder. Once you have found the GCF, you factor it out, effectively "unpacking" the expression.- For example, to factor the expression \(60x + 60\), we can see both terms are divisible by 60. Thus, \(60x + 60 = 60(x + 1)\).- Similarly, for \(180x + 180\), the GCF is 180, so \(180x + 180\) becomes \(180(x + 1)\).Factoring is essential because it simplifies complex problems and is a crucial step in solving many algebraic equations. By recognizing patterns, such as common factors or the difference of squares, you can factor algebraic expressions efficiently.

Simplifying Fractions

Simplifying fractions is like cleaning up your workspace to make ideas clear and straightforward. It's the process of making a fraction as simple as possible. Start by identifying any common factors in the numerator and denominator, which can be cancelled out. This helps create an equivalent fraction that’s easier to work with.- Say we have the fraction \( \frac{60}{180} \). Both of these numbers share a common factor of 60.- By dividing the numerator and the denominator by 60, we simplify the fraction to \( \frac{1}{3} \).Remember, the goal is to make your fraction simple but still equivalent to its original form. A simplified fraction is easier to interpret and often more useful for further calculations or applications. Think of it as placing your answer in its purest and most elegant form.

Multiplying Fractions

Multiplying fractions involves the straightforward task of multiplying across the numerators and across the denominators. The real trick lies in simplifying each step to keep the process manageable and efficient.Begin by multiplying the numerators together and then the denominators together. This creates a new fraction that represents your answer.- Consider multiplying \( \frac{6x+6}{5} \) and \( \frac{10}{36x+36} \). Multiplying the numerators gives us \(60x + 60\), and multiplying the denominators gives \(180x + 180\).Once you have your new fraction, it's time to simplify. First, factor both the numerator and the denominator, looking for the greatest common factor. This step helps reduce the fraction to its simplest terms, revealing the final, clean result. Multiplying fractions might seem tricky at first, but with practice, you can easily break down and simplify algebraic expressions to make math work for you.