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Chapter 1: Problem 2

$$\frac{3 x}{x+5} \div \frac{6}{4 x+20}$$ Which of the following is equivalent to the expression above, given that \(x \neq-5\) ? $$\begin{array}{l}{\text { (A) } 2 x} \\ {\text { (B) } \frac{x}{2}} \\\ {\text { (B) } \frac{9 x}{2}} \\ {\text { (D) } 2 x+4}\end{array}$$

Short Answer

Expert verified
All steps simplified to obtain the expression as \(2x\). Therefore, Option (A) is correct.

Step by step solution

01

- Rewrite the Division as Multiplication

Rewrite the division of fractions as multiplication by the reciprocal. The given expression is \[ \frac{3x}{x+5} \div \frac{6}{4x+20} \] This can be rewritten as \[ \frac{3x}{x+5} \times \frac{4x+20}{6} \]
02

- Simplify the Second Fraction

Simplify the second fraction \( \frac{4x+20}{6} \). Factor the numerator: \[4x + 20 = 4(x + 5)\] Now we have: \[ \frac{4(x+5)}{6} \]
03

- Multiplication of the Fractions

Multiply the two fractions: \[ \frac{3x}{x+5} \times \frac{4(x+5)}{6} \] This can be written as one single fraction: \[ \frac{3x \times 4(x+5)}{(x+5) \times 6} \]
04

- Simplify the Expression

Simplify the expression by canceling out common terms in the numerator and the denominator. The \(x+5\) terms cancel out, and the expression simplifies to: \[ \frac{3x \times 4}{6} \] Further simplifying: \[ \frac{12x}{6} = 2x \]
05

- Identify the Correct Option

We simplified the expression to \(2x\). Comparing this with the given options, we find that the equivalent expression is: \(2x\) Thus, the correct option is (A).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

fraction division
To divide fractions, the process involves converting the division operation into a multiplication operation. This is done by multiplying the first fraction by the reciprocal of the second fraction.
The reciprocal of a fraction is created by swapping its numerator and denominator. For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
In the given exercise: \(\frac{3x}{x+5} \div \frac{6}{4x+20}\), we rewrite it to multiply by the reciprocal of \(\frac{6}{4x+20}\):
Therefore, \(\frac{3x}{x+5} \div \frac{6}{4x+20}\) becomes \(\frac{3x}{x+5} \times \frac{4x+20}{6}\). This transforms the division problem into a multiplication, making it simpler to solve.
fraction simplification
Fraction simplification is about reducing a fraction to its simplest form. This involves factoring out the greatest common divisor (GCD) from both the numerator and the denominator.
In our exercise, we simplify the fraction \(\frac{4x+20}{6}\).
First, factor out common terms in the numerator. Factoring \((4x + 20)\) gives \((4(x + 5))\).
Now, the fraction becomes \(\frac{4(x+5)}{6}\).
Simplification helps in later steps, particularly in canceling out common factors. This makes computations more straightforward and avoids handling larger numbers unnecessarily.
algebraic expressions
Algebraic expressions are mathematical phrases that include numbers, variables, and operation symbols. In this problem, \(\frac{3x}{x+5}\) and \(\frac{4(x+5)}{6}\) are algebraic expressions involving the variable \(x\).
Such expressions can be simplified, combined, or factorized depending on the operation required.
For instance, during multiplication, the expressions \(\frac{3x}{x+5} \times \frac{4(x+5)}{6}\) become \(\frac{3x \times 4(x+5)}{6(x+5)}\). Understanding the properties of algebraic expressions, such as how to factor and simplify them, is crucial for problem-solving.
multiplication of fractions
Multiplying fractions is straightforward compared to addition or subtraction. To multiply two fractions, multiply the numerators together and the denominators together.
In the exercise, we had \(\frac{3x}{x+5} \times \frac{4(x+5)}{6}\).
Multiply their numerators: \(3x \times 4(x+5)\).
Multiply their denominators: \(x+5 \times 6\).
This results in a single fraction: \- \(\frac{3x \times 4(x+5)}{(x+5) \times 6})\). Before calculating, we simplify by canceling out common terms, ultimately leading to \(\frac{12x}{6} = 2x\).
Ensuring clarity in fractions simplification and multiplication facilitates reaching the correct solution efficiently.

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