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Magazine Chapter 7: Problem 87
Simplify. \(\frac{\frac{x}{6}}{\frac{2 x}{3}+\frac{1}{2}}\)
Short Answer
Expert verified
\( \frac{x}{4x + 3} \)
Step by step solution
01
Simplify the denominator
First, we need to simplify the denominator of the fraction, which is \( \frac{2x}{3} + \frac{1}{2} \). To do this, find a common denominator. The common denominator of 3 and 2 is 6.
02
Adjust each term in the denominator to have the same denominator
Convert each fraction in the denominator to have 6 as the denominator:\( \frac{2x}{3} = \frac{2x \times 2}{3 \times 2} = \frac{4x}{6} \)\( \frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} \)
03
Combine the fractions in the denominator
Add the fractions \( \frac{4x}{6} \) and \( \frac{3}{6} \) to form the unified denominator:\[ \frac{4x}{6} + \frac{3}{6} = \frac{4x + 3}{6} \]
04
Divide the numerator by the simplified denominator
Now that the denominator is simplified to \( \frac{4x + 3}{6} \), divide the numerator, \( \frac{x}{6} \), by this expression.This is equivalent to multiplying by the reciprocal of the denominator:\[ \frac{x}{6} \div \frac{4x + 3}{6} = \frac{x}{6} \times \frac{6}{4x + 3} \]
05
Simplify the resulting expression
We can now multiply the fractions:\[ \frac{x}{6} \times \frac{6}{4x + 3} = \frac{x \times 6}{6 \times (4x + 3)} = \frac{x}{4x + 3} \]The 6 in the numerator and denominator cancel out.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Denominator
When working with fractions, finding a common denominator is essential, especially when you need to add or subtract them. A common denominator is a shared multiple of the denominators of the fractions involved.
Here's how you find one:
Here's how you find one:
- Identify the denominators of the fractions.
- Determine the least common multiple (LCM) of these denominators.
- Adjust each fraction so that its denominator matches this LCM.
Reciprocals
Reciprocals are frequently used in division problems involving fractions. The reciprocal of a fraction is simply flipping the numerator and the denominator.
For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \). When dividing by a fraction, multiplying by its reciprocal simplifies the process:
For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \). When dividing by a fraction, multiplying by its reciprocal simplifies the process:
- The division \( \frac{a}{b} \div \frac{c}{d} \) becomes \( \frac{a}{b} \times \frac{d}{c} \).
- This means swapping and then multiplying gives the same result.
Fractions
Fractions are a way to represent parts of a whole. They are written as \( \frac{a}{b} \) where \( a \) is the numerator and \( b \) is the denominator. The key with fractions is understanding how to manipulate them for operations like addition, subtraction, multiplication, and division.
These operations often involve:
These operations often involve:
- Finding a common denominator for addition and subtraction.
- Multiplying numerators and denominators separately.
- Using reciprocals for division by fractions.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions using various algebra techniques. This includes distributing, combining like terms, and factoring, to name a few.
Considerations while simplifying include:
Considerations while simplifying include:
- Carefully observing the operations to be executed.
- Applying reciprocal and common denominator strategies.
- Canceling common factors in numerators and denominators.
