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Magazine Chapter 6: Problem 20
Simplify each complex fraction. Use either method. $$ \frac{6+\frac{2}{r}}{\frac{3 r+1}{4}} $$
Short Answer
Expert verified
\(\frac{24r + 8}{r(3r + 1)}\)
Step by step solution
01
- Simplify the numerator
The numerator is given as \(6 + \frac{2}{r}\). To simplify this, find a common denominator. The common denominator for \(6\) (which is \(6/1\)) and \(\frac{2}{r}\) is \(r\):\[6 + \frac{2}{r} = \frac{6r}{r} + \frac{2}{r} = \frac{6r + 2}{r}\]
02
- Simplify the denominator
The denominator is given as \(\frac{3r + 1}{4}\). It is already in its simplest form, so no further simplification is necessary.
03
- Combine simplified forms
Combine the simplified forms of the numerator and the denominator to rewrite the complex fraction:\[\frac{\frac{6r + 2}{r}}{\frac{3r + 1}{4}} \]
04
- Simplify the complex fraction
To simplify a complex fraction, multiply the numerator by the reciprocal of the denominator:\[\frac{\frac{6r + 2}{r}}{\frac{3r + 1}{4}} = \frac{6r + 2}{r} \times \frac{4}{3r + 1} \]
05
- Perform the multiplication
Multiply the numerators and the denominators:\[\left(\frac{6r + 2}{r}\right) \times \left(\frac{4}{3r + 1}\right) = \frac{(6r + 2) \cdot 4}{r(3r + 1)} \]
06
- Simplify the result
Distribute and simplify the numerator, while leaving the denominator as is:\[\frac{(6r + 2) \cdot 4}{r(3r + 1)} = \frac{24r + 8}{r(3r + 1)} \]
07
- Factor and simplify further if possible
Factor out any common terms in the numerator. In this case, there is no further simplification possible, so the final simplified form is:\[\frac{24r + 8}{r(3r + 1)}\]
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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 a crucial step for addition or subtraction. The common denominator is the least common multiple (LCM) of the denominators.
For instance, in the problem \(6 + \frac{2}{r}\), we need a common denominator to combine the terms. Here, the common denominator for \(6 = \frac{6}{1}\) and \(\frac{2}{r}\) is \(r\).
You rewrite each term so their denominators match:
For instance, in the problem \(6 + \frac{2}{r}\), we need a common denominator to combine the terms. Here, the common denominator for \(6 = \frac{6}{1}\) and \(\frac{2}{r}\) is \(r\).
You rewrite each term so their denominators match:
- \(6 = \frac{6r}{r}\)
- So, \(6 + \frac{2}{r} = \frac{6r}{r} + \frac{2}{r} = \frac{6r + 2}{r}\)
Reciprocal
The reciprocal of a fraction flips the numerator and the denominator.
For example, the reciprocal of \(\frac{3r + 1}{4}\) is \(\frac{4}{3r + 1}\). In simplifying complex fractions, we often use the reciprocal to eliminate the nested fraction by multiplying.
In our example, when we're simplifying \(\frac{\frac{6r + 2}{r}}{ \frac{3r + 1}{4}} , \) we multiply by the reciprocal of the denominator: \(\frac{6r + 2}{r} \times \frac{4}{3r + 1}\).
This step turns division into multiplication, making the fraction simpler to handle.
For example, the reciprocal of \(\frac{3r + 1}{4}\) is \(\frac{4}{3r + 1}\). In simplifying complex fractions, we often use the reciprocal to eliminate the nested fraction by multiplying.
In our example, when we're simplifying \(\frac{\frac{6r + 2}{r}}{ \frac{3r + 1}{4}} , \) we multiply by the reciprocal of the denominator: \(\frac{6r + 2}{r} \times \frac{4}{3r + 1}\).
This step turns division into multiplication, making the fraction simpler to handle.
Multiplying Fractions
Multiplying fractions is straightforward: multiply the numerators together and the denominators together.
Let's look at \(\frac{\left(6r + 2\right)}{r} \times \frac{4}{3r + 1}\):
Understanding this step helps in simplifying the entire complex fraction.
Let's look at \(\frac{\left(6r + 2\right)}{r} \times \frac{4}{3r + 1}\):
- Numerators: \(\left(6r + 2\right) \times 4 = 24r + 8 \)
- Denominators: \(r \times \left(3r + 1\right) = r\left(3r + 1\right) \)
Understanding this step helps in simplifying the entire complex fraction.
Factoring
Factoring involves breaking down expressions into their constituent parts.
This can help in simplifying expressions, particularly when looking for common factors that can be canceled out.
In our problem, although \(24r + 8\) doesn't have a common factor, understanding factoring is critical for problems where it does apply.
Here's a quick reminder on factoring:
This can help in simplifying expressions, particularly when looking for common factors that can be canceled out.
In our problem, although \(24r + 8\) doesn't have a common factor, understanding factoring is critical for problems where it does apply.
Here's a quick reminder on factoring:
- Find the greatest common factor (GCF) of the terms.
- Rewrite each term as a product of the GCF and another factor.
- Apply the distributive property in reverse.
