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Chapter 6: Problem 34

Use either method to simplify each complex fraction. \(\frac{p-\frac{p+2}{4}}{\frac{3}{4}-\frac{5}{2 p}}\)

Short Answer

Expert verified
\( \frac{p(3p - 2)}{3p - 10} \)

Step by step solution

01

Simplify the Numerator

First, simplify the numerator of the complex fraction. The numerator is \[ p - \frac{p + 2}{4} \] To combine these terms, get a common denominator. The common denominator is 4: \[ p = \frac{4p}{4} \] Now we can write the numerator as: \[ \frac{4p}{4} - \frac{p+2}{4} = \frac{4p - (p + 2)}{4} = \frac{4p - p - 2}{4} = \frac{3p - 2}{4} \]
02

Simplify the Denominator

Next, simplify the denominator of the complex fraction. The denominator is \[ \frac{3}{4} - \frac{5}{2p} \] To combine these terms, get a common denominator. The common denominator is 4p: \[ \frac{3}{4} = \frac{3p}{4p} \] and \[ \frac{5}{2p} = \frac{10}{4p} \] Now we can write the denominator as: \[ \frac{3p}{4p} - \frac{10}{4p} = \frac{3p - 10}{4p} \]
03

Write the Complex Fraction

Now that the numerator and denominator have been simplified, the complex fraction can be written as: \[ \frac{\frac{3p - 2}{4}}{\frac{3p - 10}{4p}} \]
04

Simplify the Complex Fraction

Simplify the complex fraction by multiplying by the reciprocal of the denominator: \[ \frac{\frac{3p - 2}{4}}{\frac{3p - 10}{4p}} = \frac{3p - 2}{4} \times \frac{4p}{3p - 10} \] The 4's cancel out, leaving: \[ \frac{(3p - 2) \times p}{3p - 10} \] which simplifies to: \[ \frac{p(3p - 2)}{3p - 10} \]

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Key Concepts

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

Common Denominator
When simplifying complex fractions, a crucial first step is finding a common denominator. This helps us combine fractions by making sure they all share a common base.
For instance, look at the fraction \(\frac{p-\frac{p+2}{4}}{\frac{3}{4}-\frac{5}{2p}}\). To simplify the numerator, we notice that \(\frac{p+2}{4}\) and \(\frac{4p}{4}\) share a common denominator of 4.
To combine these, we rewrite p as \(\frac{4p}{4}\). Now our numerator can be simplified to:
\[ \frac{4p}{4} - \frac{p + 2}{4} = \frac{4p - (p + 2)}{4} = \frac{4p - p - 2}{4} = \frac{3p - 2}{4} \]
Numerator and Denominator Simplification
Simplifying both the numerator and denominator of a complex fraction is essential before further operations.
For our fraction, the complex numerator simplifies to \(\frac{3p-2}{4}\). Next, we simplify the complex denominator \(\frac{3}{4} - \frac{5}{2p}\).
Finding the common denominator for \(\frac{3}{4}\) and \(\frac{5}{2p}\) is important. Here, it is 4p:
\[ \frac{3}{4} = \frac{3p}{4p} \] and \[ \frac{5}{2p} = \frac{10}{4p} \]
We can now combine these fractions:
\[ \frac{3p}{4p} - \frac{10}{4p} = \frac{3p - 10}{4p} \]
This simplified denominator enables the next step in handling our complex fraction.
Multiplying by Reciprocal
In the final step of simplifying complex fractions, we use the reciprocal. A complex fraction like \(\frac{\frac{3p - 2}{4}}{\frac{3p - 10}{4p}}\) simplifies by multiplying by the reciprocal of its denominator: \(\frac{3p-10}{4p}\). Effectively, we switch the numerator and denominator of the fraction we are dividing by, making it easier to handle.
This leads to:
\[ \frac{\frac{3p - 2}{4}}{\frac{3p - 10}{4p}} = \frac{3p - 2}{4} \times \frac{4p}{3p - 10} \]
Notice that the 4s cancel out, simplifying to:
\[ \frac{(3p - 2) \times p}{3p - 10} \]
which results in our final answer:
\[ \frac{p(3p - 2)}{3p - 10} \]
Understanding these steps helps simplify any complex fraction effectively.

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Most popular questions from this chapter

Multiply or divide as indicated. $$ \frac{p^{2}-25}{4 p} \cdot \frac{2}{5-p} $$

A student was asked to simplify the rational expression $$ \frac{2 x+y}{4 x+2 y} $$ and wrote the incorrect answer \(\frac{x+y}{4 x+y}\) WHAT WENT WRONG? What is the correct answer?

Add or subtract as indicated. $$\frac{3}{x^{2}+4 x+4}+\frac{7}{x^{2}+5 x+6}$$

As review, multiply or divide the rational numbers as indicated. Write answers in lowest terms. $$ \frac{4}{21} \cdot \frac{7}{10} $$

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