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Chapter 1: Problem 95

Simplify. $$ \frac{27 p q r s}{9 p r s t} $$

Short Answer

Expert verified
The simplified expression is \( \frac{3 q}{t} \).

Step by step solution

01

Identify Common Factors

Look at the numerator and the denominator to identify common factors. In the given fraction, both the numerator and the denominator contain the variables \( p \), \( r \), and \( s \).
02

Simplify the Coefficients

Divide the coefficients in the numerator and the denominator. In this case, divide 27 by 9: \[ \frac{27}{9} = 3 \]
03

Simplify the Variables

Divide the common variables in both the numerator and the denominator: \[ \frac{p q r s}{p r s t} = \frac{3 q}{t} \] because \( p \), \( r \), and \( s \) cancel each other out.
04

Write the Simplified Expression

Combine the simplified coefficients and the remaining variables to write the final simplified fraction: \[ \frac{3 q}{t} \]

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Key Concepts

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

identifying common factors
When simplifying algebraic fractions, the first step is identifying common factors in both the numerator and the denominator. Look for shared variables and coefficients. In our example, the fraction \(\frac{27 p q r s}{9 p r s t}\), we notice that the variables \( p \), \( r \), and \( s \) appear in both parts of the fraction. It's crucial to spot these common elements, as they can be divided out to simplify the fraction. This step makes the following steps in the simplification process more straightforward.
simplifying coefficients
After identifying common factors, the next step is to simplify the coefficients鈥攖he numerical parts of the fraction. In our case, we have 27 in the numerator and 9 in the denominator. To simplify, divide 27 by 9: \[ \frac{27}{9} = 3 \]. Now, our fraction looks more manageable, and we write it as: \(\frac{3 p q r s}{p r s t}\). Simplifying coefficients helps in reducing the numerical complexity, making it easier to work with the fraction.
canceling variables
In this step, we focus on canceling out common variables in the numerator and the denominator. Look at \(\frac{3 p q r s}{p r s t}\). We see that \( p \), \( r \), and \( s \) are present in both parts. By dividing these out, we effectively 'cancel' them: \[ \frac{p r s}{p r s} = 1 \] leaving us with \[ \frac{3 q}{t} \]. Canceling variables is key to reducing the fraction to its simplest form by eliminating redundant parts.
fraction simplification
Bringing all steps together, fraction simplification is the process of reducing a fraction to its simplest form. For \(\frac{27 p q r s}{9 p r s t}\), after identifying common factors, simplifying coefficients, and canceling variables, we end up with \[ \frac{3 q}{t} \]. Always remember to tackle one element at a time鈥攕tarting with coefficients, then moving to variables. This detailed yet straightforward approach ensures accurate and easy simplification for any algebraic fraction.

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