- Simplify the numerator and denominator by finding a common denominator for each: the numerator becomes \(\frac{17-4a}{a-3}\) and the denominator becomes \(\frac{2a-5}{a-3}\).
- Perform the division of the two fractions by multiplying the simplified numerator by the reciprocal of the simplified denominator.
- Observe that the term \((a-3)\) cancels out, leaving the final simplified form: \(\frac{17-4a}{2a-5}\).
- Conclude that the first option describes the correct algebraic steps: "Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerator by the reciprocal of the denominator."
- Final Answer: \(\boxed{\text{Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerator by the reciprocal of the denominator.}}\)
Explanation
- Analyzing the Expression
The expression to simplify is \(\frac{\frac{5}{a-3}-4}{2+\frac{1}{a-3}}\). This is a complex fraction because both the numerator and the denominator contain their own fractional components. To simplify this type of expression, we must perform operations within the numerator and denominator individually and then address the division of the two resulting simplified fractions.
- Simplifying Numerator and Denominator
To simplify the numerator \(\frac{5}{a-3} - 4\), we find a common denominator, which is \(a-3\).
\[ \frac{5}{a-3} - 4 = \frac{5}{a-3} - \frac{4(a-3)}{a-3} = \frac{5 - 4a + 12}{a-3} = \frac{17 - 4a}{a-3} \]
Next, we simplify the denominator \(2 + \frac{1}{a-3}\) by using the same common denominator \(a-3\).
\[ 2 + \frac{1}{a-3} = \frac{2(a-3)}{a-3} + \frac{1}{a-3} = \frac{2a - 6 + 1}{a-3} = \frac{2a - 5}{a-3} \]
Now the entire expression is:
\[ \frac{\frac{17 - 4a}{a-3}}{\frac{2a - 5}{a-3}} \]
- Performing the Division
To divide two fractions, we multiply the numerator by the reciprocal of the denominator.
\[ \frac{\frac{17 - 4a}{a-3}}{\frac{2a - 5}{a-3}} = \frac{17 - 4a}{a-3} \cdot \frac{a-3}{2a - 5} \]
Canceling the common term \((a-3)\) in the numerator and denominator:
\[ \frac{17 - 4a}{2a - 5} \]
This confirms that the procedure of writing both parts with a common denominator and then multiplying the numerator by the reciprocal of the denominator is a correct and efficient way to simplify the expression.
- Selecting the Correct Procedure
Comparing this result with the given options, we identify the first option: "Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerator by the reciprocal of the denominator." This matches the logic performed. Other options describe incorrect algebraic procedures for division or less standard simplification methods. Thus, the first choice is correct.
Examples
Simplifying complex fractions is a fundamental skill in algebra that appears frequently in calculus and physics. For example, if you are analyzing a system where a variable represents a changing rate, such as a velocity or density formula involving fractions, you will often arrive at expressions that need to be simplified before you can solve for a specific value or take a derivative. Efficiently managing these algebraic steps ensures that complex engineering or scientific models remain readable and solvable.