Problem 65
Question
For exercises \(25-68\), evaluate or simplify. $$ \frac{3}{3+\frac{3}{3+x}} $$
Step-by-Step Solution
Verified Answer
\( \frac{3 + x}{4 + x} \)
1Step 1 - Simplify the Inner Fraction
Simplify the inner fraction inside the denominator. Start with \ \ \ \ \ \[ \frac{3}{3 + \frac{3}{3 + x}} \] \ \ \ \ Simplify the inner fraction \ \[ \frac{3}{3 + x} \]
2Step 2 - Combine Fractions
Combine the fractions in the denominator. Rewrite \ \ \ \ \[ 3 + \frac{3}{3 + x} \] using a common denominator: \ \ \ \ \[ 3 + \frac{3}{3 + x} = \frac{3(3 + x) + 3}{3 + x} = \frac{9 + 3x + 3}{3 + x} = \frac{12 + 3x}{3 + x} \]
3Step 3 - Divide by the Fraction
Divide the numerator by the fraction obtained in the denominator. You have: \ \ \ \ \[ \frac{3}{\frac{12 + 3x}{3 + x}} \] which is equivalent to multiplying by the reciprocal: \ \ \ \ \[ 3 \cdot \frac{3 + x}{12 + 3x} = \frac{3(3 + x)}{12 + 3x} \]
4Step 4 - Simplify the Fraction
Simplify the resulting fraction: \ \ \ \ \[ \frac{3(3 + x)}{3(4 + x)} = \frac{3 + x}{4 + x} \]
Key Concepts
FractionsCommon DenominatorReciprocalSimplification
Fractions
Fractions represent a part of a whole. They consist of a numerator and a denominator. The numerator (top number) indicates how many parts we have, while the denominator (bottom number) denotes the total number of parts the whole is divided into. For instance, in the fraction \(\frac{3}{4}\), 3 is the numerator, and 4 is the denominator.
Understanding fractions is crucial when simplifying complex fractions, such as \(\frac{3}{3 + \frac{3}{3 + x}}\). Recognizing that each fraction is a division problem will help you approach the simplification process step by step.
In complex fractions, we deal with fractions within fractions, requiring a structured approach to simplify them effectively.
Understanding fractions is crucial when simplifying complex fractions, such as \(\frac{3}{3 + \frac{3}{3 + x}}\). Recognizing that each fraction is a division problem will help you approach the simplification process step by step.
In complex fractions, we deal with fractions within fractions, requiring a structured approach to simplify them effectively.
Common Denominator
When combining fractions, a common denominator is essential. It means we convert each fraction to have the same denominator, allowing us to add or subtract them conveniently.
In the step-by-step solution, after identifying the inner fraction \(\frac{3}{3 + x}\), notice the next step: \(\frac{12 + 3x}{3 + x}\). This involves finding a common denominator ---- in this case, \(3 + x\).
Combining \(3 + \frac{3}{3 + x}\) into a single fraction requires expressing 3 as \(\frac{3(3 + x)}{3 + x}\). So, always focus on rewriting terms to showcase the common denominator, making the math easier to handle.
In the step-by-step solution, after identifying the inner fraction \(\frac{3}{3 + x}\), notice the next step: \(\frac{12 + 3x}{3 + x}\). This involves finding a common denominator ---- in this case, \(3 + x\).
Combining \(3 + \frac{3}{3 + x}\) into a single fraction requires expressing 3 as \(\frac{3(3 + x)}{3 + x}\). So, always focus on rewriting terms to showcase the common denominator, making the math easier to handle.
Reciprocal
The reciprocal of a fraction simply flips its numerator and denominator. For example, the reciprocal of \(\frac{5}{7}\) is \(\frac{7}{5}\).
In the simplification process, at the step where we have \(\frac{3}{\frac{12 + 3x}{3 + x}}\), we multiply by the reciprocal of the fraction in the denominator. This converts the division into multiplication:
\(\frac{3}{\frac{12 + 3x}{3 + x}} \rightarrow 3 \cdot \frac{3 + x}{12 + 3x}\).
Multiplying by the reciprocal helps us eliminate the complex fraction structure and proceed to simple fraction operations, streamlining our path to the solution.
In the simplification process, at the step where we have \(\frac{3}{\frac{12 + 3x}{3 + x}}\), we multiply by the reciprocal of the fraction in the denominator. This converts the division into multiplication:
\(\frac{3}{\frac{12 + 3x}{3 + x}} \rightarrow 3 \cdot \frac{3 + x}{12 + 3x}\).
Multiplying by the reciprocal helps us eliminate the complex fraction structure and proceed to simple fraction operations, streamlining our path to the solution.
Simplification
Simplification is the process of making an expression easier to understand or work with. In fractions, it involves reducing the numerator and denominator to their smallest terms by finding any common factors.
In our example, after multiplying by the reciprocal, we end up with: \(\frac{3(3 + x)}{12 + 3x}\). Notice here that the term 3 is common in the numerator and the denominator. We can factor out the 3, leading to: \(\frac{3(3 + x)}{3(4 + x)} = \frac{3 + x}{4 + x}\).
Removing common factors is a key part of simplification, making the fraction easier to interpret and work with in subsequent mathematical problems.
In our example, after multiplying by the reciprocal, we end up with: \(\frac{3(3 + x)}{12 + 3x}\). Notice here that the term 3 is common in the numerator and the denominator. We can factor out the 3, leading to: \(\frac{3(3 + x)}{3(4 + x)} = \frac{3 + x}{4 + x}\).
Removing common factors is a key part of simplification, making the fraction easier to interpret and work with in subsequent mathematical problems.
Other exercises in this chapter
Problem 64
For exercises 39-82, simplify. $$ \frac{3 a+6}{a^{2}+4 a+4} \div \frac{a+4}{a+2} $$
View solution Problem 65
For exercises \(65-68\), evaluate. $$ \sqrt{16} $$
View solution Problem 65
For exercises 39-82, simplify. $$ \frac{3 p-1}{8 p} \div \frac{3 p^{2}+14 p-5}{6 p^{2}} $$
View solution Problem 67
For exercises \(67-82\), use the five steps and a proportion. About five of 100 pregnant women have pre-eclampsia, a condition that results in high blood pressu
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