Problem 49
Question
Perform the indicated operations. $$ \frac{x+3}{4} \div \frac{2 x-1}{4} $$
Step-by-Step Solution
Verified Answer
\( \frac{x+3}{2x-1} \)
1Step 1: Understand the Division of Fractions
To divide fractions, we need to multiply by the reciprocal. The problem can be rewritten by flipping the second fraction as follows: \( \frac{x+3}{4} \cdot \frac{4}{2x-1} \).
2Step 2: Simplify the Expression
Before multiplying the numerators and denominators, observe that the 4 in the numerator of the second fraction and the 4 in the denominator of the first fraction will cancel each other out: \( \frac{x+3}{1} \cdot \frac{1}{2x-1} \).
3Step 3: Multiply the Remaining Fractions
After canceling the 4s, multiply the remaining numerators and denominators: \( \frac{(x+3) \cdot 1}{1 \cdot (2x-1)} = \frac{x+3}{2x-1} \).
4Step 4: Simplify If Possible
Check for further simplification. The fraction \( \frac{x+3}{2x-1} \) cannot be simplified further, as there are no common factors or terms that can cancel out.
Key Concepts
Understanding Reciprocals in Fraction DivisionSimplifying Expressions Before MultiplyingCanceling Terms for Simplification
Understanding Reciprocals in Fraction Division
When dividing fractions, the term 'reciprocal' is key. The reciprocal of a fraction is formed by flipping its numerator and denominator. For instance, the reciprocal of \( \frac{2x - 1}{4} \) is \( \frac{4}{2x - 1} \). This concept is crucial because dividing by a fraction is the same as multiplying by its reciprocal.
This means, in our exercise, to perform the division \( \frac{x+3}{4} \div \frac{2x-1}{4} \), we first rewrite it using multiplication by the reciprocal: \( \frac{x+3}{4} \times \frac{4}{2x-1} \). This step is often the most challenging part for students, but remembering that division can be recast as multiplication can simplify the process.
Therefore, having a solid grasp of finding and using reciprocals will help make fraction division seem less daunting.
This means, in our exercise, to perform the division \( \frac{x+3}{4} \div \frac{2x-1}{4} \), we first rewrite it using multiplication by the reciprocal: \( \frac{x+3}{4} \times \frac{4}{2x-1} \). This step is often the most challenging part for students, but remembering that division can be recast as multiplication can simplify the process.
Therefore, having a solid grasp of finding and using reciprocals will help make fraction division seem less daunting.
Simplifying Expressions Before Multiplying
Simplifying expressions is an essential step in mathematical operations. Simplification makes calculations easier and often prevents mistakes. In our fraction division, simplifying the expression involves recognizing that the number 4 appears both in the numerator of the second fraction and the denominator of the first fraction.
When you see this number appearing in both positions, you can "cancel" it because dividing by 4 and multiplying by 4 undo each other. The expression \( \frac{x+3}{4} \times \frac{4}{2x-1} \) simplifies to \( \frac{x+3}{1} \times \frac{1}{2x-1} \). This step makes it clear that the expression becomes simpler and indicates readiness for the next multiplication step.
Simplifying expressions can involve canceling numbers, like the 4s here, or spotting common factors. This makes your calculations cleaner and more straightforward.
When you see this number appearing in both positions, you can "cancel" it because dividing by 4 and multiplying by 4 undo each other. The expression \( \frac{x+3}{4} \times \frac{4}{2x-1} \) simplifies to \( \frac{x+3}{1} \times \frac{1}{2x-1} \). This step makes it clear that the expression becomes simpler and indicates readiness for the next multiplication step.
Simplifying expressions can involve canceling numbers, like the 4s here, or spotting common factors. This makes your calculations cleaner and more straightforward.
Canceling Terms for Simplification
Canceling terms is a technique used to simplify fractions by reducing them to their simplest form. It’s essential to accurately identify terms that can be canceled.
In the context of our exercise, canceling the common term 4 in the expression \( \frac{x+3}{4} \times \frac{4}{2x-1} \) is a mathematical process of removing equivalent numbers from the numerator and denominator across the fractions. This results in \( \frac{x+3}{1} \times \frac{1}{2x-1} \), where the number 1 indicates that the terms have been fully reduced.
It's important to note that canceling terms simplifies the calculation but doesn't change the mathematical meaning or value of the original expression. By canceling terms, we reduce complexity, making it easier to perform further operations, like multiplication here. Always be cautious to only cancel terms that are truly common so the mathematical integrity is maintained.
In the context of our exercise, canceling the common term 4 in the expression \( \frac{x+3}{4} \times \frac{4}{2x-1} \) is a mathematical process of removing equivalent numbers from the numerator and denominator across the fractions. This results in \( \frac{x+3}{1} \times \frac{1}{2x-1} \), where the number 1 indicates that the terms have been fully reduced.
It's important to note that canceling terms simplifies the calculation but doesn't change the mathematical meaning or value of the original expression. By canceling terms, we reduce complexity, making it easier to perform further operations, like multiplication here. Always be cautious to only cancel terms that are truly common so the mathematical integrity is maintained.
Other exercises in this chapter
Problem 48
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