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Chapter 7: Problem 34

Alternate Methods How can you simplify a complex fraction without inverting the divisor?

Short Answer

Expert verified
The complex fraction \(\frac{3}{4} / \frac{1}{2}\) can be simplified without resort to inverting the divisor. This is done by multiplying both fractions by the least common denominator (LCD), 4, resulting in simplified fraction \(\frac{3}{2}\).

Step by step solution

01

Understanding Complex Fractions

A complex or compound fraction is a fraction where the numerator, denominator, or both contain a fraction. For example, \(\frac{3}{4} / \frac{1}{2}\) is a complex fraction.
02

Traditional Method 鈥� Inverting and Multiplying

In ordinary fraction division operations, we follow the rule of inverting the divisor (the fraction on the bottom) and then multiplying. Using the example above, we would convert it to \(\frac{3}{4} * \frac{2}{1} = \frac{3 * 2}{4 * 1}\) . This simplifies to \(\frac{3}{2}\).
03

Alternative Method 鈥� Multiplication by the Least Common Denominator

An alternative method to simplify complex fractions is by multiplying the numerator and the denominator by the LCD. The LCD of the smaller denominators, in the case of our example, is 4. Hence, our original complex fraction will look like: \( (\frac{3}{4} * 4) / (\frac{1}{2} * 4) = \frac{3 * 4}{2 * 4}\). Once simplified, this results in \(\frac{3}{2}\). Thus, both methods arrive at the same solution.

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Key Concepts

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

Fraction Simplification
Simplifying fractions, whether complex or straightforward, always makes algebra easier. A complex fraction is a situation where there's a fraction in either the numerator or the denominator. When we simplify these, our goal is to make them look just like regular fractions again. For example, if we start with something like \(\frac{3/4}{1/2}\), we want to end up with something simpler like \(\frac{3}{2}\).
  • Start by identifying what part of the given fraction is complex.
  • Look for patterns or common factors that might help in simplifying the whole expression.
By practicing fraction simplification regularly, your overall grasp of fractions will become stronger, making future problems easier to tackle.
Least Common Denominator
The Least Common Denominator (LCD) is incredibly useful when dealing with fractions, especially complex ones. The LCD is the smallest number that each of the denominators in the fractions can divide into without leaving a remainder.
  • Identify the denominators in your complex fraction.
  • Find the smallest number that all these denominators divide evenly into 鈥� this is your LCD.
  • Use the LCD to transform the fractions into equivalent fractions that have a common base, simplifying the computation process.
When you multiply both the numerator and the denominator of a complex fraction by this LCD, it eliminates the fraction within fractions, simplifying the overall expression significantly. In our example, multiplying by the LCD of 4 helped simplify \(\frac{3/4}{1/2}\) to \(\frac{3}{2}\).
Alternative Methods in Algebra
Algebra often presents more than one approach to solving a problem, which can be both an advantage and a challenge. In the context of simplifying complex fractions, one alternative to the "invert and multiply" approach is using the Least Common Denominator method.
  • This approach removes the need for inverting fractions, by directly tackling the nested fractions to simplify the entire expression.
  • It's particularly useful for those who prefer dealing with smaller denominators directly, as it might offer more intuitive steps for some learners.
Exploring different methods in algebra not only strengthens understanding but also boosts flexibility in problem-solving, preparing you for a broader range of mathematical challenges.

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Most popular questions from this chapter

\(\frac{a+3}{4}-\frac{a-1}{6}=\frac{4}{3}\)

\(\frac{20-x}{x}=x\)

Writing A student submits the following argument to prove that 1 is equal to 0 . Discuss what is wrong with the argument. \(\begin{aligned} x=1 & \text { Original equation } \\ x^{2}=x & \text { Multiply each side by } x . \\ \frac{x(x-1)=0}{x-1)}=\frac{\text { Subtract } x \text { from each side. }}{x-1} & \text { Factor. } \\ x=0 & \text { Divide each side by } x-1 . \\ \text { Simplify. } \end{aligned}\)

\(\frac{3}{x-1}+\frac{5}{x+1}=\frac{8 x+5}{x^{2}-1}\)

\(\frac{3}{x}+\frac{1}{4}=\frac{2}{x}\)
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