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Chapter 6: Problem 17

Simplify each complex fraction. Use either method. $$ \frac{\frac{m+2}{3}}{\frac{m-4}{m}} $$

Short Answer

Expert verified
\( \frac{m^2 + 2m}{3(m-4)} \)

Step by step solution

01

Understand the Problem

The given complex fraction is \(\frac{\frac{m+2}{3}}{\frac{m-4}{m}}\). The goal is to simplify this by eliminating the complex fraction.
02

Rewrite the Complex Fraction

Express the complex fraction as a single fraction: \(\frac{m+2}{3} \div \frac{m-4}{m}\). Remember that dividing by a fraction is the same as multiplying by its reciprocal.
03

Multiply by the Reciprocal

Rewrite the division as a multiplication: \(\frac{m+2}{3} \cdot \frac{m}{m-4}\).
04

Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together: \(\frac{(m+2) \cdot m}{3 \cdot (m-4)} = \frac{m(m+2)}{3(m-4)}\).
05

Simplify the Expression

Distribute the \(m\) in the numerator: \(\frac{m^2 + 2m}{3(m-4)}\). The fraction is now simplified.

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Key Concepts

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

Division by a Fraction
One of the key skills for simplifying complex fractions is understanding how to handle division by a fraction. When faced with a division problem involving fractions, such as \(\frac{a}{b} \div \frac{c}{d}\), it's important to remember that dividing by a fraction is equivalent to multiplying by its reciprocal. This means you can convert the division into a multiplication problem by flipping the second fraction.
Multiplying by the Reciprocal
Once you've rewritten your division problem as a multiplication one, the next step is to multiply by the reciprocal. For instance, in the step where \(\frac{m+2}{3} \div \frac{m-4}{m}\) is converted to \(\frac{m+2}{3} \cdot \frac{m}{m-4}\), the division by \(\frac{m-4}{m}\) is changed to multiplication by \(\frac{m}{m-4}\). After this transformation, you can simply proceed by multiplying the numerators and denominators across.
Simplifying Algebraic Expressions
After transforming and multiplying the fractions, the final step involves simplifying the algebraic expression. This usually involves distributing terms and combining like terms. In our example, \(\frac{(m+2) \cdot m}{3 \cdot (m-4)}\) becomes \(\frac{m(m+2)}{3(m-4)}\). Then, by distributing the \m\ in the numerator, we simplify it to \(\frac{m^2 + 2m}{3(m-4)}\), resulting in a simpler, more manageable fraction.

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Most popular questions from this chapter

\(\frac{3 r}{5 r-5}=\frac{?}{15 r-15}\)

Find the slope of the line that passes through each pair of points. This will involve simplifying complex fractions. $$ \left(\frac{1}{2}, \frac{5}{12}\right) \text { and }\left(\frac{1}{4}, \frac{1}{3}\right) $$

Simplify each expression, using only positive exponents in the answer. $$ \frac{x^{-1}+2 y^{-1}}{2 y+4 x} $$

The fractions here are continued fractions. Simplify by starting at "the bottom" and working upward. $$ 1+\frac{1}{1+\frac{1}{1+1}} $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{x}{y}-\frac{y}{x}}{\frac{x}{y}+\frac{y}{x}} $$
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