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Chapter 6: Problem 30

Multiply or divide. Write each answer in lowest terms. $$ \frac{m-2}{4} \div \frac{2-m}{6} $$

Short Answer

Expert verified
-\frac{3}{2}

Step by step solution

01

- Rewrite the Division as Multiplication

To divide fractions, first rewrite the division operation as multiplication by the reciprocal. So, rewrite \ \ \( \frac{m-2}{4} \div \frac{2-m}{6} \ \ \) as \ \( \frac{m-2}{4} \times \frac{6}{2-m} \ \).
02

- Simplify the Reciprocal

Recall that \(2 - m = -(m - 2) \) and rewrite the expression. This changes our multiplication to \ \ \(\frac{m-2}{4} \times \frac{6}{-(m-2)} \ \ \).
03

- Cancel Common Factors

Notice that \(m - 2\) in \( \frac{m-2}{4}\) and \(m - 2\) in \( \frac{6}{-(m-2)}\) are common factors. Cancel these out: \ \ \( \frac{(m-2)}{4} \times \frac{6}{-(m-2)} = \frac{1}{4} \times \frac{6}{-1} \ \). Simplify to \ \ \( \frac{6}{-4} \ = -\frac{3}{2} \ \).
04

- Write the Final Answer in Lowest Terms

The simplified form of the fraction \( \frac{6}{-4} \) in lowest terms is \ \ \(-\frac{3}{2} \ \).

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Key Concepts

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

Fraction Division
When you need to divide one fraction by another, you actually have to multiply by the reciprocal of the second fraction. This might sound complex at first, but it鈥檚 a simple switch. For example, in the problem \(\frac{m-2}{4} \div \ \frac{2-m}{6}\), we rewrite it as \(\frac{m-2}{4} \times \frac{6}{2-m}\). This is the core concept behind fraction division. When we switch from division to multiplication, our job becomes simpler. We just replace the division sign with a multiplication sign and flip the second fraction.
Reciprocals
A reciprocal is what you get when you flip a fraction. For example, the reciprocal of \(\frac{2-m}{6}\) is \(\frac{6}{2-m}\). Knowing this helps when you need to divide fractions. When we flip the second fraction, we're essentially making it easier to work with. In our example, after flipping \(\frac{2-m}{6}\), we simplify it by rewriting \(2-m\) as \(-(m-2)\). So our expression becomes \(\frac{m-2}{4} \times \frac{6}{-(m-2)}\).
Simplifying Fractions
Simplifying fractions helps to make complex problems much simpler. To simplify, you divide both the numerator and the denominator by any common factors. In our example, \(\frac{m-2}{4} \times \frac{6}{-(m-2)}\), we see that \(m-2\) is a common factor in both fractions. By canceling these common factors, the expression simplifies to \(\frac{1}{4} \times \frac{6}{-1}\), which is much easier to handle. From here, we multiply the fractions normally to get \(\frac{6}{-4} = -\frac{3}{2}\). Simplifying fractions steps up our math game and makes solving problems more straightforward.

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Most popular questions from this chapter

Multiply or divide. Write each answer in lowest terms. $$ \frac{r^{2}+r-6}{r^{2}+4 r-12} \div \frac{r+3}{r-1} $$

involve factoring by grouping (Section 5.1) and factoring sums and differences of cubes (Section 5.4). Write each rational expression in lowest terms $$ \frac{k m+4 k-4 m-16}{k m+4 k+5 m+20} $$

Multiply or divide. Write each answer in lowest terms. $$ \frac{z^{2}-3 z+2}{z^{2}+4 z+3} \div \frac{z-1}{z+1} $$

Multiply or divide. Write each answer in lowest terms. $$ \frac{3 x^{2}-5 x-2}{x-2} \cdot \frac{x-3}{x+1} $$

Only one of these choices is equal to \(\frac{\frac{1}{3}+\frac{1}{12}}{\frac{1}{2}+\frac{1}{4}} .\) Which one is it? Answer this question without showing any work, and explain your reasoning. A. \(\frac{5}{9}\) B. \(-\frac{5}{9}\) C. \(-\frac{9}{5}\) D. \(-\frac{1}{12}\)
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