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Chapter 7: Problem 47

Divide and, if possible, simplify. $$\frac{x}{4} \div \frac{5}{x}$$

Short Answer

Expert verified
\( \frac{x^2}{20} \)

Step by step solution

01

Rewrite the Division as Multiplication

When dividing by a fraction, you can multiply by its reciprocal. Rewrite \(\frac{x}{4} \div \frac{5}{x}\) as \(\frac{x}{4} \times \frac{x}{5}\).
02

Multiply the Fractions

Multiply the numerators together and the denominators together. So, \(\frac{x}{4} \times \frac{x}{5}\) becomes \(\frac{x \cdot x}{4 \cdot 5}\).
03

Simplify the Multiplication

Simplify the result by carrying out the multiplication. \(\frac{x \cdot x}{4 \cdot 5} \) simplifies to \(\frac{x^2}{20}\).

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Key Concepts

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

Reciprocal
Understanding the concept of a reciprocal is key to dividing fractions in algebra. The reciprocal, or multiplicative inverse, of a fraction is found by flipping its numerator and denominator. For example, the reciprocal of \(\frac{5}{x}\) is \(\frac{x}{5}\). This is because multiplying a fraction by its reciprocal results in 1: \(\frac{5}{x} \times \frac{x}{5} = 1\).

Knowing how to find reciprocals allows us to convert a division problem into a multiplication problem, making it easier to solve.
Multiplying Fractions
Multiplying fractions involves a straightforward process. To multiply two fractions, you multiply the numerators together and the denominators together.

For example, in the problem \(\frac{x}{4} \div \frac{5}{x}\), we first convert the division into multiplication: \(\frac{x}{4} \div \frac{5}{x} = \frac{x}{4} \times \frac{x}{5}\). Next, we multiply the numerators: \(x \times x = x^2\), and the denominators: \(4 \times 5 = 20\). This gives us the fraction \(\frac{x^2}{20}\).

This step is essential for simplifying complex algebraic expressions.
Simplifying Algebraic Expressions
Simplifying algebraic expressions helps to make them more manageable and easier to understand. After performing operations such as multiplying fractions, we look to reduce the expression to its simplest form.

In our example, after converting the division to multiplication and doing the multiplication, we ended up with \(\frac{x^2}{20}\). Always check if the numerator and the denominator have common factors that can be canceled out.

In this case, \(x^2\) and 20 have no common factors, so \(\frac{x^2}{20}\) is already in its simplest form. Simplification is crucial for clarity in algebraic calculations.

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Most popular questions from this chapter

Write an equivalent expression without parentheses. $$-(3-a)$$

For each pair of functions fand \(g,\) find all values of a for which \(f(a)=g(a)\). $$f(x)=\frac{x+3}{x+2}-\frac{x+4}{x+3}, g(x)=\frac{x+5}{x+4}-\frac{x+6}{x+5}$$

If two quantities vary directly, does this mean that one is "caused" by the other? Why or why not?

Find equivalent expressions that have the LCD. $$\frac{3}{10 a^{3}}, \frac{b}{5 a^{6}}$$

Find the LCM. $$9 x^{3}-9 x^{2}-18 x, 6 x^{5}-24 x^{4}+24 x^{3}$$
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