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Chapter 4: Problem 150

Divide, and write the answer in simplified form. \(\frac{3}{4} \div \frac{x}{11}\)

Short Answer

Expert verified
\frac{33}{4x}

Step by step solution

01

Understand the Problem

We need to divide the fraction \(\frac{3}{4}\) by \( \frac{x}{11} \) and simplify the result.
02

Reciprocal of the Second Fraction

To divide by a fraction, multiply by its reciprocal. The reciprocal of \( \frac{x}{11} \) is \( \frac{11}{x} \).
03

Multiply the Fractions

Multiply \( \frac{3}{4} \) by the reciprocal of the second fraction \( \frac{11}{x} \).\( \frac{3}{4} \times \frac{11}{x} = \frac{3 \times 11}{4 \times x} = \frac{33}{4x} \).
04

Simplify the Result

We now have the fraction \( \frac{33}{4x} \), which is in its simplest form.

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Key Concepts

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

reciprocal
When dividing fractions, the reciprocal is a key concept. The reciprocal of a fraction is what you get when you flip the numerator (top number) and the denominator (bottom number). For example, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
Using the reciprocal makes dividing fractions easier. Instead of dividing, you multiply by the reciprocal of the fraction you鈥檙e dividing by.
In our problem, the reciprocal of \( \frac{x}{11} \) is \( \frac{11}{x} \).
This step lays the groundwork for multiplying fractions, which we'll explore next.
multiplying fractions
Once we have the reciprocal, we can easily multiply fractions. Multiplying fractions is straightforward.
You multiply the numerators together and the denominators together. For instance, with \( \frac{a}{b} \) and \( \frac{c}{d} \), the result of multiplying these fractions is \( \frac{ac}{bd} \).
In our current problem, the fractions to multiply are \( \frac{3}{4} \) and \( \frac{11}{x} \).
When we multiply \( \frac{3}{4} \) by \( \frac{11}{x} \, \) we get:
\[ \frac{3 \times 11}{4 \times x} = \frac{33}{4x} \].
Notice how we did not change the orders of multiplications since they follow fraction multiplication rules.
simplifying fractions
After multiplying, always check if the fraction can be simplified. A fraction is simplified when you cannot divide the numerator and the denominator by the same number any further.
Here鈥檚 a quick tip: Look for the greatest common divisor (GCD) of both the numerator and the denominator. Divide both by this number.
In our fraction \( \frac{33}{4x} \), there is no common factor between 33 and 4x that we can simplify with.
Thus, \( \frac{33}{4x} \) is already in its simplest form. Always confirm if additional content simplifications is needed to avoid any missing opportunities.

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Most popular questions from this chapter

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. 4\(p^{2} q\) when \(p=-\frac{1}{2}\) and \(q=\frac{5}{9}\)

Convert the improper fraction to \(a\) mixed number. $$ \frac{19}{5} $$

In the following exercises, solve the equation. $$ a-\frac{3}{10}=-\frac{9}{10} $$

In the following exercises, simplify. $$ -\frac{3}{13}+\left(-\frac{4}{13}\right) $$

In the following exercises, perform the indicated operations and simplify. $$\frac{11}{12}-\frac{2}{3}$$
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