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Chapter 1: Problem 377

Divide. $$ \frac{3}{4} \div \frac{x}{11} $$

Short Answer

Expert verified
The result is \(\frac{33}{4x}\).

Step by step solution

01

- Understand the Division of Fractions

To divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
02

- Find the Reciprocal

The reciprocal of \(\frac{x}{11}\) is \(\frac{11}{x}\).
03

- Set Up the Multiplication

Rewrite the division problem as a multiplication problem: \(\frac{3}{4} \times \frac{11}{x}\).
04

- Multiply the Fractions

Multiply the numerators together and the denominators together: \(\frac{3 \times 11}{4 \times x} = \frac{33}{4x}\).

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Key Concepts

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

Reciprocal of a Fraction
Dividing fractions might seem a bit tricky at first. But once you understand the concept of the reciprocal, it becomes much easier. The reciprocal of a fraction is simply obtained by swapping its numerator (the top number) and its denominator (the bottom number).
For example, the reciprocal of \(\frac{3}{5}\) is \(\frac{5}{3}\). When we divide by a fraction, it is the same as multiplying by its reciprocal. This is key. So, if you see \(\frac{3}{4} \div \frac{x}{11}\), you should think of it as \(\frac{3}{4}\ \times\ \frac{11}{x}\).
Fraction Multiplication
Now that we know division by a fraction is the same as multiplication by its reciprocal, we can focus on multiplying the fractions. To multiply two fractions, follow these steps:
  • Multiply the numerators (top numbers) together.
  • Multiply the denominators (bottom numbers) together.
If we use the example from the exercise, we rewrite the division \(\frac{3}{4} \div \frac{x}{11}\) as multiplication, \(\frac{3}{4}\ \times\ \frac{11}{x}\).
Following the steps for multiplication, we get:
  • Numerator: \3 \times 11 = 33\
  • Denominator: \4 \times x = 4x\
Therefore, \(\frac{3}{4} \times \frac{11}{x}\) equals \(\frac{33}{4x}\).
Simplifying Fractions
After performing the multiplication of fractions, it's good to simplify the result if possible. Simplifying a fraction involves reducing it to its smallest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD).
In our example, the fraction we got is \(\frac{33}{4x}\). If there were a common factor between 33 and 4x, we would divide them by that factor to simplify.
However, in this particular case, 33 and 4x often have no common factors other than 1. So, \(\frac{33}{4x}\) is already in its simplest form. Remember to always check for common factors after multiplying to ensure your result is fully simplified.

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