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Chapter 7: Problem 48

Divide and, if possible, simplify. $$\frac{5}{x} \div \frac{x}{12}$$

Short Answer

Expert verified
The simplified expression is \(\frac{60}{x^2}\).

Step by step solution

01

Rewrite the Division as Multiplication

To divide by a fraction, multiply by its reciprocal. Rewrite the expression \(\frac{5}{x} \div \frac{x}{12}\) as \(\frac{5}{x} \times \frac{12}{x}\).
02

Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together. The expression becomes \(\frac{5 \times 12}{x \times x} = \frac{60}{x^2}\).
03

Simplify the Expression

Check if the expression can be further simplified. The fraction \(\frac{60}{x^2}\) is already in its simplest form because there are no common factors between the numerator and the denominator.

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Key Concepts

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

Reciprocal
When dividing fractions, one crucial idea is the concept of a reciprocal. The reciprocal of a number is what you multiply that number by to get 1. For a fraction, the reciprocal is simply reversing its numerator and denominator.

For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).

In the exercise, we divided \(\frac{5}{x}\) by \(\frac{x}{12}\). To do this, we took the reciprocal of \(\frac{x}{12}\), which is \(\frac{12}{x}\). This converts the division problem into a multiplication problem.

  • Original fraction: \(\frac{x}{12}\)
  • Reciprocal: \(\frac{12}{x}\)
Multiplication
Once we have the reciprocal, we switch from dividing to multiplying. So, instead of \(\frac{5}{x}\div\frac{x}{12}\), we multiply \(\frac{5}{x}\) by \(\frac{12}{x}\).

This step is straightforward: just multiply the numerators together and the denominators together.

  • Numerators: \(5\times12\)
  • Denominators: \(x\times x\)
Hence, we get \(\frac{5 \times 12}{x \times x} = \frac{60}{x^2}\).

This conversion from division to multiplication simplifies the fraction division process significantly.
Simplification
After multiplication, it's crucial to check if the resulting fraction can be simplified. Simplification means reducing the fraction to its simplest form. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD).

In our exercise, we ended up with \(\frac{60}{x^2}\). Here:
  • The numerator is 60
  • The denominator is \(x^2\)
Since there are no common factors between 60 and \(x^2\), this fraction is already in its simplest form. Always try to find common factors to simplify your results when possible.
Numerator and Denominator
Fractions consist of two parts: the numerator and the denominator. These terms are fundamental when dealing with fractions.

  • Numerator: The top number of a fraction, representing the part of the whole.
  • Denominator: The bottom number of a fraction, representing the total number of equal parts the whole is divided into.
For example, in the fraction \(\frac{5}{x}\), 5 is the numerator, and x is the denominator.

Understanding these basics helps in operations such as addition, subtraction, multiplication, and division of fractions. In our case, knowledge of numerators and denominators was key in the multiplication and simplification steps.

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