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Chapter 7: Problem 117

This will help you prepare for the material covered in the next section. In each exercise, perform the indicated operation. $$\frac{3}{4} \div \frac{1}{2}$$

Short Answer

Expert verified
The quotient of \(\frac{3}{4}\) and \(\frac{1}{2}\) is \(\frac{3}{2}\)

Step by step solution

01

Setup the problem

Rewrite the problem as \(\frac{3}{4} \div \frac{1}{2}\)
02

Convert to multiplication

Replace the division operation with a multiplication operation. The concept here is 'dividing by a fraction is the same as multiplying by its reciprocal'. Hence, the problem becomes \(\frac{3}{4} \times \frac{2}{1}\)
03

Perform multiplication

Multiply the fractions normally. Multiply the numerators together and the denominators together, resulting in a new fraction \(\frac{3 \times 2}{4 \times 1}\) which simplifies to \(\frac{6}{4}\)
04

Simplify

Lastly, we simplify the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This yields \(\frac{6}{4} = \frac{3}{2}\)

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Key Concepts

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

Reciprocal
A reciprocal is a special type of fraction that plays a crucial role when dividing fractions. To find the reciprocal of a fraction, simply flip the numerator and the denominator. For example, the reciprocal of \( \frac{1}{2} \) is \( \frac{2}{1} \). This method of flipping fractions helps in converting division problems into multiplication problems, which are usually simpler to solve.
  • Start with the original fraction.
  • Swap the places of the numerator and the denominator.
  • The new fraction is the reciprocal.
For example, if you have \( \frac{3}{4} \) divided by \( \frac{1}{2} \), convert \( \frac{1}{2} \) to \( \frac{2}{1} \). By using the reciprocal, dividing fractions becomes a process of multiplying fractions. This technique simplifies the calculation, making it easier to handle.
Simplifying Fractions
Simplifying fractions is the process of reducing them to their simplest form. A fraction is considered simplified when the numerator and the denominator have no common factors other than 1. Simplifying involves a few straightforward steps:
  • Determine if there are any common factors between the numerator and the denominator.
  • Divide both numbers by their greatest common divisor (GCD).
  • The result is the fraction in its simplest form.
For our example, with \( \frac{6}{4} \), you'll find the GCD of 6 and 4, which is 2. Divide both the numerator and the denominator by 2. Thus, \( \frac{6}{4} \) simplifies to \( \frac{3}{2} \). Remember, a simplest form fraction is not only more aesthetically pleasing but also easier to work with in more complex calculations.
Multiplying Fractions
Multiplying fractions is a fundamental operation that involves combining the fractions by multiplying their numerators together and their denominators together. This process is largely straightforward and involves these steps:
  • Take the numerators of both fractions and multiply them together. This product becomes the new numerator.
  • Take the denominators of both fractions and multiply them together. This product becomes the new denominator.
  • Simplify the resulting fraction if possible.
In our exercise, \( \frac{3}{4} \times \frac{2}{1} \) involves multiplying 3 and 2 to get the new numerator of 6, and 4 and 1 for the new denominator of 4. This results in \( \frac{6}{4} \), which we then simplify to \( \frac{3}{2} \). Multiplying fractions can seem complex, but by following these steps, you'll master it in no time.

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Most popular questions from this chapter

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{5 x-2}{3 x-4}+\frac{2 x-3}{4-3 x}$$

It normally takes 2 hours to fill a swimming pool. The pool has developed a slow leak. If the pool were full, it would take 10 hours for all the water to leak out. If the pool is empty, how long will it take to fill it?

Subtract: \(\frac{13}{15}-\frac{8}{45}\) (Section 1.2, Example 9)

Explain how to add rational expressions when denominators are opposites. Use an example to support your explanation.

Add or subtract as indicated. Simplify the result, if possible. $$\frac{7}{x}+4$$
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