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Chapter 1: Problem 44

Find each quotient. $$ -\frac{6}{5} \div\left(-\frac{1}{3}\right) $$

Short Answer

Expert verified
\(\frac{18}{5}\)

Step by step solution

01

Understand the division of fractions

To divide fractions, you multiply by the reciprocal of the divisor. So, for \(-\frac{6}{5} \, \div \, -\frac{1}{3}\), convert \(-\frac{1}{3}\) to its reciprocal.
02

Find the reciprocal of the divisor

The reciprocal of \(-\frac{1}{3}\) is \(-3\).
03

Multiply the fractions

Now multiply \(-\frac{6}{5}\) by \(-3\):\[-\frac{6}{5} \, \times \, -3 = \frac{18}{5}\].
04

Simplify (if needed)

In this case, \(\frac{18}{5}\) is already 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 first important concept to understand is the reciprocal. A reciprocal is what you get when you flip a fraction upside down.
For example:
  • The reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\), which is simply 2.
  • The reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).
To find the reciprocal, the numerator becomes the denominator and the denominator becomes the numerator.
In our example exercise, the divisor is -\(\frac{1}{3}\). So, its reciprocal would be -3 because flip \(\frac{1}{3}\) and multiply by -1 gives -3.
Multiplying Fractions
Once you have the reciprocal, the next step is multiplying fractions. It involves straightforward steps:
  • Multiply the numerators (the top numbers).
  • Multiply the denominators (the bottom numbers).
Here鈥檚 a simple example:
If you want to multiply \(\frac{2}{3} \) and \(\frac{4}{5}\), you multiply 2 and 4 to get 8, and 3 and 5 to get 15, resulting in \(\frac{8}{15}\).
In the exercise, after finding the reciprocal of the divisor, you multiply \(-\(\frac{6}{5}\)\times-3\).
To do that:
  • Multiply -6 and -3 to get 18.
  • You keep the denominator of 5 the same.
The fraction becomes \(\frac{18}{5}\).
Simplifying Fractions
The final concept is simplifying fractions. This involves reducing the fraction to its simplest form. To simplify:
  • Find the greatest common divisor (GCD) of the numerator and denominator.
  • Divide both the numerator and the denominator by the GCD.
For example:
If you have \(\frac{8}{12}\), the GCD of 8 and 12 is 4. Dividing both the numerator and the denominator by 4 gives you \(\frac{2}{3}\). In the exercise, \(\frac{18}{5}\) is already in its simplest form because 18 and 5 have no common divisors other than 1. Therefore, nothing more needs to be done and the answer remains \(\frac{18}{5}\).
Simply follow these steps for any fractions problem and it will be much easier to understand and solve.

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Most popular questions from this chapter

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