Problem 32 - Algebra 1: Concepts and Skills

Question

Find the quotient. $$ \frac{3}{-\frac{5}{6}} $$

Step-by-Step Solution

Answer

The quotient of \(\frac{3}{-\frac{5}{6}}\) is \(-3 \frac{3}{5}\).

Step 1: Recognize the problem

The exercise asks for the quotient of \(\frac{3}{-\frac{5}{6}}\). Note that dividing by a fraction is equivalent to multiplying by the reciprocal of the fraction.

Step 2: Multiply by reciprocal

Change the division operation to multiplication by the reciprocal. Replace \(-\frac{5}{6}\) with \(-\frac{6}{5}\), so the problem becomes \(3 * (-\frac{6}{5})\).

Step 3: Perform multiplication

Next, perform the multiplication: \(3 * (-\frac{6}{5}) = - \frac{18}{5}\).

Step 4: Convert to mixed number

To convert the improper fraction \(- \frac{18}{5}\) to a mixed number, we divide the numerator by the denominator. -18 divided by 5 equals -3 with a remainder of 3, so our result is \(-3 \frac{3}{5}\).

Key Concepts

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is essential for dividing fractions effectively. A reciprocal is simply a flipped version of the original fraction. For example, the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\). To find the reciprocal, you interchange the numerator (top number) and the denominator (bottom number).
When dividing fractions, instead of dividing by the second fraction, you multiply by its reciprocal. This turns a division problem into a multiplication problem, making it easier to solve. It's important to remember that the reciprocal of a whole number, such as 5, is \(\frac{1}{5}\), since any whole number can be written as a fraction with 1 as the denominator.

• Identify the second fraction that you are dividing by.
• Flip the second fraction to find its reciprocal.
• Change the division operation to multiplication with the reciprocal.

By using this method, you always work with multiplication, which can simplify the process considerably. In our exercise, the reciprocal of \(-\frac{5}{6}\) is \(-\frac{6}{5}\), making the division problem into a multiplication problem with the fractions \(3\times(-\frac{6}{5})\).

Improper Fraction to Mixed Number

Sometimes the result of dividing fractions is an improper fraction, where the numerator is larger than the denominator. In order to convert an improper fraction to a mixed number, which has a whole number and a fraction, you divide the numerator by the denominator.
This division will give you the whole number part, and the remainder will be the numerator of the fractional part. The denominator remains the same.
For instance, if we have the improper fraction \(-\frac{18}{5}\), we divide 18 by 5, which equals 3 with a remainder of 3. Thus, \(-\frac{18}{5}\) becomes \(-3\frac{3}{5}\)—a mixed number composed of a whole number \(-3\) and a fraction \(\frac{3}{5}\).

• Divide the numerator by the denominator to get the whole number.
• The remainder becomes the numerator of the fractional part.
• The original denominator stays the same for the fractional part.

It's always good to simplify the fractional part of the mixed number if possible.

Multiplying by Reciprocal

When faced with the task of dividing by a fraction, one common method is to multiply by the reciprocal of that fraction. Multiplying by the reciprocal avoids the more complex division operation and simplifies the calculation process. This is based on the mathematical property that any number multiplied by its reciprocal equals 1.
Here’s how to apply this concept:

• Replace the division sign with a multiplication sign.
• Use the reciprocal of the divisor fraction as the second operand.
• Carry out the multiplication as you would with any fractions.

In the context of our exercise, once we change division to multiplication by using the reciprocal \(-\frac{6}{5}\) instead of \(-\frac{5}{6}\), the problem simplifies to the multiplication of 3 by \(-\frac{6}{5}\). This multiplication results in \(-\frac{18}{5}\), an improper fraction which we then converted to a mixed number in the previous steps.