Problem 26 - Prealgebra

Question

Find each quotient. Use an area model if necessary. $$\frac{6}{11} \div\left(-\frac{4}{5}\right)$$

Step-by-Step Solution

Answer

The quotient is \(-\frac{15}{22}\).

Step 1: Understand Division of Fractions

When dividing fractions, such as \( \frac{6}{11} \div \left(-\frac{4}{5}\right) \), we multiply the first fraction by the reciprocal of the second. The reciprocal of a fraction is created by swapping its numerator and denominator. Thus, the reciprocal of \(-\frac{4}{5}\) is \(-\frac{5}{4}\).

Step 2: Set up the Multiplication

Now, rewrite the division problem as a multiplication problem:\[ \frac{6}{11} \div \left(-\frac{4}{5}\right) = \frac{6}{11} \times \left(-\frac{5}{4}\right) \]

Step 3: Multiply the Fractions

Multiply the numerators together and the denominators together: \[\frac{6}{11} \times \left(-\frac{5}{4}\right) = \frac{6 \times (-5)}{11 \times 4} = \frac{-30}{44}\]

Step 4: Simplify the Fraction

The result from multiplication, \(\frac{-30}{44}\), can be simplified. Find the greatest common divisor (GCD) of 30 and 44, which is 2. Divide both the numerator and denominator by their GCD:\[\frac{-30}{44} = \frac{-30 \div 2}{44 \div 2} = \frac{-15}{22}\] This is the simplest form of the fraction.

Key Concepts

Understanding Reciprocals

In the context of fractions, a reciprocal is a special transformation that makes division much easier. To find the reciprocal of a fraction, you simply swap its numerator and denominator. For instance, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \). This concept becomes particularly handy when dividing fractions.
In our problem, we needed to divide \( \frac{6}{11} \) by \( -\frac{4}{5} \). Instead of performing division directly, we used the reciprocal of \( -\frac{4}{5} \), which is \( -\frac{5}{4} \). So, instead of dividing, we multiply by its reciprocal. This trick simplifies the process and avoids the more complex operation of direct division amongst fractions.

Multiplying Fractions

Once we have the reciprocal, dividing fractions turns into a simpler problem: multiplication. Multiplying fractions is a straightforward process. All you need to do is multiply the numerators together and the denominators together.
In our example, when changing \( \frac{6}{11} \div \left(-\frac{4}{5}\right) \) into \( \frac{6}{11} \times \left(-\frac{5}{4}\right) \), we perform two simple multiplications:

• For the numerators: \(6 \times (-5) = -30 \)
• For the denominators: \(11 \times 4 = 44 \)

Connecting these results, the resulting fraction from the multiplication is \( \frac{-30}{44} \).

Simplifying Fractions

The last step is to simplify the fraction to its simplest form. A fraction is simplified when you cannot divide the numerator and the denominator by any common factor other than 1.
To simplify fractions, find the greatest common divisor (GCD) of the numerator and denominator. In our case, the GCD of 30 and 44 is 2. By dividing both parts of the fraction \( \frac{-30}{44} \) by 2, we simplify it:

• Numerator: \(-30 \div 2 = -15 \)
• Denominator: \(44 \div 2 = 22 \)

Therefore, the fraction simplifies to \( \frac{-15}{22} \), which cannot be reduced further. Simplification ensures the fraction is as concise and readable as possible.