Problem 69 - Elementary and Intermediate Algebra

Question

Divide. See Example 5. $$ -\frac{9}{16} \div\left(-\frac{3}{20}\right) $$

Step-by-Step Solution

Answer

The result of the division is \(\frac{15}{4}\).

Step 1: Rewrite the Division as Multiplication

When dividing by a fraction, you can multiply by its reciprocal. The given expression is \(-\frac{9}{16} \div \left(-\frac{3}{20}\right)\). This can be rewritten as \(-\frac{9}{16} \times \left(-\frac{20}{3}\right)\).

Step 2: Multiply the Numerators

Next, multiply the numerators of the fractions: \(-9\) times \(-20\). Calculate \(-9\times -20 = 180\).

Step 3: Multiply the Denominators

Now, multiply the denominators: \(16\) times \(3\). Calculate \(16 \times 3 = 48\).

Step 4: Form the Resulting Fraction

Combine the results of the previous steps to form the resulting fraction: \(\frac{180}{48}\).

Step 5: Simplify the Fraction

Simplify \(\frac{180}{48}\) by dividing both the numerator and the denominator by their greatest common divisor, which is \(12\). So, \(\frac{180\div12}{48\div12} = \frac{15}{4}\).

Key Concepts

Understanding Reciprocals

To divide fractions, understanding the concept of a reciprocal is crucial. Essentially, the reciprocal of a fraction involves flipping its numerator and denominator. For instance, the reciprocal of \(-\frac{3}{20}\) is \(-\frac{20}{3}\).
This is because the fraction \(-\frac{3}{20}\) means "3 parts out of a whole divided into 20," whereas its reciprocal \(-\frac{20}{3}\) means "20 parts out of a whole divided into 3." By swapping, you enable the division of fractions to become multiplication, which is much more straightforward to perform.

When dividing fractions like \(-\frac{9}{16} \div \left(-\frac{3}{20}\right)\), you "flip" the second fraction to multiply, hence transforming it into a multiplication problem: \(-\frac{9}{16} \times \left(-\frac{20}{3}\right)\). This fundamental understanding simplifies the operation entirely.

The Process of Multiplying Fractions

Once you transform a division problem into a multiplication problem using reciprocals, the next step is to handle the multiplication itself.
Multiplying fractions involves a straightforward rule:

• Multiply the numerators (the top numbers) together to make a new numerator.
• Multiply the denominators (the bottom numbers) together to make a new denominator.

In our example, multiply the numerators \(-9\) and \(-20\) to get \(180\). Then, multiply the denominators \(16\) and \(3\) to get \(48\).

The resulting fraction before any further simplification is then \(\frac{180}{48}\). This procedure keeps the process organized and ensures that the multiplication of fractions stays simple and clear.

Simplifying Fractions for Clarity

Often when multiplying fractions, the outcome is not in its simplest form. Simplification is the step where you make the fraction as simple as possible.

The fraction \(\frac{180}{48}\) can be simplified by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD here is \(12\).
Divide both the numerator and the denominator by \(12\) to simplify:

• \(\frac{180}{12} = 15\)
• \(\frac{48}{12} = 4\)

Thus, the simplified fraction is \(\frac{15}{4}\).
Simplifying fractions not only makes them easier to understand, but it also presents the solution in its most efficient form.