Problem 97 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ -\frac{16}{25} \div \frac{64}{15} $$

Step-by-Step Solution

Answer

The result is \(-\frac{3}{20}\).

Step 1: Rewrite the Division Problem as Multiplication

To divide by a fraction, multiply by its reciprocal. So, rewrite \(-\frac{16}{25} \div \frac{64}{15}\) as \(-\frac{16}{25} \times \frac{15}{64}\).

Step 2: Multiply the Numerators

Multiply the numerators of the fractions. The numerators are \(-16\) and \(15\). Thus, \(-16 \times 15 = -240\).

Step 3: Multiply the Denominators

Multiply the denominators of the fractions. The denominators are \(25\) and \(64\). Thus, \(25 \times 64 = 1600\).

Step 4: Simplify the Fraction

Combine the results to get the fraction \(-\frac{240}{1600}\). The greatest common divisor of \(240\) and \(1600\) is \(80\). Divide both by 80 to simplify the fraction: \(-\frac{240}{1600} = -\frac{3}{20}\).

Key Concepts

Reciprocal Multiplication

When dividing fractions, you are actually performing something known as a reciprocal multiplication. It can seem confusing at first, but once you grasp this idea, dividing fractions becomes much simpler. The reciprocal of a fraction is found by swapping the numerator (top part) with the denominator (bottom part). For instance, the reciprocal of \( \frac{64}{15} \) is \( \frac{15}{64} \). When dividing fractions, you replace the division sign with a multiplication sign and use the reciprocal of the divisor fraction.

Here's how it works in the example exercise:

• Original problem: \(-\frac{16}{25} \div \frac{64}{15}\)
• Turn the division into multiplication by using the reciprocal: \(-\frac{16}{25} \times \frac{15}{64}\)

This approach changes the problem into a simple multiplication of fractions.

Numerators and Denominators

Understanding numerators and denominators is crucial in solving any fraction problem, whether it’s adding, subtracting, multiplying, or dividing fractions. In a fraction such as \( \frac{a}{b} \), \( a \) is the numerator, and \( b \) is the denominator.

For the exercise, the numerators of the fractions are \(-16\) and \(15\); the denominators are \(25\) and \(64\). When you multiply fractions:

• Multiply the numerators together to get the new numerator: \(-16 \times 15 = -240\).
• Multiply the denominators together to get the new denominator: \(25 \times 64 = 1600\).

This results in the fraction \(-\frac{240}{1600}\). Recognizing these components can help avoid mistakes and simplify the process.

Simplifying Fractions

After performing operations with fractions, sometimes it is necessary to simplify. Simplifying fractions means reducing them to their most basic form, where the numerator and denominator have no common factors other than one.

To simplify \(-\frac{240}{1600}\):

• Find the greatest common divisor (GCD) of 240 and 1600, which is 80.
• Divide both the numerator and the denominator by the GCD: \(-\frac{240}{80} \div \frac{1600}{80} = -\frac{3}{20}\).

The fraction \(-\frac{3}{20}\) is in its simplest form because 3 and 20 have no common factors apart from one. Simplifying helps in easily understanding and comparing fractions.