Problem 55 - Elementary Algebra

Question

For exercises \(23-74\), evaluate. $$ -\frac{9}{10} \div \frac{1}{12} $$

Step-by-Step Solution

Answer

-\frac{54}{5}

Step 1: Understand the Division of Fractions

To divide two fractions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Reciprocal of the Second Fraction

The reciprocal of \(\frac{1}{12}\) is \(\frac{12}{1}\).

Step 3: Set Up the Multiplication

Rewrite the division problem as the multiplication problem \(-\frac{9}{10} \times \frac{12}{1}\).

Step 4: Multiply the Fractions

Multiply the numerators and the denominators: \(-\frac{9 \times 12}{10 \times 1} = -\frac{108}{10}\).

Step 5: Simplify the Fraction

Simplify \(-\frac{108}{10}\) by dividing the numerator and the denominator by their greatest common divisor, which is 2: \(-\frac{108 \div 2}{10 \div 2} = -\frac{54}{5}\).

Key Concepts

Reciprocal

A reciprocal is essentially what you get when you **flip** a fraction. If you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). For example, the reciprocal of \(\frac{1}{12}\) is \(\frac{12}{1}\). Reciprocals are key when dividing fractions because we turn the division into a multiplication problem. This makes solving division of fractions much easier.

Multiplication of Fractions

After finding the reciprocal of the second fraction, you need to **multiply** the first fraction by this reciprocal. Here's how:

• Multiply the numerators (top numbers) together.
• Multiply the denominators (bottom numbers) together.

For example, in the problem \(-\frac{9}{10} \times \frac{12}{1}\), you multiply \(-9\) by \12\, and \10\ by \1\, to get \(-\frac{108}{10}\).

Simplifying Fractions

Once you get your product from the multiplication, you might need to **simplify** the fraction. Simplifying a fraction means making it as simple as possible by ensuring the numerator and the denominator have no common factors except 1. For \(-\frac{108}{10}\), we see both numbers can be divided by their greatest common divisor. Simplifying helps in making fractions easier to understand and work with.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. To simplify \(-\frac{108}{10}\), we find the GCD of 108 and 10, which is 2. So, we divide both 108 and 10 by 2:

• \108 \div 2 = 54\
• \10 \div 2 = 5\

This gives us the simplified fraction \(-\frac{54}{5}\). Knowing how to find the GCD is crucial for simplifying fractions correctly.