Problem 9 - Elementary Algebra

Question

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{2}{3}}{\frac{5}{6}} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{4}{5}\).

Step 1 - Understand the Problem

Identify the numerator and the denominator. The problem is to divide the fraction \(\frac{2}{3}\) by the fraction \(\frac{5}{6}\).

Step 2 - Convert Division to Multiplication

To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{5}{6}\) is \(\frac{6}{5}\). Thus, \(\frac{\frac{2}{3}}{\frac{5}{6}} = \frac{2}{3} \times \frac{6}{5}\).

Step 3 - Multiply the Fractions

Multiply the numerators together and the denominators together: \(\frac{2}{3} \times \frac{6}{5} = \frac{2 \times 6}{3 \times 5}\).

Step 4 - Simplify the Expression

Calculate the multiplication in the numerator and the denominator: \(\frac{12}{15}\). Then, simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3. Thus, \(\frac{12 \div 3}{15 \div 3} = \frac{4}{5}\).

Key Concepts

Reciprocal

To solve fraction division problems, understanding the concept of the reciprocal is essential. The reciprocal of a fraction is simply swapping its numerator and denominator. For example, the reciprocal of \(\frac{5}{6}\) is \(\frac{6}{5}\).
When you divide by a fraction, you actually multiply by its reciprocal. For instance, instead of dividing \(\frac{2}{3}\) by \(\frac{5}{6}\), you multiply \(\frac{2}{3}\) by \(\frac{6}{5}\). This makes the process of fraction division much simpler and more straightforward.
- Step 1: Identify the fraction whose reciprocal you need (in this case \(\frac{5}{6}\)).
- Step 2: Swap its numerator and denominator to get the reciprocal (now \(\frac{6}{5}\)).
Once you have the reciprocal, you're set to move forward and convert the division problem into a multiplication one.

Simplifying Fractions

Simplifying fractions is an important step that comes after multiplying or dividing them. It's all about reducing the fraction to its simplest form. A fraction is in its simplest form when the greatest common divisor (GCD) of the numerator and denominator is 1.
Consider the fraction \(\frac{12}{15}\). Both 12 and 15 can be divided by 3, which is their GCD. By dividing both the numerator and denominator by 3, we get \(\frac{4}{5}\).
Here are the steps to simplify a fraction:

• Find the GCD of the numerator and denominator.
• Divide both the numerator and denominator by the GCD.
• Write down the resulting fraction.

By following these steps, you ensure the fraction is as simple as possible.

Multiplying Fractions

Multiplying fractions is straightforward when you follow the correct steps. First, multiply the numerators of the two fractions together and then multiply their denominators together. For example, to multiply \(\frac{2}{3}\) and \(\frac{6}{5}\), you will:

• Multiply the numerators: \2 \times 6 = 12\
• Multiply the denominators: \3 \times 5 = 15\

This results in the fraction \(\frac{12}{15}\).
After performing the multiplication, you'll often need to simplify the fraction (as covered in the previous section). Remember, the key steps in multiplying fractions involve:

• Numerator times numerator.
• Denominator times denominator.

This gives you the product of the fractions, which you may then simplify to achieve the final result.