Problem 49 - Elementary Algebra

Question

For exercises \(23-74\), evaluate. $$ \frac{11}{12} \div \frac{2}{3} $$

Step-by-Step Solution

Answer

The solution to \(\frac{11}{12} \div \frac{2}{3}\) is \(\frac{11}{8}\).

Step 1: Understand the division of fractions

In order to divide one fraction by another, you need to multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping the numerator and the denominator.

Step 2: Find the reciprocal

The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).

Step 3: Multiply the fractions

Now, multiply \(\frac{11}{12}\) by \(\frac{3}{2}\): \[ \frac{11}{12} \times \frac{3}{2} \]

Step 4: Multiply the numerators

Multiply the numerators: \[ 11 \times 3 = 33 \]

Step 5: Multiply the denominators

Multiply the denominators: \[ 12 \times 2 = 24 \]

Step 6: Simplify the resulting fraction

The resulting fraction is \(\frac{33}{24}\). Simplify by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 3: \[ \frac{33 \div 3}{24 \div 3} = \frac{11}{8} \]

Key Concepts

Reciprocal

To understand the reciprocal, remember that it's just flipping a fraction. When you have a fraction like \(\frac{2}{3}\), its reciprocal is \(\frac{3}{2}\). This means the numerator (top number) and the denominator (bottom number) swap places.

Reciprocals are important in division of fractions. Instead of directly dividing fractions, we multiply by the reciprocal of the second fraction.

For example, if you need to evaluate \(\frac{11}{12} \/ \frac{2}{3}\), first find the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\). Then, instead of division, you move to multiplication. This makes the calculation easier.

Multiplying Fractions

Multiplying fractions is straightforward. You multiply the numerators together and the denominators together.

Let's take the fractions \(\frac{11}{12}\) and \(\frac{3}{2}\). To multiply them, follow these steps:

• Multiply the numerators: \11 \times 3 = 33\
• Multiply the denominators: \12 \times 2 = 24\

So, \(\frac{11}{12} \times \frac{3}{2} = \frac{33}{24}\).

This process is much simpler than dividing fractions directly and it works for any two fractions.

Simplifying Fractions

Simplifying fractions means making the fraction as simple as possible. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD).

For our earlier product \(\frac{33}{24}\), we can find the GCD of 33 and 24, which is 3. Now, divide both the numerator and the denominator by 3:

• Numerator: \(33 \div 3 = 11\)
• Denominator: \(24 \div 3 = 8\)

Thus, \(\frac{33}{24}\) simplifies to \(\frac{11}{8}\), which is the simplest form of the fraction.

Simplification makes it easier to understand and use fractions in further calculations.