Problem 67 - Beginning and Intermediate Algebra

Question

Perform the following operations. Write answers in lowest terms. $$ \frac{3}{4} \div \frac{7}{12} $$

Step-by-Step Solution

Answer

\(\frac{9}{7}\)

Step 1: Understand the Division of Fractions

To divide fractions, you need to multiply the first fraction by the reciprocal of the second fraction. Therefore, the expression \( \frac{3}{4} \div \frac{7}{12} \) becomes \( \frac{3}{4} \times \frac{12}{7} \).

Step 2: Multiply the Numerators and Denominators

Multiply the numerators of the fractions together and the denominators together: \( \frac{3 \times 12}{4 \times 7} = \frac{36}{28} \).

Step 3: Simplify the Fraction

Find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 36 and 28 is 4. Divide both the numerator and the denominator by their GCD: \( \frac{36}{28} = \frac{36 \div 4}{28 \div 4} = \frac{9}{7} \).

Key Concepts

Division of Fractions

Dividing fractions might seem tricky, but it's something you can master easily with practice.
The key to dividing fractions is to remember that dividing by a fraction is the same as multiplying by its reciprocal.
For the expression \( \frac{3}{4} \div \frac{7}{12} \), instead of dividing, you multiply \( \frac{3}{4} \) by the reciprocal of \( \frac{7}{12} \). The reciprocal is \( \frac{12}{7} \).

• Replace the division sign with a multiplication sign.
• Change the second fraction to its reciprocal.

This change makes the expression easier to solve.
Now, the problem becomes \( \frac{3}{4} \times \frac{12}{7} \), which is a straightforward multiplication of fractions.

Reciprocal

The reciprocal concept is pivotal in fraction division. It might sound fancy, but it's simple: the reciprocal of a fraction is just swapping its numerator and denominator.
For example, the reciprocal of \( \frac{7}{12} \) is \( \frac{12}{7} \).
Using reciprocals is what transforms a division problem into a multiplication problem.

• The reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).
• Multiplying a fraction by its reciprocal always results in 1.

Knowing this helps keep the division of fractions straightforward and manageable.

Simplifying Fractions

Simplifying fractions means making them as simple as possible. It's about finding a fraction which looks cleaner, without changing its value. When you simplify, the process involves removing any common factors in the numerator and the denominator.
For the example \( \frac{36}{28} \), after multiplying the fractions in our problem, the fraction can still be simplified.

• Identify any common factors between 36 and 28.
• Divide both by their greatest common factor to reduce it to \( \frac{9}{7} \).

This smaller fraction represents the same value but is easier to use.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is crucial for simplifying fractions.
It is the largest number that divides both the numerator and the denominator without leaving a remainder.
In our example, we need the GCD of 36 and 28 to simplify \( \frac{36}{28} \) efficiently.

• List the factors of each number.
• Identify the greatest factor they share.

In this case, the GCD is 4, since 4 is the largest number that divides both 36 and 28.
By dividing both the numerator and the denominator by 4, we get the simplified fraction \( \frac{9}{7} \).
Once you practice finding the GCD, simplifying fractions becomes a much easier task.