Problem 76 - Prealgebra

Question

Multiply or divide as indicated. \(\frac{3}{4} \div \frac{5}{6}\)

Step-by-Step Solution

Answer

\( \frac{3}{4} \div \frac{5}{6} = \frac{9}{10} \).

Step 1: Understand the Operation

The operation given is division of two fractions: \( \frac{3}{4} \div \frac{5}{6} \). We need to divide the first fraction by the second fraction.

Step 2: Invert the Second Fraction

To divide by a fraction, we multiply by its reciprocal. The reciprocal of \( \frac{5}{6} \) is \( \frac{6}{5} \).

Step 3: Convert Division to Multiplication

Rewrite the expression as a multiplication problem using the reciprocal from Step 2: \( \frac{3}{4} \times \frac{6}{5} \).

Step 4: Multiply the Fractions

Multiply the numerators together and the denominators together: \( 3 \times 6 = 18 \) and \( 4 \times 5 = 20 \). So, \( \frac{3}{4} \times \frac{6}{5} = \frac{18}{20} \).

Step 5: Simplify the Result

Simplify the fraction \( \frac{18}{20} \) by finding the greatest common divisor (GCD) of 18 and 20, which is 2. Divide both the numerator and the denominator by 2: \( \frac{18 \div 2}{20 \div 2} = \frac{9}{10} \).

Key Concepts

Reciprocal

When learning about fractions, understanding the reciprocal is essential. The reciprocal of a fraction is simply created by swapping its numerator and denominator. For example, the reciprocal of \( \frac{5}{6} \) is \( \frac{6}{5} \). Swapping parts of the fraction may seem peculiar at first, but it has a useful purpose.

In fraction division, the reciprocal plays a crucial role. Instead of directly dividing by a fraction, you multiply by its reciprocal. This is a standard mathematical operation that simplifies the process of division. In the exercise, we wanted to divide \( \frac{3}{4} \) by \( \frac{5}{6} \). By using the reciprocal of \( \frac{5}{6} \), which is \( \frac{6}{5} \), the division becomes a straightforward multiplication problem.

Simplifying Fractions

Simplifying fractions is a process to make fractions as simple as possible. It involves reducing the numbers in the fraction's numerator and denominator to their smallest possible values while still retaining the fraction's original value.

In our example, we ended with \( \frac{18}{20} \) after multiplication. Though correct, this fraction is not in its simplest form. To simplify \( \frac{18}{20} \), we reduce it by dividing both the numerator and the denominator by their greatest common factor. The simplified fraction \( \frac{9}{10} \) is easier to interpret and is often preferred for clarity.

• Ensures ease of comparison with other fractions.
• Makes equations easier to solve.

Simplified fractions are more intuitive and cleaner, so mastering simplification is a key skill.

Greatest Common Divisor (GCD)

Calculating the greatest common divisor (GCD) is a vital step in simplifying fractions. The GCD is the largest number that divides both the numerator and denominator without a remainder.

For instance, in the fraction \( \frac{18}{20} \), we need to find the GCD of 18 and 20. Breaking down both numbers into their prime factors helps:

• 18: 2 \( \times \) 3 \( \times \) 3
• 20: 2 \( \times \) 2 \( \times \) 5

Both numbers share the prime factor 2. Thus, the GCD is 2. By dividing both 18 and 20 by 2, we get the simplified fraction \( \frac{9}{10} \).

Knowing how to determine the GCD not only helps with simplifying fractions, but also proves useful in a variety of mathematical tasks.