Problem 96 - Elementary Algebra

Question

Evaluate: \(\frac{3}{4} \div \frac{5}{6}\)

Step-by-Step Solution

Answer

\(\frac{9}{10}\)

Step 1 - Understand the Division of Fractions

Recall that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, \(\frac{3}{4} \div \frac{5}{6}\) can be rewritten as \(\frac{3}{4} \times \frac{6}{5}\).

Step 2 - Find the Reciprocal

Identify the reciprocal of the second fraction \(\frac{5}{6}\). Its reciprocal is \(\frac{6}{5}\).

Step 3 - Multiply the Fractions

Multiply the two fractions: \(\frac{3}{4} \times \frac{6}{5} = \frac{3 \times 6}{4 \times 5} = \frac{18}{20}\).

Step 4 - Simplify the Fraction

Simplify the fraction \(\frac{18}{20}\). Find 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 of a Fraction

To understand how to divide fractions, you first need to know about the reciprocal of a fraction. The reciprocal of a fraction is simply a flipped version of the fraction. So, to find the reciprocal of \( \frac{5}{6} \), you turn it into \( \frac{6}{5} \). The numerator (top number) becomes the denominator (bottom number) and vice versa. Understanding reciprocals is crucial because dividing by a fraction is the same as multiplying by its reciprocal.

Multiplying Fractions

Once you have the reciprocal, you can proceed to multiplication. Multiplying fractions is straightforward. You multiply the numerators together and the denominators together. In our exercise, we turned division into multiplication: \( \frac{3}{4} \) divided by \( \frac{5}{6} \) becomes \( \frac{3}{4} \times \frac{6}{5} \). So, we multiply 3 by 6 and 4 by 5 to get: \( \frac{3 \times 6}{4 \times 5} = \frac{18}{20} \).

Simplifying Fractions

After multiplying, you often need to simplify the resulting fraction. Simplifying a fraction means making it as simple as possible by ensuring that the numerator and the denominator have no common factors other than 1. In our case, we got \( \frac{18}{20} \). To simplify, we need to find the greatest common divisor (GCD) of 18 and 20 and divide both the numerator and the denominator by this number.

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder. For 18 and 20, their GCD is 2. So, we divide both by 2. \( \frac{18 \div 2}{20 \div 2} = \frac{9}{10} \). Therefore, \( \frac{9}{10} \) is the simplified form of \( \frac{18}{20} \). This step ensures the fraction is in its simplest form.