Problem 21 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{1}{2} \div \frac{3}{4} \div \frac{5}{6} $$

Step-by-Step Solution

Answer

\(\frac{1}{2} \div \frac{3}{4} \div \frac{5}{6} = \frac{4}{5}\).

Step 1: Change Division to Multiplication

To simplify the expression \(\frac{1}{2} \div \frac{3}{4} \div \frac{5}{6}\), we first convert divisions into multiplications by flipping the divisors. So, \(\frac{1}{2} \div \frac{3}{4}\) becomes \(\frac{1}{2} \times \frac{4}{3}\) and \(\frac{1}{2} \div \frac{3}{4} \div \frac{5}{6}\) becomes \(\frac{1}{2} \times \frac{4}{3} \times \frac{6}{5}\).

Step 2: Multiply Fractions

Now we multiply these fractions together. First multiply \(\frac{1}{2}\times \frac{4}{3}\). Here's how: the numerator \(1 \times 4 = 4\), and the denominator \(2 \times 3 = 6\). This results in \(\frac{4}{6}\).

Step 3: Simplify Intermediate Fraction

Simplify \(\frac{4}{6}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2. This results in \(\frac{2}{3}\).

Step 4: Continue Multiplication

Now, take \(\frac{2}{3}\) and multiply by \(\frac{6}{5}\). Multiply the numerators: \(2 \times 6 = 12\), and multiply the denominators: \(3 \times 5 = 15\). This gives \(\frac{12}{15}\).

Step 5: Simplify Final Result

Simplify \(\frac{12}{15}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This results in \(\frac{4}{5}\). This is the simplest form of the fraction.

Key Concepts

Multiplication of Fractions

Understanding how to multiply fractions is a fundamental skill in simplifying complex fractional expressions. When multiplying fractions, the process involves handling both the numerators and denominators separately. Here's how it works:

• Multiply the numerators together to obtain the new numerator.
• Multiply the denominators together to get the new denominator.

For example, consider the fractions \(\frac{1}{2}\) and \(\frac{4}{3}\). To multiply them, calculate:\[\text{Numerator: } 1 \times 4 = 4\]\[\text{Denominator: } 2 \times 3 = 6\]Thus, \(\frac{1}{2} \times \frac{4}{3} = \frac{4}{6}\). It is crucial to check if the resulting fraction can be simplified, which involves finding the greatest common divisor. Multiplying fractions is straightforward once you break it down into these simple steps.

Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is key to simplifying fractions. It is the largest number that can divide both the numerator and the denominator of a fraction without leaving a remainder. Simplifying fractions means rewriting them in the simplest form where no common factors exist between the numerator and the denominator (apart from 1).To determine the GCD:

• List the factors of each number involved.
• Identify the largest factor common to both the numerator and the denominator.

For instance, take \(\frac{4}{6}\). The factors of 4 are 1, 2, and 4, while the factors of 6 are 1, 2, 3, and 6. The common factor is 2, making it the GCD. Dividing both the numerator and denominator by 2 gives you \(\frac{2}{3}\), which is its simplest form. Ensuring the fraction is simplified not only makes calculations easier but also provides cleaner solutions when dealing with mathematical problems.

Division of Fractions

Division of fractions can initially seem complicated, but with the right approach, it becomes manageable. The key concept here is to flip the divisor fraction and change the operation to multiplication. This process is known as "multiplying by the reciprocal."

• Identify the fraction you are dividing by. This is the divisor.
• Flip the divisor to find its reciprocal. For example, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).
• Change the division operation to multiplication using this reciprocal.

For our exercise, start with \(\frac{1}{2} \div \frac{3}{4}\). Flip \(\frac{3}{4}\) to get \(\frac{4}{3}\), and multiply to obtain:\[\frac{1}{2} \times \frac{4}{3} = \frac{4}{6}\]By applying this method, division of fractions transforms into a simpler multiplication task, allowing for straightforward simplification via multiplication of fractions, followed closely by reducing through the greatest common divisor.