Problem 101 - Elementary Algebra

Question

\(\frac{1}{3} \cdot \frac{1}{8} \div \frac{5}{6}\)

Step-by-Step Solution

Answer

\( \frac{1}{20} \)

Step 1: Multiply the first two fractions

Start by multiplying \(\frac{1}{3}\) and \(\frac{1}{8}\). To multiply fractions, multiply the numerators together and the denominators together. Therefore: \[ \frac{1}{3} \times \frac{1}{8} = \frac{1 \times 1}{3 \times 8} = \frac{1}{24} \]

Step 2: Invert and multiply

Since division by a fraction is the same as multiplying by its reciprocal, invert the fraction \(\frac{5}{6}\) to get \(\frac{6}{5}\). Then, multiply \(\frac{1}{24}\) by the inverted fraction: \[ \frac{1}{24} \times \frac{6}{5} = \frac{1 \times 6}{24 \times 5} = \frac{6}{120} \]

Step 3: Simplify the fraction

Simplify \(\frac{6}{120}\) by finding the greatest common divisor (GCD) of 6 and 120, which is 6. Then, divide both the numerator and the denominator by their GCD: \[ \frac{6 \text{ (numerator)}}{120 \text{ (denominator)}} = \frac{6 \text{ (numerator)} \: \ 6}{120 \text{ (denominator)} \: \ 6} = \frac{1}{20} \]

Key Concepts

multiplying fractions

Multiplying fractions is an essential skill in elementary algebra. When you multiply fractions, you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together.
This keeps the process straightforward and manageable.

For example, in the exercise \(\frac{1}{3} \cdot \frac{1}{8}\), we follow these steps:

• Multiply the numerators: 1 \times 1 = 1
• Multiply the denominators: 3 \times 8 = 24

So, \(\frac{1}{3} \cdot \frac{1}{8}\) becomes \(\frac{1}{24}\).

Always remember this fundamental property: XYZ. Every fraction multiplication can be simplified by following these steps, ensuring clarity and accuracy.

reciprocal of a fraction

A key concept in dividing fractions is the reciprocal. The reciprocal of a fraction is simply flipping the numerator and denominator.
For example, the reciprocal of \( \frac{5}{6} \) is \( \frac{6}{5} \).

When dividing by a fraction, you multiply by its reciprocal. Let's look at the exercise where we have \( \frac{1}{24} \div \frac{5}{6} \):

• Identify the reciprocal of \(\frac{5}{6}\) which is \(\frac{6}{5}\).
• Multiply the current result \( \frac{1}{24} \) by \( \frac{6}{5} \):
  \[ \frac{1}{24} \times \frac{6}{5} = \frac{1 \cdot 6}{24 \cdot 5} = \frac{6}{120} \]

Using reciprocals simplifies division of fractions into a multiplication problem, making the process easier.

simplifying fractions

Simplifying fractions helps in reducing them to their simplest form. This step ensures that the fraction is as clear and concise as possible.
To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).

In our exercise, we simplified \(\frac{6}{120} \) by finding the GCD of 6 and 120, which is 6:

• Divide the numerator (6) by 6: 6 \div 6 = 1
• Divide the denominator (120) by 6: 120 \div 6 = 20

So, \(\frac{6}{120}\) simplifies to \(\frac{1}{20}\).

Always checking if a fraction can be simplified ensures you are working with the simplest form, making it easier to understand and compare with other fractions.