Problem 93 - Algebra 1: Concepts and Skills

Question

Divide. Write the answer in simplest form. $$ \frac{3}{4} \div \frac{2}{9} $$

Step-by-Step Solution

Answer

The simplest form of \( \frac{3}{4} \div \frac{2}{9} \) is \( \frac{27}{8} \)

Step 1: Understand Division of Fractions

In dividing fractions, it's important to remember the process involves multiplying the first fraction by the reciprocal of the second fraction. So in this case, the problem \( \frac{3}{4} \div \frac{2}{9} \) will become \( \frac{3}{4} \times \frac{9}{2} \).

Step 2: Multiply the Fractions

Once the division problem is converted into a multiplication problem, the next step is to multiply the fractions together. Multiply the numerators together for the new numerator, and the denominators together for the new denominator: \( \frac{3}{4} \times \frac{9}{2} = \frac{27}{8} \).

Step 3: Writing the Answer in Simplest Form

The fraction \( \frac{27}{8} \) is already in its simplest form since the numerator and the denominator do not have any common factors other than 1.

Key Concepts

Reciprocal

When dividing fractions, a fundamental concept is understanding the reciprocal. The reciprocal of a fraction is simply what you multiply the fraction by to get 1. In simpler terms, to find the reciprocal of any given fraction, you just flip the numerator and the denominator. For instance, in the exercise above, you are dividing \( \frac{3}{4} \) by \( \frac{2}{9} \).
In order to divide, you first need to take the reciprocal of \( \frac{2}{9} \), which is \( \frac{9}{2} \). "Switch the places!" might be an easy mantra to remember. By flipping it, you transform the division problem into a multiplication problem.
Reciprocals are essential when dividing fractions because they simplify the process into something more straightforward. Remember: *Divide with the reciprocal in your game plan!*

Multiplication of Fractions

Once you have switched over to a multiplication problem using the reciprocal, you are ready to multiply the fractions. Multiplying fractions is easier than it seems!
Here’s a quick method:

• Multiply the numerators (the top numbers) together to get the new numerator.
• Multiply the denominators (the bottom numbers) together to get the new denominator.

For the fraction problem \( \frac{3}{4} \times \frac{9}{2} \), multiply \( 3 \times 9 \) to get 27 and \( 4 \times 2 \) to get 8. The result of the multiplication is \( \frac{27}{8} \).
Keep practicing multiplying fractions by using this method. It can really make fraction multiplication feel as easy as pie!

Simplest Form

After multiplying, you want to check if your fraction is in its simplest form. This means the fraction cannot be reduced any further, as its numerator and denominator have no common factor except 1.
To simplify a fraction:

• Look for the greatest common divisor (GCD) of the numerator and the denominator.
• If needed, divide both by their GCD.

In our case, the fraction \( \frac{27}{8} \) is already in simplest form. Sometimes, it means eyeballing whether a smaller number exists that both top and bottom can be divided by evenly and noticing there’s none. As a quick tip, when you compute and nothing reduces, like here, it means you're already at your destination - the simplest form!
Never forget to check for simplest form. It helps keep your answers neat and universally understandable.