Problem 9 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Example \(1 .\) $$ \frac{\frac{2}{3}}{\frac{3}{4}} $$

Step-by-Step Solution

Answer

The simplified complex fraction is \(\frac{8}{9}\).

Step 1: Understand the Problem

We are asked to simplify the complex fraction \(\frac{\frac{2}{3}}{\frac{3}{4}}\). This involves dividing one fraction by another.

Step 2: Invert the Divisor

To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).

Step 3: Perform the Multiplication

Multiply the numerator \(\frac{2}{3}\) by the reciprocal of the denominator \(\frac{4}{3}\):\[\frac{2}{3} \times \frac{4}{3} = \frac{8}{9}\]

Step 4: Simplify the Fraction

Check to see if the resulting fraction \(\frac{8}{9}\) can be simplified further. In this case, it is already in its simplest form as 8 and 9 have no common factors other than 1.

Key Concepts

Complex Fractions

Complex fractions might sound tricky, but they are just fractions all around! A complex fraction is one in which the numerator, the denominator, or both, are also fractions. Think of them like double-decker buses with layers of fractions inside. In the given problem, we have \(\frac{\frac{2}{3}}{\frac{3}{4}}\). So, we need to find a way to simplify this top-bottom fraction ratio into something easier to digest!

Understanding complex fractions means recognizing that simplifying them boils down to division. Translating them into simpler expressions often involves working step by step to eventually obtain a regular, straightforward fraction. The key here is to break them down, just as we did in our exercise, which leads us to our next step, dealing with reciprocals.

Reciprocal

Next, let’s talk about reciprocals. The reciprocal of a number is like its mirror image that helps in division. For a fraction, you find the reciprocal by flipping the numerator and the denominator. If you have a fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\).

In our problem, we need to divide by \(\frac{3}{4}\), and a cool trick in math is to multiply by its reciprocal instead. So, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\). Neat, huh?

This flipping simplifies the division process and makes complex fractions easier to handle, converting it into a multiplication problem.

Multiplication of Fractions

Now, we're getting to the heart of simplifying complex fractions—multiplication. Why do we multiply instead of divide? Because working with reciprocals turns division into a more straightforward multiplication task.

Take the numerator fraction and multiply it by the reciprocal of the denominator fraction. It's pretty straightforward! Following our exercise: multiply \(\frac{2}{3}\) by \(\frac{4}{3}\).

• Write the fractions: \(\frac{2}{3} \times \frac{4}{3}\)
• Multiply the numerators: \(2 \times 4 = 8\)
• Multiply the denominators: \(3 \times 3 = 9\)

This gives us \(\frac{8}{9}\), a much friendlier fraction! Multiplying fractions is as simple as "top times top, bottom times bottom." Just make sure you multiply straight across and not sideways.

Simplest Form

Our last step in fraction land is determining if the resulting fraction is in its simplest form. A fraction is in simplest form when the greatest common divisor (GCD) of the numerator and the denominator is 1. It means the fraction cannot be reduced any further without turning it back into decimals or integers.

In our specific problem, after multiplying, we arrived at \(\frac{8}{9}\). So, let's check:

• The GCD of 8 and 9 is 1 because they have no common factors other than 1.

Therefore, \(\frac{8}{9}\) is already as simple as it gets!

Checking for simplest form helps in ensuring the final answer is as tidy and reduced as possible, making it easy to interpret and use in further math problems.