Problem 10 - Basic College Mathematics

Question

Divide and simplify. \(\frac{2}{3} \div \frac{3}{4}\)

Step-by-Step Solution

Answer

\(\frac{2}{3} \div \frac{3}{4} = \frac{8}{9}\).

Step 1 - Understand the division of fractions

When dividing fractions, we convert the operation to multiplication by using the reciprocal of the divisor.

Step 2 - Find the reciprocal of the divisor

The given fraction to divide by is \(\frac{3}{4}\). Its reciprocal is obtained by swapping the numerator and the denominator. Thus, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).

Step 3 - Set up the multiplication problem

Now we convert the division problem into a multiplication problem by multiplying the first fraction by the reciprocal of the second fraction. This changes \(\frac{2}{3} \div \frac{3}{4}\) to \(\frac{2}{3} \times \frac{4}{3}\).

Step 4 - Perform the multiplication

Multiply the numerators together and the denominators together: \(\frac{2 \times 4}{3 \times 3} = \frac{8}{9}\).

Step 5 - Simplify if necessary

Check to see if the fraction can be simplified further. In this case, \(\frac{8}{9}\) is already in its simplest form.

Key Concepts

Reciprocal

To divide fractions, the first step is to find the reciprocal of the fraction we are dividing by.
The reciprocal of a fraction is simply swapping its numerator and denominator.
For instance, if we have the fraction \(\frac{3}{4}\), its reciprocal would be \(\frac{4}{3}\). Why do we find the reciprocal? This transformation lets us change the division problem into a multiplication problem, which is typically easier to handle.
To summarize:

• The reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
• Finding the reciprocal simplifies the process of dividing fractions.

Multiplication of Fractions

Once we have the reciprocal, we can change our division problem into a multiplication problem.
Multiplying fractions is much simpler and follows straightforward rules.
To multiply two fractions, you multiply the numerators together and the denominators together.
For example, when converting \(\frac{2}{3} \times \frac{4}{3}\), you do:

• Numerators: 2 \times 4 = 8
• Denominators: 3 \times 3 = 9

This gives us \(\frac{8}{9}\). Remember, changing a division problem to multiplication using reciprocals streamlines the process and makes it more manageable.

Fraction Simplification

The final step is to simplify the fraction, if possible.
A fraction is simplified when the numerator and the denominator have no common factors other than 1.
To check, we look for the greatest common divisor (GCD) of both numbers.
In our case, \(\frac{8}{9}\), 8 and 9 have no common factors other than one. Thus, the fraction is already simplified.
Simplifying is important because it present clearer and more understandable results.
Let's review what you should recall:

• Check for common factors between the numerator and the denominator.
• If no common factors exist, the fraction is in its simplest form.
• Simplification reduces fractions to their most basic form.