Problem 12 - Elementary Algebra

Question

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{4 p}{9}}{\frac{2 p^{2}}{3}} $$

Step-by-Step Solution

Answer

The simplified form is \[ \frac{2}{3p} \].

Step 1: Rewrite the Complex Fraction

The given expression is a complex fraction. Rewrite it as a division of two fractions: \[ \frac{\frac{4p}{9}}{\frac{2p^2}{3}} = \frac{4p}{9} \times \frac{3}{2p^2} \]

Step 2: Simplify the Expression

Multiply the numerators and the denominators: \[ \frac{4p \times 3}{9 \times 2p^2} = \frac{12p}{18p^2} \]

Step 3: Cancel Common Factors

Observe that both the numerator and the denominator can be divided by 6p. Rewrite the fraction: \[ \frac{12p}{18p^2} = \frac{12 \times p}{18 \times p \times p} = \frac{2}{3p} \]

Key Concepts

Fraction Division

To begin with, let’s talk about dividing fractions. When we see a complex fraction which is a fraction within a fraction, we can simplify this by using the concept of division. For example, say we have \(\frac{a}{b} \div \frac{c}{d}\). This is the same as \(\frac{a}{b} \times \frac{d}{c}\). Notice we 'flip' the second fraction and change the division sign to a multiplication sign.
When dealing with algebraic expressions in a complex fraction, the same rule applies. This means that in our given exercise \(\frac{\frac{4p}{9}}{\frac{2p^2}{3}}\), we change it to: \(\frac{4p}{9} \times \frac{3}{2p^2}\).
Simplifying these kinds of expressions step-by-step makes solving them much easier.

Cancelling Factors

In algebra, cancelling factors is a key step in simplifying expressions. After rewriting a complex fraction as a multiplication problem, the next step involves cancelling common factors. We observe the numerators and denominators to identify and remove any common factors.
In our example, after rewriting the fraction as: \(\frac{4p}{9} \times \frac{3}{2p^2}\), we multiply the numerators together and the denominators together: \(\frac{4p \times 3}{9 \times 2p^2} = \frac{12p}{18p^2}\).
At this stage, we identify common factors in the numerator and the denominator. Here, both 12 and 18 share a factor of 6, and both terms also include a 'p'. Therefore, we can divide the numerator and the denominator by 6p.
This results in: \(\frac{12p}{18p^2} = \frac{2}{3p}\). The fraction is now simplified.

Algebraic Expressions

Understanding algebraic expressions is crucial in simplifying complex fractions. An algebraic expression is a mathematical phrase that can include numbers, variables (like p), and operators (like +, -, \times, ÷).
Let's consider our problem: \(\frac{\frac{4p}{9}}{\frac{2p^2}{3}}\). Here, \(\frac{4p}{9}\) and \(\frac{2p^2}{3}\) are algebraic fractions.
Simplifying such fractions involves rewriting and then factoring or cancelling common terms. In our final simplified form, \(\frac{2}{3p}\), we can see how understanding each part of the expression and applying basic rules help achieve the simplest form.
Whether you're dealing with polynomials, monomials, or more complex structures, mastering the simplification of algebraic expressions is a fundamental skill in algebra.