Problem 37 - Elementary and Intermediate Algebra

Question

Multiply, and then simplify, if possible. \(24\left(\frac{3 a-5}{2 a}\right)\)

Step-by-Step Solution

Answer

The simplified form is \(36 - \frac{60}{a}\).

Step 1: Distribute the 24

To multiply, we distribute the 24 across the fraction. This involves multiplying each term inside the fraction's numerator by 24. The expression becomes:\[ 24 \times \frac{3a - 5}{2a} = \frac{24(3a - 5)}{2a} \].

Step 2: Distribute the 24 across the numerator

Next, we distribute the 24 across each term in the numerator. This gives us:\[ 24 \times 3a = 72a \] and \[ 24 \times (-5) = -120 \]. Thus, the expression becomes:\[ \frac{72a - 120}{2a} \].

Step 3: Simplify the expression

Now, we simplify the expression. First, notice that each term in the numerator and the denominator can be divided by their greatest common divisor, which is 2. Simplifying gives:\[ \frac{72a - 120}{2a} = \frac{72a \div 2 - 120 \div 2}{2a \div 2} \]which simplifies to:\[ \frac{36a - 60}{a} \].

Step 4: Further Simplify

As a final step, check if further simplification is possible. Divide each term in the numerator by \(a\):\[ \frac{36a}{a} - \frac{60}{a} = 36 - \frac{60}{a} \]. This is the final simplified form.

Key Concepts

Distributive Property

The distributive property is an essential algebraic process that helps us break down complex expressions into more manageable parts. When using the distributive property, you distribute—or multiply—a single term across terms within a parenthesis. This means you multiply the term outside of the parentheses by each term inside.For example, in the expression \[ 24\left(\frac{3a - 5}{2a}\right) \], we distribute the "24" to both "3a" and "-5" within the fraction. Hence, 24 is multiplied by "3a" and then by "-5". This distributes the 24 uniformly across each term, making it possible to simplify.What you achieve through distribution is not just multiplying, but setting the stage for simplifying expressions more efficiently. This property not only applies to whole numbers but also to fractions, as demonstrated in the step-by-step solution.

Simplifying Fractions

Simplifying fractions is a valuable skill in algebra that helps reduce expressions to their simplest form, making them easier to work with and understand. The process of simplifying involves reducing the numbers in a fraction to their smallest possible form, ensuring that they are easy to interpret and solve.In the exercise, after distributing, we end up with an expression like \[ \frac{72a - 120}{2a} \].To simplify it, we must find the greatest common divisor (GCD) of the terms in the numerator and the denominator. Here, **2** is the GCD, so we divide every term by **2**:

• \( \frac{72a}{2} = 36a \)


• \( \frac{120}{2} = 60 \)


• \( \frac{2a}{2} = a \)

Simplifying fractions in this manner not only cleans up the expression but also paves the way for further simplification if possible, such as cancelling variables or constants, thus leading to a more straightforward solution.

Multiplication of Algebraic Terms

The multiplication of algebraic terms is a fundamental concept in algebra. It involves multiplying coefficients and variables separately, adhering to algebraic rules and property principles.Consider the terms within our expression where we had to multiply 24 by the terms inside the parentheses:

• \( 24 \times 3a = 72a \)
• \( 24 \times (-5) = -120 \)

Here’s some important points about algebraic multiplication:- **Coefficients**: Multiply the numerical part of the expressions.- **Variables**: Keep the variable consistent when multiplying, unless there's a specific need to apply exponents.Multiplication in algebra involves meticulous organization and application. Ensuring accuracy in this step is crucial, as it sets the stage for correctly manipulating and solving algebraic expressions later on. Keeping this in mind helps in dealing effectively with a variety of algebraic challenges.