Problem 115 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ 24\left(-\frac{5}{6} r\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \\(-20r\\).

Step 1: Distribute the Constant

To simplify the expression, we need to multiply the constant 24 with the terms inside the parentheses. Begin by multiplying 24 by \(-\frac{5}{6}r\).

Step 2: Multiply the Fractions and Coefficient

Multiply the constant 24 by \(-\frac{5}{6}\). When multiplying, you perform the multiplication as follows: \[ 24 \times \left(-\frac{5}{6}\right) = -\frac{120}{6} \].

Step 3: Simplify the Fraction

Simplify the fraction obtained from the previous step. \(-\frac{120}{6}\) simplifies to -20, because \(120 \div 6 = 20\).

Step 4: Combine the Results

Now attach the variable \r\ back to the numerical result from the multiplication. The simplified result of the expression is \(-20r\).

Key Concepts

Distributive Property

When you encounter an algebraic expression with a constant multiplied by terms in parentheses, like \(24\left(-\frac{5}{6} r\right)\), the distributive property is what allows you to simplify it. This property states that you can "distribute" the multiplication over each term in the parenthesis separately. It is especially useful when dealing with both constants and variables. In simple terms, you multiply the constant by each term within the bracket:

• Multiply 24 by \(-\frac{5}{6}\) to apply the distributive property.

The purpose of using this property is to eliminate parentheses and simplify expressions into a single algebraic term, making calculations easier to perform.

Multiplying Fractions

Multiplying fractions is a fundamental skill in algebra. When you multiply a whole number by a fraction, you aim to distribute the number evenly across that fraction. Here's how it works using our example:

• Convert the whole number into a fraction by using it as the numerator over a denominator of 1. For 24, think of it as \(\frac{24}{1}\).
• Multiply straight across: the numerators with each other and the denominators with each other. In our example, \(-\frac{5}{6}\) becomes \(-5/6\), and when multiplied by \(\frac{24}{1}\), results in \(-\frac{120}{6}\).

This results in a new fraction, which simplifies the process of handling large or complicated multiplications, aiding in our mission of simplifying algebraic expressions.

Simplifying Expressions

Simplifying expressions is the final step in many algebra problems, including in our example. After applying the distributive property and managing multiplication of fractions, you're often left with a fraction like \(-\frac{120}{6}\). Here, you simplify by dividing the numerator by the denominator:

• Perform the division: 120 divided by 6 is 20.
• Since the numerator of the fraction was originally negative, the result carries that negative sign, giving us -20.

Lastly, you reintroduce the variable to this simplified number: attaching \(r\) to -20 results in a final simplified expression of \(-20r\). This process reduces complex expressions to simpler, more manageable forms, essential for solving more intricate mathematical problems.