Problem 354 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ 18 \cdot \frac{5}{6}(15 h+24) $$

Step-by-Step Solution

Answer

225h + 360

Step 1: Apply the Distributive Property Inside the Parentheses

First, distribute the \(\frac{5}{6}\) inside the parentheses to both terms. This gives \[ 18 \times \frac{5}{6} \times 15h + 18 \times \frac{5}{6} \times 24 \]

Step 2: Simplify Each Term

Next, simplify the expression by performing the multiplications. Calculate \[ 18 \times \frac{5}{6} \times 15h \] and \[ 18 \times \frac{5}{6} \times 24 \].

Step 3: Simplify the Coefficients

First, simplify \[ 18 \times \frac{5}{6} \]. We get: \[ 18 \times \frac{5}{6} = 18 \times \frac{5}{6} = 15 \]. Now multiply: \[ 15 \times 15h = 225h \] and \[ 15 \times 24 = 360 \]

Step 4: Combine the Results

Combine the simplified terms to get: \[ 225h + 360 \].

Key Concepts

Simplifying Expressions

Simplifying expressions is a key skill in algebra. It makes complex expressions easier to understand and work with. To start, let's break down each component of the expression step by step. Simplifying is often about seeing how terms can be combined or factored to make equations less complicated.

In our exercise, the expression to simplify is $$18 \times \frac{5}{6}(15h + 24).$$
Using the distributive property, we distribute $$\frac{5}{6}$$ into $$15h + 24.$$
Example: $$a(b+c) = ab + ac.$$
So here: $$18 \times \frac{5}{6} \times 15h + 18 \times \frac{5}{6} \times 24.$$ The next step involves simplifying each term individually by performing the multiplications.

Algebraic Properties

Algebraic properties like the distributive property make it easier to manipulate and simplify algebraic expressions. The distributive property states that:
$$a(b+c) = ab + ac.$$
This property helps in breaking down our given expression into more manageable parts.
First, we applied the distributive property to $$18 \times \frac{5}{6} \times (15h + 24),$$ resulting in: $$18 \times \frac{5}{6} \times 15h + 18 \times \frac{5}{6} \times 24.$$
This step-by-step method of expanding and then simplifying is crucial for solving more complicated problems. Additionally, understanding this property also helps in reverse operations, such as factoring.

Coefficient Multiplication

Coefficient multiplication involves multiplying the numerical values in front of the variables within an expression. In our exercise, we simplify the coefficients:
$$18 \times \frac{5}{6}.$$ Start by simplifying $$18 \times \frac{5}{6}.$$
Notice that: 18 simplifies with $$6,$$ from $$\frac{5}{6},$$ leaving us with $$15.$$ Hence, $$18 \times \frac{5}{6} = 15.$$ Next, multiply $$15$$ by the remaining terms in the equation: $$15 \times 15h$$ and $$15 \times 24,$$ yielding $$225h + 360.$$
This final step delivers a simpler, yet equivalent form: $$225h + 360.$$
By mastering coefficient multiplication, solving algebraic expressions becomes more straightforward.
Always remember to check your work to ensure each step is simplified correctly!