Problem 120 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ 72\left(\frac{7}{8} f-\frac{8}{9}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( 63f - 64 \).

Step 1: Distribute the 72

Multiply 72 by each term inside the parentheses. This gives you: \[ 72 \cdot \left( \frac{7}{8} f \right) - 72 \cdot \left( \frac{8}{9} \right) \]

Step 2: Simplify Each Multiplication

Carry out the multiplication for each term:1. Multiply \( 72 \cdot \frac{7}{8} f \): - Simplify \( \frac{72}{8} = 9 \), so you have \( 9 \cdot 7f = 63f \) 2. Multiply \( 72 \cdot \frac{8}{9} \): - Simplify \( \frac{72}{9} = 8 \), so you have \( 8 \cdot 8 = 64 \)

Step 3: Write the Simplified Expression

Combine the simplified terms to form the final expression: \[ 63f - 64 \]

Key Concepts

Distributive Property

Understanding the distributive property is key to working with algebraic expressions. It involves multiplying a single term by two or more terms inside a set of parentheses. Essentially, this property allows you to "distribute" the multiplication over each term within the brackets.
For example, in our original exercise, we see it with the expression \( 72( \frac{7}{8}f - \frac{8}{9} ) \). To apply the distributive property:

• Multiply 72 by \( \frac{7}{8}f \)
• Then multiply 72 by \( -\frac{8}{9} \)

This step breaks down complex expressions into simpler, more manageable parts, making them easier to handle in subsequent steps.

Simplification

Simplifying expressions is a crucial part of solving algebra problems. After applying the distributive property, the next step is to simplify each term. This involves reducing fractions and performing multiplications.

• First, divide 72 by 8 to simplify \( \frac{72}{8} \) to obtain 9. Consequently, \( 9 \times 7f \) results in \( 63f \).


• Second, divide 72 by 9, simplifying \( \frac{72}{9} \) to 8. Hence, \( 8 \times 8 \) yields 64.

This method of breaking each term into simple numbers allows for easier calculation and ensures accuracy. Always check if there are whole numbers or simple fractions that can further reduce components of your expression.

Fractions in Algebra

Fractions are a regular component of algebra and can often complicate expressions if not dealt with properly. Managing fractions involves understanding division and multiplication well, and sometimes simplifying them for ease.
In the given exercise, fractions such as \( \frac{7}{8} \) and \( \frac{8}{9} \) appear within a larger expression. Handling these properly involves:

• Multiplying the numerator of the fraction with the outside number
• Dividing by the denominator when necessary

In our case, simplifying \( \frac{72}{8} = 9 \) and \( \frac{72}{9} = 8 \) is crucial. This step helps to convert fractions into integers, which are easier to work with in algebraic computations. With practice, managing these fractions becomes second nature.