Problem 29 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. 7 times the total of \(77, h,\) and 88

Step-by-Step Solution

Answer

The expression is \(1155 + 7h\) or \(7(165 + h)\).

Step 1: Identify the Components of the Phrase

The phrase "7 times the total of 77, h, and 88" can be broken down into two main components: (a) the total of 77, h, and 88, and (b) multiplication of this total by 7.

Step 2: Translate "The Total of 77, h, and 88"

To translate "the total of 77, h, and 88" into an algebraic expression, write it as the sum: \(77 + h + 88\).

Step 3: Translate "7 Times the Total"

The phrase "7 times the total" means multiplying the sum obtained in Step 2 by 7. This is written as \(7 \times (77 + h + 88)\).

Step 4: Simplify the Expression

Simplify the expression by combining the constants inside the parentheses first: \(77 + 88 = 165\). Thus the expression becomes \(7 \times (165 + h)\).

Step 5: Final Algebraic Expression

Write the final expression by distributing the 7 or leaving it factored: This results in \(7(165 + h)\) or you can expand it to \(7 \times 165 + 7h\), which is \(1155 + 7h\).

Key Concepts

Translation of Phrases

When dealing with algebraic expressions, translating phrases into mathematical language is an essential skill. It allows us to represent real-world situations with symbols, making calculations easier and more systematic. In this exercise, we begin by understanding each part of the phrase "7 times the total of 77, h, and 88".

• "The total of 77, h, and 88" suggests an operation of addition, where we sum these three elements. This part is expressed algebraically as the sum: \(77 + h + 88\).
• "7 times the total" requires us to multiply the entire sum by 7, creating a product. This translates to the algebraic expression \(7 \times (77 + h + 88)\).

By translating phrases into algebraic expressions, we connect words to operations, simplifying complex verbal expressions into straightforward mathematical equations.

Simplification

Simplification of algebraic expressions is a key process to make equations more manageable and easier to use. It involves combining like terms and reducing expressions to their simplest form. In our example, simplifying the expression \(7 \times (77 + h + 88)\) involves a couple of straightforward steps.
First, we sum the constant numbers within the parentheses:

• \(77 + 88\) equals \(165\).

This modifies the expression to \(7 \times (165 + h)\). By simplifying, we make it more elegant and easier to handle in further calculations.

Multiplication of Sums

The multiplication of sums in algebra commonly involves applying the distributive property. This principle helps to expand expressions and uncover each component's contribution to the total. In this scenario, considering the expression \(7 \times (165 + h)\), we can "distribute" the 7.
Distribution involves multiplying each term inside the parentheses by 7:

• \(7 \times 165\) gives us \(1155\).
• \(7 \times h\) becomes \(7h\).

Putting it all together, the expanded form of the expression is \(1155 + 7h\). Understanding multiplication of sums through distribution is crucial for solving and simplifying many algebraic problems.