Problem 63 - Algebra 1

Question

Give the correct algebraic translation of "Howard's hourly wage \(h\) is \(\$ 2\) greater than Marla's hourly wage \(m\)." (A)\(h < m+2\) (B) \(h=m+2\) (C) \(m=h+2\) (D) \(h > m+2\)

Step-by-Step Solution

Answer

The correct algebraic translation is \(h = m + 2\), so the answer is (B).

Step 1: Analyze the Statement

The statement says 'Howard's hourly wage \(h\) is \$2 greater than Marla's hourly wage \(m\)'. This indicates that Howard's wage is \$2 more than Marla's wage. 'More than' in mathematics often implies addition or a greater value.

Step 2: Translate to Math

We can represent Howard’s wage as \(h\), and Marla’s wage as \(m\). When we say Howard's wage is \$2 'more than' Marla's wage, we mean that if we take Marla's wage and add \$2 to it, we get Howard's wage. Mathematically, this is expressed as \(h = m + \$2\).

Step 3: Match to Options

In the options provided, this is expressed as option (B), \(h = m + 2\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of mathematics. They involve numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. In this context, variables such as \( h \) and \( m \) represent unknown or changeable quantities. These expressions are used to model real-world situations mathematically, providing a way to encode information about relationships and values.
For instance, in the problem, Howard's hourly wage \( h \) and Marla's hourly wage \( m \) form part of an algebraic expression, especially when joined by the operation "plus". This connection indicates that the difference between Howard's and Marla's wages can be expressed as \( h = m + 2 \). Such expressions allow us to solve and analyze the relationship between the two wages mathematically.

Mathematical Inequality

Mathematical inequalities are expressions that show a relationship between quantities that are not necessarily equal. They use symbols such as \(<\), \(>\), \(\leq\), and \(\geq\) to indicate whether one expression is less than, greater than, or equal to another.
In the original exercise, while option (B) reflects an equation showing equality (\( h = m + 2 \)), options like \( h < m + 2 \) or \( h > m + 2 \) depict inequalities. Inequalities express conditions where the relationship might range over certain values instead of having one exact solution as seen with equalities.
In practical scenarios, inequalities help determine ranges and are essential in various fields like economics, science, and engineering, where not every situation can be modeled with precise equality.

Word Problems

Word problems turn real-world situations into mathematical questions. They require you to read a scenario, identify the mathematics involved, and then solve the equation or inequality that represents the scenario.
In this setup, understanding the language used is crucial. Phrases like "greater than" can imply addition or a related inequality, dictating how you will set up your equation or inequality. The challenge often lies in correctly translating words into a mathematical form.
The integration of numbers, operations, and careful reading can help bridge everyday scenarios with mathematics. This layer of interpretation is what connects the problem statement—Howard's wage—and the mathematical expression, which we can analyze to find the solution.

Translating Words to Equations

Translating words into equations is at the heart of solving algebraic problems. This process involves reading words and identifying their arithmetic counterparts. It includes recognizing keywords or phrases like "sum of," "more than," "difference between," which correspond to operations like addition, subtraction, multiplication, etc.
In the problem provided, the phrase "is \$2 greater than" acts as a hint, directing us to use an addition operation. This understanding helps us set up the correct mathematical statement: \( h = m + 2 \).
By breaking down sentences into basic components and focusing on these keyword cues, translating between words and mathematical statements becomes easier and more intuitive. Mastery of this skill allows students to tackle even more complex equations and real-life problems effectively.