Problem 38 - Algebra 1

Question

Write the verbal sentence as an equation, or an inequality. Fifty multiplied by the quantity twenty divided by a number \(n\) is greater than or equal to two hundred fifty.

Step-by-Step Solution

Answer

The verbal sentence 'Fifty multiplied by the quantity twenty divided by a number n is greater than or equal to two hundred fifty' can be written as the inequality \(50*(20/n) ≥ 250\).

Step 1: Identify the Mathematical Operations

Transform the verbal phrases into mathematical operations. 'Fifty multiplied by' converts to '50*', 'the quantity twenty divided by a number n' converts to '20/n', and 'is greater than or equal to' converts to '≥'.

Step 2: Write the Expression

Combine these operations into a single mathematical expression: \(50*(20/n)\).

Step 3: Write the Inequality

Now, write the full inequality statement. This implies showing that the expression \(50*(20/n)\) is greater than or equal to 250: \(50*(20/n) ≥ 250\).

Key Concepts

Understanding Inequalities

Inequalities are mathematical statements that show the relationship between two expressions. Unlike equations, which assert that two expressions are equal, inequalities communicate that one expression is either less than, greater than, less than or equal to, or greater than or equal to another expression.
Inequalities use symbols such as:

• "<": less than
• ">": greater than
• "≤": less than or equal to
• "≥": greater than or equal to

In the exercise we are solving, the relationship is expressed as "greater than or equal to," represented by "≥." This tells us that the value on the left side of the inequality can be equal to or larger than the value on the right. Understanding this concept is crucial because it helps us determine the range of possible values that a variable, such as \( n \), can take to satisfy the inequality.

Verbal to Symbolic Translation

Translating verbal sentences into symbolic form is an important skill in mathematics. It involves identifying specific words and phrases that correspond to mathematical symbols and operations. In this exercise, several key phrases are translated:

• "Fifty multiplied by" is translated to the multiplication operation "50 \times".
• "The quantity twenty divided by a number \( n \)" is interpreted as the division operation "20/n".
• "Is greater than or equal to" is symbolized by "≥".

As you translate verbal sentences, remember to carefully interpret the operations and how they relate to each other in the larger structure of the expression or inequality. This accuracy ensures correct formulation of mathematical expressions.

Mathematical Operations in Algebra

Mathematical operations such as addition, subtraction, multiplication, and division are the building blocks of algebraic expressions and equations. They help us construct and comprehend complex relationships between numbers and variables.
In this specific problem, we're dealing with multiplication and division:

• "Multiplication" is used when we see phrases like "multiplied by." For instance, "Fifty multiplied by" results in "50 \times".
• "Division" is apparent in phrases like "divided by a number \( n \)," transforming into "20/n."

Understanding these operations helps us to simplify and arrange expressions correctly. By arranging the problem's operations, we form the expression \(50\times(20/n)\), which is integral to setting up our inequality \(50\times(20/n) ≥ 250\). Mastery of these operations is crucial for solving problems and simplifies the process of converting verbal statements into symbolic expressions.