Problem 41 - Intermediate Algebra

Question

Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen. The quotient of the number of clients and seventy-five gives the number of social workers needed.

Step-by-Step Solution

Answer

The mathematical model is \( \frac{n}{75} = s \).

Step 1: Understanding the Verbal Model

The statement says: 'The quotient of the number of clients and seventy-five gives the number of social workers needed.' This indicates we are dividing the number of clients by 75 to determine the number of social workers needed.

Step 2: Define Variables

Let's define suitable variables for the problem. Let the variable be the 'number of clients', and let the variable be the 'number of social workers needed'.

Step 3: Translate the Verbal Model

Now that we have defined the variables, we translate the verbal expression into a mathematical one. The phrase 'the quotient of the number of clients and seventy-five' translates mathematically to \( \frac{n}{75} \). According to the problem, this quotient 'gives the number of social workers needed,' meaning that \( \frac{n}{75} = s \).

Key Concepts

Verbal to Mathematical Translation

Understanding how to translate verbal expressions into mathematical models is an important skill in problem-solving. The process allows us to solve real-world problems by using the language of mathematics. Let’s break it down:

• Begin by carefully reading the verbal statement. Identify key phrases that indicate mathematical operations, like 'the quotient of', 'sum', 'product' or 'difference.'
• Understand the context of each phrase. For example, 'the quotient of the number of clients and seventy-five' suggests a division operation.
• Note any comparisons or conditions within the statement. Sometimes words like 'is equal to', 'is more than', or 'results in' can help determine the structure of the equation.
• Finally, use the identified operation to write a mathematical expression. In this instance, the word 'quotient' tells us to divide the number of clients by 75, resulting in the expression: \( \frac{n}{75} \).

These steps are crucial for accurately converting a word problem into a mathematical equation that can then be manipulated and solved.

Quotient in Algebra

In algebra, the term 'quotient' is used to describe the result of the division of one number by another. It's essential to grasp what a quotient represents when interpreting algebraic expressions:

• The quotient is what you get when dividing one quantity by another. It can be written as \( \frac{a}{b} \), where \( a \) is the dividend, and \( b \) is the divisor.
• Understanding the quotient helps when translating verbal models to mathematical ones. Phrases like 'the quotient of x and y' typically mean \( \frac{x}{y} \).
• The quotient can be an integer, a fraction, or a decimal, depending on the values involved. For instance, with \( \frac{n}{75} = s \), if we know \( n \) (number of clients) and solve for \( s \) (number of social workers), we calculate the exact quotient reflecting how many social workers are needed per 75 clients.

By comprehending this concept, you can better navigate problem statements using division and set up equations correctly.

Defining Variables in Equations

Defining variables is a fundamental step when creating equations from verbal problems. This process simplifies problem-solving by turning words into symbols that can be more easily manipulated:

• First, identify the quantities that are mutable or unknown in the problem. These are your candidates for variables.
• A variable is typically indicated by a letter, often \( x \), \( n \), or \( s \), to represent specific unknowns. In our exercise, \( n \) is the 'number of clients', and \( s \) is the 'number of social workers.'
• This approach allows you to write concise equations like \( \frac{n}{75} = s \), making it much clearer to solve for unknown quantities.
• Carefully defining variables helps avoid confusion and errors, ensuring clarity throughout the problem-solving process.

Defining variables transforms a complex problem into a simplified mathematical model, paving the way for effective analytical solutions.