Problem 84 - Introductory Algebra for College Students

Question

Write as an algebraic expression in which \(x\) represents the number: the quotient of 9 and a number, decreased by 4 times the number. (Section 1.1, Example 3)

Step-by-Step Solution

Answer

The algebraic expression is \(\frac{9}{x} - 4x\).

Step 1: Understanding the Terms

First, clarify what the mathematical terminology means. The 'quotient of 9 and a number' means that we divide 9 by the given number. The term 'decreased by 4 times the number' means that we subtract 4 times the given number from the quotient.

Step 2: Formulating the expression

Second, translate the English sentence into an algebraic expression. Here, the number we are dealing with is represented as \(x\), hence the expression for 'quotient of 9 and a number' is \(\frac{9}{x}\). 'Decreased by 4 times the number \(x\)' translates into \(4x\). Therefore, the final algebraic expression is \(\frac{9}{x} - 4x\).

Step 3: Check for Understanding

Finally, review the algebraic expression to make sure it accurately reflects the given English sentence. The expression \(\frac{9}{x} - 4x\) does indeed represent the quote 'the quotient of 9 and a number, decreased by 4 times the number'.

Key Concepts

Quotient

When you come across the word 'quotient', it simply means the result of division. It is the answer you get when one number is divided by another. In our exercise, the phrase 'the quotient of 9 and a number' suggests dividing 9 by a number. The number is represented by a variable, which we'll call \( x \). So, when we talk about the algebraic expression for this, it is written as \( \frac{9}{x} \).
Understanding how to find the quotient is essential because it is a basic mathematical concept that is used in many other complex operations. Remember:

• Quotient involves division.
• It tells us how many times one number is contained within another.
• In this context, the number that divides 9 is represented by \( x \).

Having a solid grasp of what a quotient means will help you solve similar problems smoothly.

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and values. It allows us to create expressions and equations that can solve real-world problems. In this exercise, we use algebra to express the relationship described in the sentence using symbols and operations.
Our sentence translates to an algebraic expression where \( x \) is a variable representing the number. The expression for 'decreased by 4 times the number' means taking away or subtracting \( 4x \) from the initial quotient \( \frac{9}{x} \). Therefore, the final expression is \( \frac{9}{x} - 4x \).
Key features of algebra include

• Variables like \( x \) that stand in for unknown values.
• Operators such as "minus" (decreased by) and "times" (multiplication).
• Expressions simplify complex problems by using symbols.

In algebra, understanding how to effectively translate situations into expressions is crucial for solving them.

Translation of Word Problems

Translating word problems into algebraic expressions or equations is a critical skill in algebra. It involves reading the problem carefully, identifying key phrases, and then expressing them mathematically. This exercise helps demonstrate that process.
Firstly, you identify key terms like 'quotient' and actions such as 'decreased by.' Each term has a mathematical operation associated with it:

• 'Quotient of 9 and a number' tells us to divide 9 by \( x \).
• 'Decreased by 4 times the number' suggests subtracting \( 4x \).

After identifying these terms and understanding their meanings, arrange them in sequence to match the problem's requirements, which results in the expression \( \frac{9}{x} - 4x \).
In translating word problems, remember:

• Focus on what each word or phrase represents mathematically.
• Ensure that the sequence of operations aligns with the problem statement.
• Practice makes perfect, the more problems you translate, the more intuitive it becomes.

This skill is not only helpful in algebra but also valuable in many real-world situations where translating language into mathematics is necessary.