Problem 13 - Algebra for College Students

Question

Translate each verbal phrase into \(a\) mathematical expression using \(x\) as the variable. The quotient of three times a number and 10.

Step-by-Step Solution

Answer

\( \frac{3x}{10} \)

Step 1: Identify the variable

The variable is represented by the letter x.

Step 2: Express 'three times a number'

'Three times a number' means multiplying the variable x by 3, which gives us the expression: \( 3x \)

Step 3: Express 'the quotient of...and 10'

'The quotient of' means division. So, 'the quotient of three times a number and 10' can be written as the fraction of 3x divided by 10, which gives us the expression: \( \frac{3x}{10} \)

Key Concepts

Quotient

In mathematics, the term 'quotient' refers to the result of a division operation. For example, if we divide 20 by 4, the quotient is 5. In algebraic expressions, quotient helps to transform verbal phrases into mathematical statements.
To translate verbal phrases to algebraic expressions, identifying keywords like 'quotient' is key. In the exercise, the phrase is 'the quotient of three times a number and 10'. Here, 'quotient' signifies division.
We represent the division operation using a fraction form. Therefore, when the phrase speaks about the quotient of 'three times a number and 10,' it means we will represent this as \(\frac{3x}{10}\). It's like baking: 'quotient' is your recipe's ingredient that tells you to divide the dough.

Variables

Variables are symbols that stand for unknown values in mathematical expressions. They are usually letters like x, y, or z. In the given exercise, the variable is represented by the letter x.
Variables allow us to formulate general rules and equations. For instance, in the phrase 'three times a number,' the 'number' is unknown. We use x to represent it. Therefore, 'three times a number' becomes 3x.
It's like having a blank space in a puzzle. The variable is the piece that can fit into various spots, making the equation adaptable and versatile. When you see a term like 'three times a number,' know that the number you're multiplying (x in this case) is your variable.

Division

Division is a basic arithmetic operation that involves splitting a number into equal parts. It's one of the four main operations in math, alongside addition, subtraction, and multiplication.
In the exercise, the division operation is crucial for translating the phrase 'the quotient of three times a number and 10.' Division helps simplify real-world problems into digestible mathematical pieces. For instance, if you have 15 apples and want to divide them equally among 3 friends, each friend gets \(\frac{15}{3} = 5\) apples.
When it comes to algebraic expressions, division is often shown using fractions. For example, 'the quotient of three times a number and 10' translates to \(\frac{3x}{10}\). Think of the division symbol as a tool that helps you break down and understand how quantities are distributed or divided.