Problem 36 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. When a number divided by six is increased by nine, the result is one.

Step-by-Step Solution

Answer

The expression is \( \frac{x}{6} + 9 = 1 \). The number is \(-48\).

Step 1: Identify the Variable

Let's use a variable to represent the unknown number. We will call this number \( x \).

Step 2: Translate "a number divided by six"

The phrase "a number divided by six" can be translated into the mathematical expression \( \frac{x}{6} \).

Step 3: Translate "increased by nine"

The phrase "increased by nine" means we will add nine to our expression. So, our expression becomes \( \frac{x}{6} + 9 \).

Step 4: Translate "the result is one"

The phrase "the result is one" indicates that the expression \( \frac{x}{6} + 9 \) is equal to 1. This gives us the equation \( \frac{x}{6} + 9 = 1 \).

Step 5: Solve the Equation

To find \( x \), we first subtract 9 from both sides of the equation: \( \frac{x}{6} = 1 - 9 \). Simplifying, this becomes \( \frac{x}{6} = -8 \). Next, we multiply both sides by 6 to solve for \( x \): \( x = -8 \times 6 \). Therefore, \( x = -48 \).

Key Concepts

Mathematical Expressions

Mathematical expressions form the foundation for translating everyday language into the language of mathematics. Expressions are combinations of numbers, variables, and operations that come together to convey a specific idea or calculation. In this exercise, the phrase to be translated is "a number divided by six increased by nine equals one."

When we encounter such a phrase, it's essential to pinpoint the operations and the numbers involved. Here, "divided by six" indicates division, "increased by nine" implies addition, and "equals one" shows it as part of an equation.

By translating these components accurately, we generate the mathematical expression \( \frac{x}{6} + 9 = 1 \). This expression captures the problem's essence and allows us to move forward to solve for the unknown variable.

Variable Identification

Mathematics often deals with unknown values that we need to find, which we represent with variables. A variable is a symbol, usually a letter, that stands for an unknown quantity in an expression or equation.

In our exercise, we chose the variable \( x \) to symbolize the unknown number being discussed in the phrase.

• The choice of \( x \) is standard, but you can use any letter.
• Variables serve as placeholders that enable us to write and solve equations.

This step is crucial for translating a verbal phrase into a mathematical form because variables give us the flexibility to express unknown or changing quantities.

Solving Equations

Once we have a complete equation, our goal is to solve for the variable. Solving equations involves finding the value of the unknown that makes the equation true. In this scenario, we need to solve \( \frac{x}{6} + 9 = 1 \).

The procedure involves isolating the variable \( x \). First, we subtract 9 from both sides to remove the addition component:

• Subtract 9: \( \frac{x}{6} = 1 - 9 \)

This simplifies to \( \frac{x}{6} = -8 \).

Next, to eliminate the division, we multiply both sides by 6:

• Multiply by 6: \( x = -8 \times 6 \)

Thus, we conclude that \( x = -48 \). Solving equations like this often involves a series of steps that reverse the operations surrounding the variable.

Arithmetic Operations

Arithmetic operations, like addition, subtraction, multiplication, and division, are the building blocks of equations. In translating phrases to equations, recognizing these operations is key.

Let's break down what operations were used in the provided exercise:

• Division: The phrase "a number divided by six" converts to \( \frac{x}{6} \).
• Addition: "Increased by nine" translates to adding 9 to the expression.
• Subtraction: To isolate the variable, 9 is subtracted from 1, and we perform \( 1 - 9 \).
• Multiplication: To solve for \( x \), multiply by 6 to undo the division.

Each of these operations plays a specific role in simplifying and solving the equation for the unknown variable. Mastering these basic arithmetic operations allows you to manipulate and solve mathematical expressions and equations more effortlessly.