Problem 9 - Fundamentals of Mathematics

Question

Translate each phrase or sentence into a mathematical expression or equation. A number divided by three, minus the same number multiplied by six, is one more than the number.

Step-by-Step Solution

Answer

The mathematical expression is \( \frac{x}{3} - 6x = x + 1 \), solved as \( x = -\frac{3}{20} \).

Step 1: Define the Variable

Let's define the number in the problem as \( x \). This is the unknown number we are trying to find.

Step 2: Translate 'A number divided by three'

The phrase 'a number divided by three' translates mathematically to \( \frac{x}{3} \).

Step 3: Translate 'The same number multiplied by six'

The phrase 'the same number multiplied by six' can be expressed as \( 6x \).

Step 4: Translate the Expression 'Minus the same number multiplied by six'

Combining the previous translations, the expression 'a number divided by three minus the same number multiplied by six' is expressed as: \[ \frac{x}{3} - 6x \]

Step 5: Translate 'Is one more than the number'

The phrase 'is one more than the number' translates to an equation. This means our expression from Step 4 should equal \( x + 1 \).

Step 6: Form the Equation

Put together the expressions from the previous steps to form the final equation: \[ \frac{x}{3} - 6x = x + 1 \]

Step 7: Solve the Equation

To find the value of \( x \), we'll solve the equation: \[ \frac{x}{3} - 6x = x + 1 \] Multiplying the entire equation by 3 to eliminate the fraction gives: \[ x - 18x = 3x + 3 \] Simplifying: \[ -17x = 3x + 3 \] Combine like terms by subtracting \( 3x \) from both sides: \[ -20x = 3 \] Finally, divide by -20: \[ x = -\frac{3}{20} \]

Key Concepts

Defining Variables

When translating word problems into mathematical expressions or equations, the first step is to define the variables. This shifts the problem from words into numbers and symbols that we can manipulate mathematically. In our given problem, the mysterious number we need to find is not specified. Here's where we assign it a variable, commonly denoted as \( x \). Defining \( x \) as our unknown allows us to structure the problem into a workable equation. This variable will consistently represent the same number throughout the entire problem, making it easier to translate various parts of the sentence into mathematical language.

Equation Solving

Once we have our equation, solving it is the next critical step. Solving equations involves several strategies such as combining like terms, removing fractions by multiplication, and isolating the variable to find its value. In our case, we started with the equation \( \frac{x}{3} - 6x = x + 1 \).

• First, we eliminate the fraction by multiplying the entire equation by 3.
• This gives us \( x - 18x = 3x + 3 \).
• Next, we simplify by gathering like terms on one side, resulting in \( -20x = 3 \).
• Finally, isolating \( x \) by dividing both sides by -20 gives us \( x = -\frac{3}{20} \).

Following these clear steps helps in solving almost any linear equation after properly setting it up from the word problem.

Mathematical Expressions

Mathematical expressions allow us to represent various components of a problem without resorting to lengthy explanations. In this exercise, we translated phrases from the problem into mathematical expressions. Here’s a quick recap:

• 'A number divided by three' became \( \frac{x}{3} \).
• 'The same number multiplied by six' transformed into \( 6x \).

These translations form the building blocks of an equation and help convey mathematical operations succinctly. Being able to change spoken or written language into these expressions is a crucial skill for solving word problems efficiently.

Fraction and Multiplication Translations

In many word problems, translating fractions and multiplication can be a common stumbling block. Understanding how to translate these concepts accurately is vital. When we talk about 'a number divided by three,' it naturally translates to the fraction \( \frac{x}{3} \), where \( x \) is our defined variable. For multiplication, 'the same number multiplied by six' easily translates to \( 6x \). These translations are necessary to correctly frame the relationship between different parts of the given problem. By mastering these translations, students can more effectively tackle word problems and set up the correct mathematical equations to solve them.