Problem 10 - Fundamentals of Mathematics

Question

When three times a quantity is decreased by five times the quantity, the result is negative twenty. What is the quantity?

Step-by-Step Solution

Answer

The quantity is 10.

Step 1: Understand the Problem

First, we need to understand the problem. We are told that three times some quantity minus five times the same quantity equals negative twenty.

Step 2: Define the Variable

Let's define the variable. Let the quantity be represented by the variable \( x \).

Step 3: Set Up the Equation

According to the problem, three times the quantity minus five times the quantity equals negative twenty. We can express this as an equation: \( 3x - 5x = -20 \).

Step 4: Simplify the Equation

Combine like terms in the equation \( 3x - 5x \), which simplifies to \( -2x \). So, the equation now is \( -2x = -20 \).

Step 5: Solve for the Quantity

To find \( x \), divide both sides of the equation by \(-2\). This gives \( x = \frac{-20}{-2} \).

Step 6: Simplify the Solution

Simplify the division to get \( x = 10 \).

Key Concepts

Solving Equations

Equations are fundamental in algebra as they represent mathematical statements using variables and constants. Solving an equation involves finding the value(s) of the variable(s) that make the equation true. In our exercise, the equation is

• \( 3x - 5x = -20 \)

This is a linear equation because it contains a variable raised to the first power. The goal is to solve for \( x \). Let’s break it down:

• Combine the like terms, which are the coefficients times the variable \( x \). After combining, only \( -2x \) remains on the left side of the equation.


• The next step is to isolate \( x \). Divide both sides by \( -2 \).
• This results in \( x = \frac{-20}{-2} \), simplifying to give you \( x = 10 \).

Variable Definition

Defining variables is crucial in algebra, as it allows us to translate word problems into solvable equations. In this problem, we started by defining the variable to represent the unknown quantity. Here's how we did it:

• The unknown quantity is what we aim to calculate, so we define it as \( x \). This step simplifies the problem because mathematical operations are easier to manage with variables.


• By using \( x \), we can directly translate the word problem into mathematical expressions. For example, "three times the quantity" becomes \( 3x \), and "five times the quantity" is \( 5x \).

This approach not only makes problems more manageable but also standardizes them. Defining variables helps organize thoughts and operations required to solve the equations.

Simplification of Expressions

Simplification is about reducing expressions into a simpler or more workable form. This helps make complex problems easier to solve. In our exercise, this occurred in several ways:

• First, identify like terms. In \( 3x - 5x \), both terms contain the variable \( x \). Therefore, they can be combined.


• Combining them results in the expression \( -2x \). This is a simpler form and consolidates our terms into a single variable term.


• Lastly, simplifying the fraction \( \frac{-20}{-2} \) by dividing the numerator by the denominator. This step reduces the expression to its simplest form, helping us find that \( x = 10 \).

Simplifying expressions is an important skill in algebra. It reduces complexity and clarifies the process necessary to reach the solutions.