Problem 6 - Fundamentals of Mathematics

Question

What number decreased by nine is fifteen?

Step-by-Step Solution

Answer

The number is 24.

Step 1: Understand the Problem

We need to find a number which, when decreased by nine, equals fifteen. The problem can be represented by the equation: \( x - 9 = 15 \) where \( x \) is the unknown number.

Step 2: Set Up the Equation

Translate the problem into a mathematical equation: \( x - 9 = 15 \). Here, \( x \) represents the unknown number we are trying to find.

Step 3: Solve the Equation

To find \( x \), we need to isolate it on one side of the equation. Add 9 to both sides of the equation to remove the \(-9\) on the left:\[x - 9 + 9 = 15 + 9\]This simplifies to:\[x = 24\]

Step 4: Verify the Solution

Substitute \( x = 24 \) back into the original equation to ensure it satisfies the condition:\[24 - 9 = 15\]The equation holds true, confirming that our solution is correct.

Key Concepts

Problem Solving

When faced with a problem like "What number decreased by nine is fifteen?", the first step is to understand the question. This involves breaking down the problem into understandable parts and identifying what you are required to find. In this case, you need to determine a number that, when reduced by nine, results in fifteen.
This type of problem usually requires critical thinking and logical reasoning. You will often use these skills to find the appropriate mathematical expressions that represent the problem.

• Identify the unknown quantity you need to find—in this example, it's the number itself.
• Look for keywords that hint at mathematical operations. Here, "decreased by" suggests subtraction.
• Formulate a plan to represent the problem as a mathematical statement or equation.

By systematically understanding each element of the problem, you'll start to see how to arrive at the solution through the proper use of algebraic techniques.

Equation Solving

Equation solving is the mathematical method used to find the unknown value in an equation. It involves performing operations to isolate the unknown variable on one side of the equation. Let's take our example equation, \( x - 9 = 15 \), and see how this works.
The goal is to solve for the unknown \( x \). You do this by performing the opposite operation of what is currently affecting the variable. In this scenario, subtracting 9 from \( x \) means you need to add 9 to both sides:

• Add 9 to both sides: \( x - 9 + 9 = 15 + 9 \).
• Simplify to \( x = 24 \).

This process of performing identical operations to both sides of an equation ensures that the equality remains balanced, ultimately leading you to the correct value of the unknown variable.

Unknown Variables

In algebra, unknown variables are often represented using letters such as \( x \), \( y \), or \( z \). These symbols stand in for values that you need to determine from an equation.
Consider the equation \( x - 9 = 15 \). Here, \( x \) is the unknown variable that represents the number you are trying to find.
Understanding unknown variables involves several key steps:

• Assign a variable to the unknown quantity you need to solve for.
• Express the relationship described in the problem using this variable.
• Follow the rules of algebra to manipulate the equation and solve for the variable.

In our example, after solving the equation, you find that \( x = 24 \), which means 24 is the unknown variable value that satisfies the original condition of the problem.