Question

One number is 3 times another and their sum is 28. What are the two numbers?

Short answer

The two numbers are 7 and 21.

Step-by-step solution

Define the variables

Let one number be denoted as \(x\). Since one number is 3 times the other, let the other number be denoted as \(3x\).

Write the equation

According to the problem, the sum of the two numbers is 28. Therefore, we write the equation: \(x + 3x = 28\).

Simplify the equation

Combine like terms to simplify the equation: \(4x = 28\).

Solve for \(x\)

Divide both sides of the equation by 4 to solve for \(x\): \(x = 7\).

Find the other number

Since the other number is 3 times \(x\), we calculate: \(3x = 3(7) = 21\).

Verify the solution

Check that the sum of the two numbers is 28: \(7 + 21 = 28\). Thus, the solution is verified.

Key concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In this exercise, algebra helps us to translate a word problem into mathematical expressions and equations. By using algebraic techniques, we can find unknown values, which are called variables.

The fundamentals of algebra include:

• Defining variables: Assign letters or symbols to represent unknown quantities
• Writing equations: Expressing the relationships between variables and known values
• Simplifying expressions: Combining like terms and reducing equations to simpler forms

In the provided example, we define one number as \(x\) and the other as \(3x\). The relationship between these numbers is given by their sum. Algebra helps us convert this relationship into a manageable equation: \(x + 3x = 28\).

Equations

An equation is a mathematical statement that asserts the equality of two expressions, typically involving variables and constants. Equations are fundamental in algebra because they allow us to model and solve real-world problems.

Steps to handle equations effectively:

• Write down the equation clearly
• Simplify the equation by combining like terms
• Isolate the variable you need to solve for
• Perform arithmetic operations to solve the equation

In this problem, the equation \(x + 3x = 28\) represents the relationship between the two numbers. By simplifying it to \(4x = 28\), we can easily solve for \(x\). After dividing both sides by 4, we get \(x = 7\). This makes solving equations straightforward and systematic.

Problem Solving

Problem-solving is a critical skill not just in math, but in everyday life. It involves understanding the problem, devising a plan, carrying out the plan, and then reviewing the solution.

The steps for effective problem-solving in math are:

• Understand the problem: Read the problem carefully and identify what is given and what needs to be found
• Translate the problem: Convert the word problem into a mathematical equation
• Solve the equation: Use algebraic techniques to solve for the unknown variable
• Verify the solution: Check the solution to ensure it satisfies the original problem

In this exercise, we start by defining variables and then translate the problem (one number is three times the other, and their sum is 28) into an equation. Solving the equation and verifying by substituting back into the problem ensures our solution is correct.

Variable Manipulation

Variable manipulation is essential in solving equations, as it involves rearranging and simplifying expressions to isolate the unknown variables. This step is crucial for arriving at the solution.

Steps for effective variable manipulation:

• Identify the variables and constants in the equation
• Combine like terms to simplify the equation
• Use arithmetic operations like addition, subtraction, multiplication, and division to isolate the variable
• Check your work by substituting the variable back into the original equation

In this exercise, we start with the equation \(x + 3x = 28\). Combining the terms leads to \(4x = 28\). Dividing both sides by 4 isolates \(x\), giving us \(x = 7\). Then, finding the second number as \(3x = 21\) and verifying the sum (\(7 + 21 = 28\)) ensures our steps were correct.