Problem 94 - Fundamentals of Mathematics

Question

For the following problems, replace the [U+2610] with the whole number that will make the subtraction true. $$ \begin{array}{r} 28 \\ -\square \\ \hline 16 \end{array} $$

Step-by-Step Solution

Answer

The missing number is 12.

Step 1: Understand the Problem

We need to find a whole number that, when subtracted from 28, gives the result 16.

Step 2: Setup the Equation

Since we want to find the missing number, denote it by \( x \). The subtraction can be represented as an equation: \[ 28 - x = 16 \]

Step 3: Solve for the Missing Number

To isolate \( x \), we can add \( x \) to both sides of the equation and then subtract 16 from both sides: \[ 28 - 16 = x \] This simplifies to: \[ x = 12 \]

Step 4: Verify the Solution

Substitute \( x = 12 \) back into the original subtraction problem to verify: \[ 28 - 12 = 16 \] The subtraction is correct, confirming that \( x = 12 \) is the correct solution.

Key Concepts

Basic Arithmetic Operations

Arithmetic is the foundation of mathematics and includes operations such as addition, subtraction, multiplication, and division. Whole number subtraction is one of these basic operations. It involves finding the difference between two whole numbers, where the result is also a whole number. When subtracting, we take away a certain value from a larger number.

• Addition: Combining two numbers to get a sum.
• Subtraction: Removing one number from another to find the difference.
• Multiplication: Repeated addition of a number, a specified number of times.
• Division: Splitting a number into equal parts.

In our exercise, we dealt with subtraction, where we had the equation set up as 28 - x = 16. Solving it helps us understand the method to find the missing value or minuend to balance the equation.

Integer Subtraction

Integer subtraction specifically deals with subtracting whole numbers, which can be positive or negative. In our scenario, however, we're focused on positive integers, as whole numbers cannot be negative in this context. The goal is to find a number that, when subtracted from another, gives a specific result.

Numbers in subtraction are known as these terms:

• Minuend: The number from which another number (subtrahend) is subtracted. In our exercise, 28 is the minuend.
• Subtrahend: The number you subtract. Here, we denote this with \( x \).
• Difference: The result of the subtraction. For our problem, the difference is 16.

To solve integer subtraction problems, we rearrange the equation to isolate the unknown. For example, transforming 28 - x = 16 to x = 12 gives us the subtrahend needed to make the equation true.

Problem-Solving Steps

Solving arithmetic problems can be broken down into simple, logical steps that aid comprehension and application of concepts. Let's look into the steps used to solve the subtraction problem:

• Step 1: Understand the Problem: Determine what is given and what needs to be found. In our case, we need to find the number to subtract from 28 to reach 16.
• Step 2: Setup the Equation: Identify the relationship between the numbers, using symbols for unknown values. We expressed it as \( 28 - x = 16 \).
• Step 3: Solve the Equation: Use algebraic manipulation to isolate the unknown. Here, subtracting 16 from 28 simplified the equation to \( x = 12 \).
• Step 4: Verify the Solution: Check the solution by substituting it back into the original equation, ensuring the subtraction holds true. For \( x = 12 \), \( 28 - 12 = 16 \), confirming correctness.

These steps create a framework to approach similar problems systematically, ensuring a clear path from problem to solution.