Problem 90 - Fundamentals of Mathematics

Question

For the following problems, replace the [U+2610] with the whole number that will make the subtraction true. $$ \begin{array}{r} 14 \\ -\square \\ \hline 3 \end{array} $$

Step-by-Step Solution

Answer

The square is 11.

Step 1: Understand the Problem

The problem is a basic subtraction problem. We are given a number (14) and need to find a whole number that, when subtracted from 14, results in 3. This can be written as an equation: \( 14 - \square = 3 \). Our task is to find the value of the square (\( \square \)).

Step 2: Set Up the Equation

Write the subtraction problem as an equation: \( 14 - x = 3 \). Here, \( x \) represents the number we are trying to find.

Step 3: Solve for the Unknown

To find the value of \( x \), solve the equation \( 14 - x = 3 \). Rearrange the equation to isolate \( x \) on one side: \( x = 14 - 3 \).

Step 4: Perform the Subtraction

Subtract 3 from 14: \( 14 - 3 = 11 \). Thus, \( x = 11 \).

Step 5: Verify the Solution

Check the solution by substituting 11 back into the original problem: \( 14 - 11 = 3 \). The equation holds true, confirming that 11 is the correct answer.

Key Concepts

Whole Number Subtraction

Subtraction involving whole numbers is one of the fundamental operations in mathematics, crucial for solving basic arithmetic problems. When we talk about whole numbers, we refer to numbers without fractions or decimals. Subtracting these types of numbers can help us understand differences between quantities. In our original exercise, we deal with subtracting a yet-to-be-determined whole number from 14 to reach 3. To visualize this, imagine you're holding 14 apples and you need to give away some to be left with exactly 3. The subtraction operation here is about determining how many apples you need to part with. To perform subtraction correctly:

• Align numbers vertically by place value if you're working with multi-digit numbers.
• Start from the rightmost digit (units) and move to the left.
• If the top digit is smaller than the bottom digit, you often "borrow" from the next left digit.

Subtraction can also be introduced using number lines, where you move left from the starting number to count the steps backwards.

Basic Arithmetic

Basic arithmetic involves understanding and working with numbers using four main operations: addition, subtraction, multiplication, and division. These are the building blocks of mathematics. Among these, subtraction is often seen as the process of finding the difference between numbers or removing objects from a collection. For our exercise, basic arithmetic comes into play to form and solve the equation: - Set the initial total: 14 - Identify what remains after subtraction: 3 - Find the number that makes this true using subtraction. Practicing subtraction helps enhance skills such as: - Solving larger math problems. - Understanding the relationship between addition and subtraction since subtraction is essentially the inverse. Subtraction is instrumental in solving real-world problems, such as balancing a budget or determining elapsed time between events.

Equation Solving

Solving equations is a critical skill in mathematics, allowing us to find unknown values that balance a mathematical statement. In this case, we use subtraction to explore equations like \( 14 - x = 3 \).Here’s the step-by-step process to solve a simple subtraction equation:

• Translate the word problem into a mathematical equation.
• Rearrange the equation to isolate the unknown value on one side.
• Perform the necessary arithmetic operations to expose the unknown variable.
• Verify the solution by substituting the found value back into the original equation.

Equation solving involves understanding equality and generally requires performing the inverse operation to "undo" a mathematical action. For subtraction equations like the one in our exercise, you add the answer to the result to verify it's correct, like this: from \( x = 14 - 3 \), we get \( x = 11 \), and check it with \( 14 - 11 = 3 \). By confirming the equality holds, the solution is verified, demonstrating that understanding and solving equations require careful arithmetic thought and practice.