Problem 65 - Fundamentals of Mathematics

Question

For the next five problems, replace the letter \(m\) with the whole number that will make the addition true. $$ \begin{array}{r} 106 \\ +\quad m \\ \hline 113 \end{array} $$

Step-by-Step Solution

Answer

The value of \(m\) is 7.

Step 1: Understand the Problem

We have the addition equation \(106 + m = 113\). Our goal is to find the value of \(m\) that makes this equation true. We need to determine what number added to 106 results in 113.

Step 2: Set Up the Equation

Start by writing down the equation: \(106 + m = 113\). This equation will guide us to find the missing value, \(m\).

Step 3: Solve for m

To isolate \(m\), subtract 106 from both sides of the equation. This gives us \(m = 113 - 106\).

Step 4: Calculate

Perform the subtraction: \(113 - 106 = 7\). Therefore, \(m = 7\).

Step 5: Verify the Solution

To ensure our solution is correct, substitute \(m = 7\) back into the original equation: \(106 + 7 = 113\). Both sides of the equation are equal, confirming our solution is correct.

Key Concepts

Solving Equations

Solving equations is a way to find the unknown value that makes an equation true. Equations are statements showing that two expressions are equal. In our example:

• The equation is written as: \(106 + m = 113\).
• The job is to find the value of \(m\) that when added to 106 equals 113.

One common method to solve such problems is to isolate the variable or unknown. In this case, the variable is \(m\). You do this by using operations that "undo" what is currently being done to the variable. Here, subtraction helps us find the solution by removing 106 from both sides of the equation, resulting in: \[m = 113 - 106\]. After the subtraction, the solution comes out: \(m = 7\). Breaking down the equation step by step is critical, as it prevents overlooking any simple but important detail.

Whole Numbers

Whole numbers are a set of numbers that include zero and all the positive numbers without fractions or decimals. They start from 0 and go on endlessly like 0, 1, 2, 3, and so on. In arithmetic problems, these numbers often serve as basic elements that need to be manipulated and calculated.

• Whole numbers are easy to recognize, as they do not have negative signs or fractional parts.
• Understanding how to work with whole numbers is fundamental to solving basic arithmetic operations.

In our exercise, the whole numbers are 106, \(m\), and 113. Finding \(m\) involves simple addition and subtraction, which are core operations when dealing with whole numbers. Remember that arithmetic problems often require an understanding of how numbers relate to each other through operations like addition and subtraction.

Mathematical Verification

Mathematical verification is the process of confirming that a solution to a problem is correct. It involves checking your work to ensure accuracy. Verification is crucial because it ensures that the solution makes sense regarding the problem statement.

• In our example, the original equation is \(106 + m = 113\).
• Once we solve for \(m\) to get 7, we plug it back into the equation as a practice of verification: \(106 + 7 = 113\).

Both sides must be equal for the solution to be considered correct. This final step helps prevent errors and builds confidence in the solution. Remember, verification is just as crucial as calculating the result, as it confirms that the intended problem is accurately solved.