Problem 66 - Fundamentals of Mathematics

Question

For the next five problems, replace the letter \(m\) with the whole number that will make the addition true. $$ \begin{array}{r} 432 \\ +\quad m \\ \hline 451 \end{array} $$

Step-by-Step Solution

Answer

The value of \(m\) is 19.

Step 1: Understand the Problem

We need to find a whole number for \(m\) such that when added to 432, the result is 451.

Step 2: Set Up the Equation

Write the equation based on the addition problem: \(432 + m = 451\).

Step 3: Solve for \(m\)

Isolate \(m\) by subtracting 432 from both sides of the equation: \[m = 451 - 432\]

Step 4: Calculate the Result

Perform the subtraction: \[m = 451 - 432 = 19\]

Step 5: Verify the Solution

Add 432 and the value we found for \(m\), 19, to confirm: \[432 + 19 = 451\] Since the result is 451, the solution is correct.

Key Concepts

Addition

Addition is one of the basic operations in arithmetic. It involves combining two or more numbers to get a total sum. When dealing with addition, it's helpful to imagine stacking up items. If you have 3 apples and you add 2 more, you then have a total of 5 apples.

In mathematical terms, when we add numbers together, we are essentially summing their individual values. For instance, in our problem:

• The number 432
• Plus a mystery number we call \( m \)
• Gives us a total sum of 451

We write this as an equation: \[ 432 + m = 451 \] This setup helps us see what needs to be added to 432 to reach 451. It’s like filling in the missing piece of the puzzle.

Whole Number

Whole numbers are a set of numbers that include zero, all the positive numbers, and no fractions or decimals. Examples are 0, 1, 2, 3, and so on. These numbers are straightforward to work with, especially in problems involving basic arithmetic.

In our scenario, we are looking for a whole number that, when added to 432, results in 451. Whole numbers are great for this kind of arithmetic exercise because they ensure that the result is clear-cut and easy to understand. Since 451 is also a whole number, we are confident that our mystery number \( m \) must also be a whole number.

Putting this into practice, if you think of a number line, the 'hole' that separates each number from the next is never filled by anything other than another whole number. When finding \( m \), you can visualize simply jumping forward from 432 to 451 directly.

Equation Solving

Solving equations is like solving a mystery where you use the information you have to find the missing number. It’s an essential skill in mathematics that allows us to find the value of unknowns using known operations. In our case, we followed these steps:

• Set up the equation from the problem: \( 432 + m = 451 \)
• Rearrange the equation to solve for \( m \). This means isolating \( m \) on one side.
• Subtract 432 from both sides to isolate \( m \) which gives us: \( m = 451 - 432 \)
• Calculate the subtraction to find \( m \).

By performing these steps, we discover that \( m = 19 \). This means 19 is the number that, when added to 432, results in 451. This problem elegantly shows how we can demystify what might seem like complex tasks using equations in daily arithmetic applications. Solving for equations like this is a foundational part of understanding math.