Problem 62 - Prealgebra

Question

Find each sum. $$24+(-12)+15$$

Step-by-Step Solution

Answer

The sum is 27.

Step 1: Identify the Operation Type

The problem involves adding together three numbers: 24, -12, and 15. This is a sum operation.

Step 2: Apply the Associative Property

Start by grouping the numbers in an advantageous way for addition. Group the first two numbers: 24 and -12. The associative property allows us to regroup these numbers without changing the sum: \[(24 + (-12)) + 15\]

Step 3: Calculate the First Sum

Calculate the sum of the first pair: \[24 + (-12) = 12\]This simplifies our expression to:\[12 + 15\]

Step 4: Add the Remaining Number

Now, complete the calculation by adding the remaining number to the intermediate sum:\[12 + 15 = 27\]

Step 5: Conclude with the Final Sum

The final sum of all three numbers is 27. Therefore, \[24 + (-12) + 15 = 27\]

Key Concepts

Associative Property

In mathematics, the associative property is an essential principle that simplifies complex calculations, particularly in arithmetic operations like addition. This property states that no matter how you group numbers when adding or multiplying, the result will be the same.

This principle is symbolized typically as \((a + b) + c = a + (b + c)\). In simpler terms, you can add numbers in any order, which is especially useful when you're dealing with more than two numbers.

• For instance, when adding \(24 + (-12) + 15\), you might choose to group the first two numbers together as \((24 + (-12)) + 15\).
• Alternatively, you could group the last two numbers: \(24 + ((-12) + 15)\).

Both approaches will yield the same result. This flexibility in grouping makes it easier to perform calculations mentally or to simplify expressions for further operations. It's particularly handy when some numbers combine to make an easier sum to work with.

Negative Numbers

Negative numbers can sometimes make arithmetic operations seem challenging, but understanding their properties offers clarity. A negative number is a number that is less than zero, typically represented with a minus sign (-).

In addition, negative numbers behave in particular ways:

• When you add a negative number, it is the same as subtracting the absolute value of that number. For example, \(24 + (-12)\) is essentially \(24 - 12\).
• Negative numbers can also affect other numbers in addition. Consider \((-3) + (-4)\), which equates to \(-7\) because you are effectively adding their absolute values (\(3 + 4 = 7\)) and attaching a negative sign to the sum.

Getting comfortable with negative numbers allows you to perform calculations confidently, even when the signs might initially confuse you.

Arithmetic Operations

Arithmetic operations include the basic mathematical processes such as addition, subtraction, multiplication, and division. Of these, addition is the process of finding the total or sum by combining numbers.

• Addition: In our example \(24 + (-12) + 15\), we are adding three numbers together to find a single total sum.
• Subtraction: Often involves the concept of adding a negative number, such as \(-12\) being subtracted from \(24\) in previously discussed terms.

In arithmetic operations, understanding the properties like associative and commutative helps to rearrange and simplify expressions. When performing these operations, it's vital to keep track of positive and negative signs, as they determine the direction of numbers on the number line.

Mastering basic arithmetic is crucial, as it forms the foundation for more advanced topics in mathematics. These operations are not only essential for solving textbook problems but also for everyday problem-solving.