Problem 100 - Prealgebra and Introductory Algebra

Question

How much larger is 5 than \(-11 ?\)

Step-by-Step Solution

Answer

5 is 16 larger than -11.

Step 1: Identifying the Numbers

First, identify the two numbers mentioned in the question. Here, they are 5 and -11.

Step 2: Applying Subtraction

Since we are asked how much larger 5 is than -11, we need to subtract -11 from 5. In mathematical terms, this would be \(5 - (-11)\). Simplified, it becomes \(5 + 11\).

Step 3: Calculating the Result

Add 5 and 11 together, which equals 16.

Key Concepts

Mathematical Subtraction

Mathematical subtraction is a fundamental operation that represents the process of taking one quantity away from another. When we subtract two numbers, the number from which the other number is subtracted is called the 'minuend', and the number that is subtracted is called the 'subtrahend'. The result of subtraction is known as the 'difference'.

In the context of our example, we're comparing two numbers to find out how much larger one is than the other—here specifically, how much larger 5 is than -11. To do this, we subtract the smaller number, or the 'subtrahend', from the larger number, the 'minuend'. This is expressed mathematically as the equation \(5 - (-11)\). It's essential to recognize that subtracting a negative is the same as adding its positive equivalent. Therefore, the operation simplifies to \(5 + 11\), leading to a difference of 16.

In everyday terms, if you think about having 5 apples and you 'owe' someone 11 apples, you effectively have 16 more apples than what you owe, which showcases the subtraction principle.

Negative Numbers

Negative numbers are a critical component of the number system, representing values less than zero. They are often used to denote a lack or deficit, such as debt or temperatures below freezing.

In our exercise, we deal with a negative number when we see -11. It's crucial to understand that negative numbers follow specific rules when involved in mathematical operations like addition and subtraction. One of the most common points of confusion arises when subtracting negative numbers. Subtraction of a negative number is equivalent to addition, because two 'negatives' make a 'positive'. This fact is part of the reason why our initial operation turned into \(5 - (-11) = 5 + 11\), and is a key concept for solving algebraic problems that involve negative values.

Tip to Remember

Always remember that when you 'owe' something (which is a negative scenario), and you 'lose' that debt (subtract it), you are effectively gaining (adding) the value of what you owed, turning the situation into a positive one.

Elementary Algebra

Elementary algebra encompasses basic algebraic operations and the application of algebraic principles. It introduces variables and the use of symbols to represent numbers and mathematical relations. While the example given does not use a variable, it utilizes algebraic thinking to understand and solve the problem.

In solving our textbook exercise, though appearing simple, we apply elementary algebraic principles such as combining like terms and understanding the rules governing the use of negative numbers. Recognizing that subtracting a negative number means we can change the operation to addition, we use these fundamental rules of algebra to simplify the problem and arrive at a solution efficiently. This algebraic manipulation demonstrates the power of algebra in transforming and solving problems that can appear more complex at first glance.

Building Blocks

The ability to manipulate numbers and understand the inherent rules in operations serves as the building blocks for more advanced mathematics. As students develop their skills in elementary algebra, they progress towards solving more intricate problems with confidence and ease.