Problem 26 - Algebra 1

Question

Find the difference. $$ 5.4-(-3.8) $$

Step-by-Step Solution

Answer

The difference is 9.2.

Step 1: Convert subtraction of negatives to addition

In this step, it might be beneficial to remember that subtracting a negative number is the same as adding its positive counterpart. So, the original expression \(5.4 - (-3.8)\) simplifies to \(5.4 + 3.8\).

Step 2: Perform the addition

Once the expression has been simplified to \(5.4 + 3.8\), it becomes a standard addition problem. When you add these two numbers together, you get 9.2.

Key Concepts

Adding Positive Counterparts

When you're faced with the task of subtracting negative numbers, the process can be simplified by turning it into an addition problem. Imagine taking a step back and removing a debt—effectively, you're gaining, which is the essence of addition. This is what we mean by 'adding positive counterparts.'

For example, if we see an expression like \( 5.4 - (-3.8) \), it's akin to saying, 'I have 5.4 and I'm eliminating a debt of 3.8.' In mathematical terms, subtracting a negative number (\( -(-3.8) \)) is equivalent to adding its positive version (\( +3.8 \)). So, our expression becomes \( 5.4 + 3.8 \). This approach not only simplifies the calculation but also aligns with our intuitive understanding of negating a negative situation—resulting in a positive outcome.

Simplifying Expressions

Simplifying expressions is a fundamental part of solving math problems efficiently. It involves identifying and removing any unnecessary complexity, making the equation easier to understand and solve.

In the context of our example, simplifying the expression starts with changing the subtraction of a negative number to the addition of its positive counterpart. To recap, \( 5.4 - (-3.8) \), after simplification, turns into \( 5.4 + 3.8 \). This transformation paves the way for a more straightforward addition, stripping down the original problem to its bare essentials. Simplification isn't just about making things easier to compute; it also clarifies the structure of mathematical problems, which is vital when you're dealing with more complex equations.

Performing Addition

Performing addition is likely the first arithmetic operation we truly internalize. When subtracting negative numbers is reframed into this familiar territory, it becomes far less intimidating. Upon simplifying \( 5.4 - (-3.8) \) to \( 5.4 + 3.8 \), we can proceed with normal addition.

Add the two positive numbers as you've been taught: start with the rightmost digits, lining up the decimal points to ensure accuracy. You combine the tenths (0.4 + 0.8 = 1.2) and the whole numbers (5 + 3 = 8), and then add those results together. The act of performing addition in this context demonstrates how simplifying expressions can streamline the process and lead to the final answer, which in this case is 9.2.