Problem 53 - Algebra 1: Concepts and Skills

Question

Find the sum. Use a calculator if you wish. $$300.3+(-22.24)+78.713$$

Step-by-Step Solution

Answer

The sum of 300.3, -22.24 and 78.713 is 356.85.

Step 1: Understand the notation

In this problem, we're adding together three separate numbers. When we see '(-22.24)', understand that this is the same as '-22.24', it means we're subtracting 22.24 from the other numbers.

Step 2: Add the Positive Numbers

First, add together the positive numbers: 300.3 and 78.713. Their sum is 378.83.

Step 3: Subtract the Negative Number

Now, subtract the negative number (-22.24) from the sum we got in the previous step. This means subtract 22.24 from 379.013. The result is 356.85.

Key Concepts

Decimal Arithmetic

Understanding decimal arithmetic is fundamental when working with numbers that have a fractional part. When we add or subtract decimals, it is crucial to line up the decimal points vertically before proceeding.

For instance, if we have the problem \(300.3 + (-22.24) + 78.713\), we start by writing the decimals in a column ensuring that the decimal points are aligned:
\[\begin{align*} &\phantom{-}300.3\ &+78.713\end{align*}\]
We then proceed to add these numbers. After that, we handle subtracting the negative number, which in decimal arithmetic, is equivalent to adding its positive counterpart. The key here is to perform each operation step by step while keeping the decimal points aligned.

Negative Numbers

Negative numbers often cause confusion, but they're simply numbers with a minus sign in front of them, representing values less than zero. When a negative sign precedes a quantity in parentheses, like \( (-22.24) \), the parentheses indicate that the negative sign applies to the entire number within.

This means that \( (-22.24) \), when added, actually subtracts from the sum of the other numbers. To effectively deal with negative numbers in equations, always remember that \( -(-a) = a \), meaning subtracting a negative number is the same as adding its positive value. For example, adding \( -22.24 \) is the same as subtracting \( 22.24 \) from the total. Applying this rule helps avoid errors when combining positive and negative decimals.

Order of Operations

The order of operations dictates the sequence in which calculations should be performed to ensure consistent results. The common acronym 'PEMDAS' reminds us to calculate Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right) in that sequential order.

In our exercise, however, we are only dealing with addition and subtraction, which means we simply proceed from left to right:
\[300.3 + (-22.24) + 78.713\]
Firstly, we ignore the parentheses and consider the sign in front of the number. As negative numbers imply subtraction, we adjust our sequence to:
\[300.3 - 22.24 + 78.713\]
This adherence to the order of operations ensures that we correctly combine all terms to reach the correct sum or difference.