Problem 76 - Elementary and Intermediate Algebra

Question

Add. $$ 57.93+(-93.27) $$

Step-by-Step Solution

Answer

The sum is -35.34.

Step 1: Identify the Numbers

We are given the numbers 57.93 and -93.27. Notice that one is a positive number and the other is negative.

Step 2: Determine Operation

Since we are adding a negative number, this is equivalent to subtracting its absolute value. Thus, we will perform the operation 57.93 - 93.27.

Step 3: Subtract the Numbers

Align the numbers by the decimal point and subtract the smaller number from the larger: \[93.27 - 57.93 = 35.34\]

Step 4: Determine the Resulting Sign

Since the absolute value of the negative number (-93.27) is larger than the positive number (57.93), the result will be negative.

Step 5: Write the Final Answer

The answer, considering the sign from Step 4, is -35.34.

Key Concepts

Integer Operations

When dealing with integer operations, it is crucial to understand the differences between positive and negative numbers. Numbers with a positive sign are greater than zero, while numbers with a negative sign are less than zero. Understanding this helps when performing basic arithmetic operations, like addition and subtraction, with these numbers. For example, when we add a negative number, it is as if we are subtracting its absolute value from the positive number. This can turn an addition problem into a subtraction one. If we go back to our original exercise, adding \( 57.93 + (-93.27) \), it transforms because of the negative sign into a subtraction \( 57.93 - 93.27 \). This is a key idea when handling operations involving signed numbers.

Absolute Value

The absolute value of a number signifies its distance from zero on the number line, irrespective of direction. This means that while the absolute value does not consider whether a number is positive or negative, it only reflects its magnitude.For example:

• The absolute value of \(-93.27\) is \(93.27\).
• The absolute value of \(57.93\) is \(57.93\).

Understanding absolute values is particularly useful in integer operations. It allows us to compare which numbers have larger magnitudes, helping in determining results like we did when subtracting larger negative numbers from positive numbers. This was essential in finding that the absolute value of \(-93.27\) being greater than \(57.93\) meant the result would be negative.

Subtraction Process

The subtraction process might seem straightforward, but it is important to tactfully handle it, especially when dealing with decimal numbers. As observed in the original step-by-step solution, aligning the decimal points is fundamental before performing the subtraction. This ensures that we properly subtract each corresponding positional value. In the example of \(57.93 - 93.27\), you align the numbers like so:

  93.27- 57.93---------

By borrowing where needed, we flawlessly perform the subtraction resulting in \(35.34\). Finally, evaluating the signs of the original numbers informs us that since we subtracted a larger number from a smaller one, the result inherits the negative sign of the greater absolute value. Thus, the final result is \(-35.34\). This makes clear the importance of a systematic approach to the subtraction process.