Problem 97 - Beginning and Intermediate Algebra

Question

Without calculating, determine whether each answer is positive or negative. Then use a calculator to find the exact difference. \(4.362-7.0086\)

Step-by-Step Solution

Answer

Negative; the exact difference is -2.6466.

Step 1: Understand the Problem

We need to determine whether the result of the subtraction \(4.362 - 7.0086\) will be positive or negative before calculating the exact answer.

Step 2: Compare the Numbers

Compare 4.362 and 7.0086. Observe that 4.362 is smaller than 7.0086.

Step 3: Determine the Sign

Since the number we are subtracting from (4.362) is smaller than the number we are subtracting (7.0086), the result will be a negative number.

Step 4: Perform the Subtraction

Using a calculator, subtract 7.0086 from 4.362 to find the exact difference. The calculation gives us \(4.362 - 7.0086 = -2.6466\).

Step 5: Interpret the Result

We confirm the calculation: the result is \(-2.6466\), which is indeed negative.

Key Concepts

Subtraction

Subtraction is one of the basic operations in mathematics. It involves taking one number, known as the subtrahend, from another number, known as the minuend. In the exercise, we deal with the subtraction problem: \(4.362 - 7.0086\). Here, \(4.362\) is the minuend, and \(7.0086\) is the subtrahend. Understanding the result of subtraction is crucial, especially when dealing with negative results. If the subtrahend is larger than the minuend—in this case, \(7.0086\) is larger than \(4.362\)—the answer will be negative. This is because you are removing more than the original number contains, resulting in an outcome less than zero. Always remember: When subtracting a larger number from a smaller one, the difference is negative. When dealing with decimals, as in this problem, the same rules apply. You can predict the sign of the result before calculating exact values by simply observing the size of the numbers involved.

Comparison of Numbers

Comparing numbers helps determine their relative sizes, which is essential before performing operations like subtraction. In the problem \(4.362 - 7.0086\), before subtracting, it’s helpful to compare the two numbers to predict the sign of the outcome.To compare numbers, particularly decimals, align the numbers by their decimal points. Do this either mentally or on paper. For example:

• Write \(4.362\)
• Underneath, write \(7.0086\)

Start comparing from the leftmost digit. Since \(4\) is less than \(7\), \(4.362\) is less than \(7.0086\). Therefore, you can conclude that the number \(4.362\) is smaller, and subtracting a larger number from it will yield a negative result. Understanding how to compare numbers can aid significantly in estimating results and checking your work quickly without a calculator, especially in exams or during quick problem-solving scenarios.

Calculator Use

Using a calculator can help simplify the process of finding exact differences in subtraction, especially when dealing with decimals. Let’s see how a calculator aids in our exercise, \(4.362 - 7.0086\).Here’s a simple guide on how to effectively use a calculator for subtraction:

• Make sure your calculator is on and set to handle decimal values.
• Enter the first number, the minuend \(4.362\).
• Press the subtraction button (usually a "−").
• Input the second number, the subtrahend \(7.0086\).
• Press the equals button (usually "=" or similar) to get the result.

The calculator will display the result, which should be \(-2.6466\) for this problem. Using calculators helps avoid mistakes in manual calculations, especially with long decimals. However, it's important to have a basic understanding of why the results appear as they do, to ensure you are inputting calculations correctly and interpreting results accurately. Practice calculator skills regularly to become fast and accurate in solving arithmetic problems intuitively.