Problem 78 - Elementary and Intermediate Algebra

Question

Add. $$ -4,061+5,000 $$

Step-by-Step Solution

Answer

The sum of -4,061 and 5,000 is 939.

Step 1: Write Down the Numbers

We need to add the two numbers: -4,061 and 5,000. Convert them to ensure they are clear: -4,061 + 5,000.

Step 2: Align the Numbers

Align the numbers vertically by place value, so that units, tens, hundreds, and thousands are all aligned:\[\begin{array}{c} \,\,\,\,\,5000 \+ (-4061) \\hline\end{array}\]

Step 3: Perform Addition

Carry out the addition by starting from the rightmost digit (units place) and moving to the left. Since we are adding a negative number, think of this as subtracting the absolute values: 1. Subtract 1 from 0 in the units column: Since you can't subtract 1 from 0, you'll need to borrow from the tens place: 10 - 1 = 9 (borrowed 1 from the tens place, which will now have 9) 2. Subtract 6 from 9 in the tens column: 9 - 6 = 3 3. Subtract 0 from 9 in the hundreds column: 9 - 0 = 9 4. Subtract 4 from 4 in the thousands column: 4 - 4 = 0 So, the sum is 939.

Step 4: Write Final Result

After performing the addition (accounting for the negative and positive signs), the result is:The result is 939, so the final answer is: \[0 + 939 = 939\]

Key Concepts

Understanding Negative Numbers

Negative numbers can be tricky, but think of them like owing something. In this exercise, we have

• -4,061 (a negative number), which suggests a deduction or loss,
• and 5,000 (a positive number), indicative of having or gaining.

These numbers are opposite in direction on the number line. When we add a negative number, it's similar to subtracting its absolute value from the positive one. Understanding this principle is essential for handling problems involving both negative and positive integers. Always consider that adding a negative is synonymous with subtracting its absolute value.

Borrowing in Subtraction

Borrowing is a critical component when subtracting a larger number from a smaller one in any given column. Imagine needing an extra push to carry out subtraction. If the number at a particular place value is smaller than the number you need to subtract, borrowing allows you to use resources from the next higher place value. For example:

• In the units column of this problem, we needed to subtract 1 from 0.
• Since 0 can't satisfy this request alone, we borrowed a digit from the tens place.

Thus, the tens place was reduced accordingly, enabling the proper calculation for that column. Borrowing spreads out the values to ensure smooth subtraction without missing any place values.

Place Value Alignment

Aligning numbers by their place values is crucial, especially when dealing with multiple digits. Always ensure that:

• Units align under units,
• Tens line up with tens,
• Hundreds are in sync with hundreds,
• Thousands nestle under thousands.

This structured approach simplifies arithmetic operations by keeping them organized. Misalignment might cause errors, as different place values carry different powers of ten. Simply, keeping columns straight ensures calculations remain accurate and manageable, making complex numbers easier to work with. Efficient alignment aids in clearly visualizing the action needed and limits mistakes in computation.