Problem 182 - Prealgebra

Question

In the following exercises, subtract and then check by adding. $$ 35,162-7,885 $$

Step-by-Step Solution

Answer

The result of 35,162 - 7,885 is 27,277. Verification: 27,277 + 7,885 = 35,162.

Step 1: Arrange the Numbers

Write the numbers in a column with the larger number on top, making sure to align the digits by place value. \[\begin{array}{c} 35,162 \ -7,885 \end{array}\]

Step 2: Subtract Each Column

Starting from the rightmost column (units), subtract each digit. If needed, borrow from the next column. \[\begin{array}{c} 35,162 \ -7,885 \ \ 27,277 \end{array}\] Here are the steps for each column: - Units: \(2 - 5\), borrow from the tens column, becomes \(12 - 5 = 7\). - Tens: \(5 - 8\), borrow from the hundreds column, becomes \(15 - 8 = 7\). - Hundreds: \(0 - 8\), borrow from the thousands column, becomes \(10 - 8 = 2\). - Thousands: \(4 - 7\), borrow from the ten-thousands column, becomes \(14 - 7 = 7\). - Ten-thousands: \(2\) remains since we borrowed.

Step 3: Verify by Adding

Add the result \(27,277\) to the subtracted number \(7,885\) and check if it equals the original number \(35,162\). \[\begin{array}{c} 27,277 \ +7,885 \ \ 35,162 \end{array}\] Since the sum matches the original number, the subtraction answer is verified.

Key Concepts

subtraction with borrowing

When subtracting numbers, especially multi-digit ones, you might face a situation where the digit in the top number is smaller than the digit in the bottom number. This is where **borrowing** comes into play. Borrowing means taking one unit from the next higher place value column to make the subtraction possible.

For instance, in the subtraction of 35,162 - 7,885:
- We start from the units column, where we have 2 - 5. Since 2 is smaller than 5, we borrow 1 from the tens place. This means the 2 becomes 12, and the next column's digit (which is 6) turns into 5.
- Now, we do 12 - 5, which equals 7. Moving to the tens column, we have 5 - 8. Again, since 5 is smaller than 8, we borrow 1 from the hundreds place. The 5 becomes 15, and the hundreds place digit (which is 1) turns into 0.
- Subtracting 15 - 8 gives us 7. In the hundreds place, we now have 0 - 8. Since 0 is smaller than 8, we borrow from the thousands column.

By understanding and practicing borrowing, you will find it easier to handle any subtraction problem without getting stuck.

checking subtraction by addition

A great way to verify your subtraction is by using **addition**. If your subtraction is correct, adding the result to the smaller number should give you the original larger number. This step ensures your answer is accurate. Let's check the subtraction 35,162 - 7,885 using this method:

1. The result from subtraction is 27,277.
2. Add 27,277 to 7,885.
3. The sum should be 35,162, which matches our original number.

Mathematically, it looks like this:
\[ \begin{array}{c} 27,277 \ + 7,885 \ \ \ \ 35,162 \end{array} \]

Since the sum matches the original number, it confirms that our subtraction was done correctly. This simple check can save you from errors, making sure your answers are always reliable.

place value alignment

For accurate subtraction and addition, it's crucial to align numbers correctly according to their **place values** (units, tens, hundreds, thousands, etc.). Misalignment can lead to mistakes.

Consider the problem 35,162 - 7,885:
- Units of both numbers should be in one column.
- Tens in another, hundreds in another, and so forth.

The subtraction is written like this:
\[ \begin{array}{c} 35,162 \ - 7,885 \ \ \ \ 27,277 \end{array} \]

Aligning each digit in the proper place ensures that each digit is subtracted from its corresponding place value.

For the same example, when we borrow and make changes in different columns, it is essential that these changes are kept within the columns they belong to. This helps in maintaining the integrity of the calculation and avoiding confusion.

Proper alignment, paired with a systematic approach, makes arithmetic operations easier and error-free, paving the way for better understanding and accurate results.