Problem 170 - Prealgebra

Question

In the following exercises, subtract and then check by adding. $$ 7,775-3,251 $$

Step-by-Step Solution

Answer

The difference is 4,524. Verification: 4,524 + 3,251 = 7,775.

Step 1: Align the Numbers

Write the numbers 7,775 and 3,251 one under the other, ensuring that each digit is aligned correctly according to place value (units, tens, hundreds, thousands).

Step 2: Subtract the Units Column

Subtract the units digit of 3,251 from the units digit of 7,775. Perform the borrowing if needed: 5 - 1 = 4

Step 3: Subtract the Tens Column

Subtract the tens digit of 3,251 from the tens digit of 7,775. Perform borrowing as necessary: 7 - 5 = 2

Step 4: Subtract the Hundreds Column

Subtract the hundreds digit of 3,251 from the hundreds digit of 7,775. Borrow if necessary: 7 - 2 = 5

Step 5: Subtract the Thousands Column

Subtract the thousands digit of 3,251 from the thousands digit of 7,775. Perform borrowing if needed: 7 - 3 = 4

Step 6: Final Subtraction Result

Combine all the resultant digits from the previous steps to obtain the final subtraction result: 7,775 - 3,251 = 4,524

Step 7: Check by Adding

To verify the answer, add the difference (4,524) to the smaller number (3,251) and see if you get the original larger number (7,775): 4,524 + 3,251 = 7,775

Step 8: Confirm the Result

Since the addition matches the original number, the subtraction result is verified to be correct.

Key Concepts

Place Value Alignment

When performing subtraction, it's essential to align the numbers according to their place value. This means that the units, tens, hundreds, and thousands places should all line up correctly. Proper alignment helps avoid mistakes while subtracting digits.

In our example, we have the numbers 7,775 and 3,251. We need to write them like this:

7,775
3,251

Each digit is in the right column: units under units, tens under tens, hundreds under hundreds, and thousands under thousands. This way, when we subtract, we start with the smallest place value (units) and move towards the largest (thousands).

Borrowing in Subtraction

Borrowing is a technique used when a digit in the minuend (the number being subtracted from) is smaller than the corresponding digit in the subtrahend (the number being subtracted). In such cases, we borrow 1 from the next higher place value to perform the subtraction.

Here's a breakdown:

• Units: 5 - 1 = 4 (no borrowing needed)
• Tens: 7 - 5 = 2 (no borrowing needed)
• Hundreds: 7 - 2 = 5 (no borrowing needed)
• Thousands: 7 - 3 = 4 (no borrowing needed)

In this example, no borrowing was needed. However, if we had a problem like 7,704 - 3,286, then borrowing would occur.

For the tens place (0-8), borrowing from the hundreds place would change the problem to:

9 (borrowed from hundreds) - 8 = 1
6 (remaining after borrowing) - 2 = 4

Checking Subtraction with Addition

After getting the result of your subtraction, you can always check your work by adding the difference to the smaller number. If the sum equals the original larger number, your subtraction is correct.

In our example, we found that 7,775 - 3,251 = 4,524. To verify this:

4,524 + 3,251
= 4,524 + 3,251
= 7,775

This proves the subtraction was done correctly because the result matches the original number. This technique ensures your subtraction is accurate, instilling confidence in your results.

Remember, checking your work is a good habit, especially when performing complex calculations or when accuracy is critical.