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Chapter 1: Problem 34

Find the difference in each case. (These problems all involve borrowing.) [Example 3] $$\begin{array}{r} 804 \\ -238 \\ \hline \end{array}$$

Short Answer

Expert verified
The difference is 566.

Step by step solution

01

Align Numbers for Subtraction

Write the numbers one on top of the other, ensuring that the digits are aligned by place value. In this case, write 804 with 8 in the hundreds place, 0 in the tens place, and 4 in the units place. Align 238 underneath, with 2 under 8, 3 under 0, and 8 under 4.
02

Subtract the Units Place

Begin with the units place. We need to subtract 8 from 4. Since 4 is less than 8, we need to borrow 1 from the tens place, but since the tens place is a 0, we'll need to adjust.
03

Borrow from the Hundreds Place

Since the tens place is 0, borrow 1 from the hundreds place, reducing it from 8 to 7, and convert the tens place to 10. Now, borrow 1 from the tens place (10 becomes 9) and add 10 to the units: 14.
04

Subtract the Units Place with Borrowed Value

Subtract 8 from 14 in the units place, resulting in 6. Write 6 in the units place of the answer.
05

Subtract the Tens Place

Move to the tens place, where we have a 9 (after borrowing). Subtract 3 from 9, which gives 6. Write 6 in the tens place of the answer.
06

Subtract the Hundreds Place

Finally, subtract the hundreds place: 7 (after borrowing) minus 2 results in 5. Write 5 in the hundreds place of the answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Place Value Alignment
When performing subtraction, aligning numbers based on their place value is crucial. This ensures that digits are paired correctly across the numbers being subtracted. For instance, let's look at the numbers 804 and 238 in the example.
  • The first digit from the right is the units place (or ones place), which holds the values 4 and 8, respectively.
  • The second digit from the right is the tens place, crucial for borrowing in many cases. Here, we have 0 and 3.
  • The third digit is the hundreds place, with 8 and 2.
To begin, stack these numbers vertically, ensuring each digit is directly above or below a digit of the same place value. This arrangement allows us to systematically approach subtraction from right to left, starting with the smallest place value.
Borrowing in Subtraction
Borrowing in subtraction is a technique used when the digit in the subtrahend (the number being subtracted) is larger than the digit in the minuend (the number from which you subtract) in a particular place value. In our example, you are required to subtract 8 from 4 in the units place. Since 4 is smaller than 8, direct subtraction isn't possible. This is where borrowing becomes necessary.
  • First, check the tens place. If it's possible to borrow from the tens, proceed. Here, the tens digit is 0, which can't be borrowed from directly.
  • Next, check the next higher place, the hundreds. By borrowing 1 from the hundreds, 800 becomes 700, and the tens place is adjusted from 0 to 10.
  • Finally, you redistribute the 10 to the units place: Now, 10 becomes 9 (after borrowing for the units), and the 4 becomes 14.
This action is like redistributing values to make subtraction feasible, allowing us to continue the operation smoothly.
Step-by-Step Subtraction Process
Subtraction is done methodically from right (units) to left (hundreds), especially when borrowing is involved. Here's how this works:
  • Subtract Units Place: After borrowing, subtract 8 from the adjusted units value, 14. This results in 6, written in the answer's units place.
  • Subtract Tens Place: With a new tens place value after borrowing, subtract 3 from 9 (again adjusted from earlier borrowing). This gives 6, placed in the tens place of the answer.
  • Subtract Hundreds Place: Now, subtract 2 from 7 in the hundreds place, resulting in 5. Write this in the hundreds place of the result.
The final answer, neatly aligned beneath the subtraction bar, is 566, which perfectly completes the subtraction operation by addressing borrowing needs at each step.

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