Subtraction (grouping / regrouping / borrowing)

Subtraction involving grouping, regrouping, or borrowing happens when the number being subtracted requires a bit more attention, especially in cases where digits in one place value are smaller than those in the other.

Grouping or Regrouping (Borrowing) in Subtraction

This technique is used when you need to subtract a larger number from a smaller number in a given place value. It's especially common when subtracting multi-digit numbers. Regrouping (also known as borrowing) helps you adjust the digits to make the subtraction possible.

Steps for Borrowing (Regrouping) in Subtraction

1. Look at the digits in the place value you're working with. If the digit in the minuend (the top number) is smaller than the digit in the subtrahend (the bottom number), you need to borrow.
2. Borrow from the next higher place value. For example, if you're working with the ones place and the digit is smaller than the subtrahend, you borrow 1 from the tens place (or 10 from the tens place) and add it to the ones place.
3. Subtract the new numbers after borrowing.
4. Continue to the next place value if necessary and repeat the process until the subtraction is complete.

Example 1: Simple Borrowing

Let's subtract ( 52 - 28 ):

  52
- 28
-----

• Start with the ones place: 2 (minuend) is smaller than 8 (subtrahend), so we need to borrow.
• Borrow 1 from the tens place (5 becomes 4), and add 10 to the ones place (2 becomes 12).
• Now, subtract: 12 - 8 = 4.

  4 (from the tens place)
  12 (in the ones place)
- 28
-----
  24

Example 2: Borrowing Across Multiple Place Values

Now, let's subtract ( 305 - 178 ):

  305
- 178
-----

• Start with the ones place: 5 (minuend) is smaller than 8 (subtrahend), so borrow 1 from the tens place (0 becomes 9), and add 10 to the ones place (5 becomes 15).
• Subtract 15 - 8 = 7.
• Move to the tens place: Now subtract 9 - 7 = 2.
• Finally, in the hundreds place: 3 - 1 = 2.

  2 (from hundreds place)
  9 (from tens place)
  15 (from ones place)
- 178
-----
  127

Key Terms in Regrouping (Borrowing):

• Minuend: The number from which you subtract (top number).
• Subtrahend: The number you subtract (bottom number).
• Difference: The result after subtraction.
• Borrow: To take 1 from the next higher place value when the digit in the current place value is too small.

Borrowing ensures that subtraction is always possible, even when digits in the minuend are smaller than those in the subtrahend at each place value.