Subtraction Within 1,000 with Regrouping - Third Grade Mathematics

Subtraction is the process of finding the difference between two quantities or taking an amount away from a whole. When the top digit in a column is smaller than the bottom digit, you cannot simply flip the numbers; instead, you must ungroup or borrow from the neighbor to the left. In this chapter, you will master the art of multi-digit subtraction within 1,000 using clear, place value regrouping.

The Principle of Regrouping in Subtraction

In subtraction, the top number is called the minuend (the total you start with), and the bottom number is called the subtrahend (the amount you are taking away).

    542   <- Minuend (Total we have)
  - 218   <- Subtrahend (Amount we take away)
  -----
    324   <- Difference (What remains)

If you have 2 single pennies in your pocket, can you hand someone 8 pennies? No! But if you have dimes in your pocket, you can trade 1 dime (1 ten) for 10 single pennies (10 ones). Now you have 12 pennies, and you can easily take 8 away!

Trade: 1 Ten -> 10 Ones
[||||||||||]  ----->  [.] [.] [.] [.] [.] [.] [.] [.] [.] [.]

Trade: 1 Hundred -> 10 Tens
+----------+          [||||||||||] [||||||||||] [||||||||||] [||||||||||] [||||||||||]
| Hundred  |  ----->  [||||||||||] [||||||||||] [||||||||||] [||||||||||] [||||||||||]
+----------+

The Golden Rule: More on the Floor? Go Next Door!

Here is an easy rhyme that mathematicians use to remember what to do:
- More on top? No need to stop! (Just subtract).

- More on the floor? Go next door and get 10 more!

- Numbers the same? Zero's the game! (For example, 7 - 7 = 0).


Step-by-Step Subtraction Algorithm

Let us walk through the process of regrouping with clarity and precision.

Case 1: Regrouping Tens to Ones

Subtract 563 - 238.

         5  13          (unbundled 1 ten: 6 tens becomes 5 tens, 3 ones becomes 13)
     5   6   3
   - 2   3   8
   -----------
     3   2   5

Step-by-step:
1. Ones column: We have 3, but need to subtract 8. Since 3 is less than 8, we go next door to the tens column.

2. Cross out the 6 tens; it becomes 5 tens.

3. Bring 10 ones over to the ones column: 3 + 10 = 13 ones.

4. Now subtract the ones: 13 - 8 = 5 ones.

5. Subtract the tens: 5 - 3 = 2 tens.

6. Subtract the hundreds: 5 - 2 = 3 hundreds.
The difference is 325.

Case 2: Regrouping Hundreds to Tens

Subtract 728 - 384.

     6  12              (unbundled 1 hundred: 7 hundreds becomes 6, 2 tens becomes 12)
     7   2   8
   - 3   8   4
   -----------
     3   4   4

Step-by-step:
1. Ones column: 8 - 4 = 4. More on top, no need to stop!

2. Tens column: We have 2 tens, but need to subtract 8 tens. Since 2 is smaller than 8, go next door to the hundreds column.

3. Cross out 7 hundreds; it becomes 6 hundreds.

4. Bring 10 tens over to the tens column: 2 + 10 = 12 tens.

5. Subtract tens: 12 - 8 = 4 tens.

6. Subtract hundreds: 6 - 3 = 3 hundreds.
The difference is 344.

Case 3: Regrouping in Both Places

Subtract 634 - 278.

         12  14         (tens unbundled, then hundreds unbundled)
     5    2  14
     6    3   4
   - 2    7   8
   ------------
     3    5   6

Step-by-step:
1. Ones column: 4 is smaller than 8. Go next door to the tens place. 3 tens becomes 2 tens. The 4 ones becomes 14 ones. 14 - 8 = 6.

2. Tens column: Now we have 2 tens, but we must subtract 7 tens. Go next door to hundreds! 6 hundreds becomes 5 hundreds. The 2 tens becomes 12 tens. 12 - 7 = 5 tens.

3. Hundreds column: 5 - 2 = 3 hundreds.
The difference is 356.

Checking Subtraction Using Addition

The safest way to verify any subtraction problem is to add the difference back to the subtrahend. If your answer is correct, you will get the minuend!

    356  (Difference)
  + 278  (Subtrahend)
  -----
    634  (Matches original Minuend exactly!)

Chapter Practice Exercises

  1. Solve each subtraction problem: a. 472 - 138 b. 681 - 347 c. 835 - 462 d. 524 - 187
  2. Solve: a. 753 - 289 b. 941 - 468 c. 615 - 378 d. 822 - 546
  3. A bookstore had 524 mystery books in stock on Monday. By Friday, customers had purchased 268 mystery books. How many mystery books remained on the shelves?
  4. Marcus subtracted 452 - 187 and got 335 because he subtracted 7 - 2 = 5 in the ones column and 8 - 5 = 3 in the tens column. Explain why Marcus's method is incorrect, and find the true answer.

Solutions and Step-by-Step Answers

  1. Subtraction practice: a. 472 - 138: Regroup tens: 7 becomes 6, 2 becomes 12. Ones: 12 - 8 = 4. Tens: 6 - 3 = 3. Hundreds: 4 - 1 = 3. Difference = 334. b. 681 - 347: Regroup tens: 8 becomes 7, 1 becomes 11. Ones: 11 - 7 = 4. Tens: 7 - 4 = 3. Hundreds: 6 - 3 = 3. Difference = 334. c. 835 - 462: Ones: 5 - 2 = 3. Regroup hundreds: 8 becomes 7, 3 becomes 13. Tens: 13 - 6 = 7. Hundreds: 7 - 4 = 3. Difference = 373. d. 524 - 187: Regroup tens (2 becomes 1, 4 becomes 14; 14 - 7 = 7). Regroup hundreds (5 becomes 4, 1 becomes 11; 11 - 8 = 3). Hundreds: 4 - 1 = 3. Difference = 337.
  2. Double regrouping practice: a. 753 - 289: 13 - 9 = 4; 14 - 8 = 6; 6 - 2 = 4. Difference = 464. b. 941 - 468: 11 - 8 = 3; 13 - 6 = 7; 8 - 4 = 4. Difference = 473. c. 615 - 378: 15 - 8 = 7; 10 - 7 = 3; 5 - 3 = 2. Difference = 237. d. 822 - 546: 12 - 6 = 6; 11 - 4 = 7; 7 - 5 = 2. Difference = 276.
  3. Books remaining: 524 - 268. Ones: 14 - 8 = 6. Tens: 11 - 6 = 5. Hundreds: 4 - 2 = 2. There are 256 mystery books remaining. Check: 256 + 268 = 524.
  4. Marcus committed the common "flipping error." In subtraction, the order cannot be changed (subtraction is not commutative). You must subtract the bottom number from the top number. Correct calculation for 452 - 187: Ones: 12 - 7 = 5. Tens: 14 - 8 = 6. Hundreds: 3 - 1 = 2. The true difference is 265.