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Add and Subtract With Regrouping: Trade, Then Check

Add and subtract 3- and 4-digit numbers by regrouping, handle zeros in the middle, and check every answer with an estimate or the inverse operation.

Grades 3–5 · 32 practice questions · Original material
Learning objectiveAdd and subtract 3- and 4-digit whole numbers in columns, regroup 10 ones as 1 ten (and so on) when a column sum reaches 10 or more, trade 1 ten for 10 ones (and so on) when a top digit is too small, regroup across zeros, explain each trade in place-value words, and check answers with an estimate and with the inverse operation.

The idea

In the last two math lessons you rounded numbers and estimated sums and differences. An estimate tells you about how big an answer should be. Today you find the exact answer with column addition and subtraction, and then use an estimate to check it.

Column methods work because of place value. Line up the ones under the ones, the tens under the tens and so on. Start with the ones column and move left. Each column can hold only one digit, from 0 to 9.

Adding: when a column adds up to 10 or more, regroup. In 468 + 275, the ones make 8 + 5 = 13 ones. Thirteen ones is 1 ten and 3 ones, so write 3 in the ones and record the 1 ten above the tens column. Then the tens make 6 + 7 + 1 = 14 tens, which is 1 hundred and 4 tens. The hundreds make 4 + 2 + 1 = 7. The sum is 743.

Subtracting: when the top digit in a column is smaller than the bottom digit, trade 1 from the next place to the left. In 724 − 386, you cannot take 6 ones from 4 ones, so trade 1 ten for 10 ones: 7 hundreds, 1 ten and 14 ones. 14 − 6 = 8. Then 1 ten minus 8 tens needs another trade: 6 hundreds and 11 tens. 11 − 8 = 3, and 6 − 3 = 3. The difference is 338.

Zeros in the middle need two trades. In 503 − 278 there are 0 tens, so trade 1 hundred for 10 tens first, then 1 ten for 10 ones. 503 becomes 4 hundreds, 9 tens and 13 ones. Now 13 − 8 = 5, 9 − 7 = 2 and 4 − 2 = 2, so the answer is 225. A quick way to say it: 503 is 50 tens and 3 ones, and 1 of those tens moves to the ones, leaving 49 tens.

Always check. Estimate first (503 − 278 is about 500 − 300 = 200, so 225 is reasonable), and then use the inverse operation: 225 + 278 should rebuild 503. An estimate catches big mistakes, and the inverse check catches small ones.

Worked examples

Column addition and subtraction with regroupingLeft panel: 468 plus 275 in hundreds, tens and ones columns. 8 plus 5 is 13 ones, so 3 is written and 1 ten is regrouped above the tens. 6 plus 7 plus 1 is 14 tens, so 4 is written and 1 hundred is regrouped above the hundreds. 4 plus 2 plus 1 is 7 hundreds. The sum is 743. Right panel: 503 minus 278. There are 0 tens, so 1 hundred is traded for 10 tens and 1 ten for 10 ones: 503 becomes 4 hundreds, 9 tens and 13 ones. 13 minus 8 is 5, 9 minus 7 is 2, 4 minus 2 is 2, so the difference is 225. A bottom band shows the checks: estimate 500 minus 300 is 200, and 225 plus 278 is 503.Regroup: trade 10 for 1, or 1 for 10Original schematic. Small numbers show each trade.1 · Add: 468 + 275HTO11468+275743ones: 8 + 5 = 13 → write 3, regroup 1 tentens: 6 + 7 + 1 = 14 → write 4, regroup 1 hundredhundreds: 4 + 2 + 1 = 72 · Subtract across a zero: 503 − 278HTO4913503−2782250 tens: trade 1 hundred → 10 tens,then 1 ten → 10 ones: 4 H, 9 T, 13 O13 − 8 = 5 · 9 − 7 = 2 · 4 − 2 = 2Check every answer two waysEstimate: 500 − 300 = 200, so 225 is reasonable · Inverse: 225 + 278 = 503 ✓
Original schematic, drawn to the labelled values. In addition, 10 in one column is regrouped as 1 in the next column to the left. In subtraction, when a top digit is too small, 1 from the next place is traded for 10; across a zero, trade from the hundreds first. Check with an estimate for size and with the inverse operation for the exact value.

Example 1 · Two regroups in addition

468 + 275

  1. Ones: 8 + 5 = 13. Write 3, regroup 1 ten.
  2. Tens: 6 + 7 + 1 = 14. Write 4, regroup 1 hundred.
  3. Hundreds: 4 + 2 + 1 = 7.
  4. Estimate check: 500 + 300 = 800, and 743 is close.

743

Example 2 · Two trades in subtraction

724 − 386

  1. Ones: 4 < 6, so trade 1 ten: 1 ten and 14 ones. 14 − 6 = 8.
  2. Tens: 1 < 8, so trade 1 hundred: 6 hundreds and 11 tens. 11 − 8 = 3.
  3. Hundreds: 6 − 3 = 3.
  4. Inverse check: 338 + 386 = 724.

338

Example 3 · Subtract across a zero

503 − 278

  1. Ones: 3 < 8, but there are 0 tens to trade.
  2. Trade 1 hundred for 10 tens, then 1 ten for 10 ones: 4 hundreds, 9 tens, 13 ones.
  3. 13 − 8 = 5, 9 − 7 = 2, 4 − 2 = 2.
  4. Inverse check: 225 + 278 = 503.

225

Example 4 · Four digits, many zeros

1000 − 364

  1. 1000 has 0 hundreds, 0 tens and 0 ones, so trade all the way from the thousand.
  2. 1000 becomes 9 hundreds, 9 tens and 10 ones.
  3. 10 − 4 = 6, 9 − 6 = 3, 9 − 3 = 6.
  4. Estimate check: 1000 − 400 = 600, and 636 is close.

636

Example 5 · Catch the mistake

Ava says 503 − 278 = 375. Is she right?

  1. Estimate: 500 − 300 = 200. 375 is far from 200, so check.
  2. She took 8 − 3, 7 − 0 and 5 − 2: the smaller digit from the bigger one.
  3. The top number must come first in every column, so trade instead.
  4. The correct answer is 225.

No. 375 is not reasonable; the answer is 225.

Common mistakes

These are tempting because they use familiar operations. Check what the question is actually measuring.

Subtracting the smaller digit from the bigger one in every column.

Why this fails: In 503 − 278, 8 − 3 is backwards. The top number is the one you take from, so trade instead: the answer is 225, not 375.

Writing two digits in one column.

Why this fails: 468 + 275 is not 6 13 13. Thirteen ones is 1 ten and 3 ones, so write 3 and regroup the 1.

Forgetting that a traded place went down by 1.

Why this fails: After trading 1 ten for 10 ones in 724 − 386, the tens digit is 1, not 2.

Trading from a zero.

Why this fails: In 806 − 357 there are 0 tens. Go to the hundreds first: 8 hundreds becomes 7 hundreds and 10 tens, then 9 tens and 16 ones.

Lining numbers up on the left.

Why this fails: In 1206 − 578, the 5 must sit under the 2 (hundreds), not under the 1. Line up the ones digits on the right.

Skipping the check.

Why this fails: A quick estimate catches big slips, and adding the answer to the number taken away catches small ones.

Foundation

Foundation practice

Line up the places, regroup when a column needs it, and record the trade.

Add with regrouping

1. 468 + 275 = ?

  1. 633
  2. 743
  3. 61313
  4. 733
Subtract with regrouping

2. 724 − 386 = ?

  1. 462
  2. 438
  3. 338
  4. 348
Which column regroups?

3. In 352 + 489, which columns need a regroup?

  1. Only the ones
  2. Only the hundreds
  3. The ones and the tens
  4. No columns
Add with regrouping

4. 586 + 347 = ?

  1. 923
  2. 833
  3. 81313
  4. 933
Name the trade

5. In 806 − 357, the ones column needs more ones but the tens digit is 0. What is the first trade?

  1. Trade 1 hundred for 10 tens, then 1 ten for 10 ones
  2. Take 1 ten from the 0 tens
  3. Subtract 6 − 7 = 1
  4. Cross out the 8 and write 9
Subtract across a zero

6. 806 − 357 = ?

  1. 551
  2. 459
  3. 449
  4. 549
Record the regroup

7. When you add 639 + 284, what small number do you write above the tens column?

  1. 13
  2. 0
  3. 2
  4. 1
Check by adding

8. Mia says 724 − 386 = 338. Which addition checks her answer?

  1. 338 + 724
  2. 724 + 386
  3. 386 + 338
  4. 338 − 386

Practice

Practice practice

Four-digit numbers, zeros in the middle and checking with an estimate.

Add four digits

9. 2457 + 1386 = ?

  1. 3843
  2. 3733
  3. 3833
  4. 3743
Subtract across zeros

10. 1000 − 364 = ?

  1. 1364
  2. 746
  3. 646
  4. 636
Subtract across zeros

11. 4005 − 1267 = ?

  1. 3262
  2. 2738
  3. 2848
  4. 2748
Estimate, then check

12. Estimate 2675 + 1849 to the nearest hundred, then choose the exact sum.

  1. About 4500; exact 4524
  2. About 4500; exact 3424
  3. About 4000; exact 4424
  4. About 4600; exact 4514
Subtract four digits

13. 3008 − 1459 = ?

  1. 2451
  2. 1549
  3. 1559
  4. 1649
Which is reasonable?

14. Sam says 915 − 468 = 553. Use an estimate to decide.

  1. Reasonable: 900 − 500 = 400 is close.
  2. Reasonable, because 553 has three digits.
  3. Not reasonable: it should be about 1400.
  4. Not reasonable: 900 − 500 = 400, and the exact answer is 447.
Add three numbers

15. 365 + 278 + 194 = ?

  1. 837
  2. 737
  3. 827
  4. 937
Word problem

16. A library had 1206 books. It lent out 578. How many are still on the shelves?

  1. 728
  2. 628
  3. 1372
  4. 1784

Think

Think practice

Find the error, explain the trade and choose the right operation.

Find the error

17. Ava wrote 503 − 278 = 375. What went wrong?

  1. She added instead of subtracting.
  2. In each column she took the smaller digit from the bigger one instead of regrouping.
  3. She forgot to line up the columns.
  4. Nothing; 375 is correct.
Find the error

18. Ben wrote 468 + 275 = 61313. What went wrong?

  1. He wrote 13 and 13 in single columns instead of regrouping.
  2. He subtracted.
  3. He lined up the numbers on the left.
  4. He forgot the hundreds.
Find the error

19. Cleo wrote 806 − 357 = 459. Which step did she miss?

  1. She forgot to trade a hundred.
  2. She added the ones.
  3. She lined up 357 one place to the left.
  4. After trading for the ones, she forgot that the 10 tens became 9.
Explain the zero

20. Why does 400 − 163 need two trades before you can subtract the ones?

  1. Because 163 is bigger than 400.
  2. Because you must always trade twice.
  3. Because 400 is an even number.
  4. Because there are 0 tens and 0 ones, so a hundred must become tens, and a ten must become ones.
Choose the operation

21. A school collected 1623 cans in October and 2798 in November. How many in all?

  1. 1623 + 2798 = 4421
  2. 2798 − 1623 = 1175
  3. 1623 + 2798 = 3311
  4. 2798 + 2798 = 5596
Choose the operation

22. A train trip is 843 km. The train has gone 397 km. How far is left?

  1. 843 + 397 = 1240
  2. 843 − 397 = 446
  3. 843 − 397 = 554
  4. 397 − 843
Compare

23. Which is greater: 2050 − 1375 or 352 + 489?

  1. 352 + 489
  2. 2050 − 1375
  3. They are equal
  4. You cannot tell without a calculator
Missing digit

24. In 5_8 + 247 = 805, what is the missing digit?

  1. 5
  2. 4
  3. 6
  4. 3

Challenge

Challenge practice

Work backwards, build numbers and check with inverse operations.

Work backwards

25. A number minus 2381 equals 2619. What is the number?

  1. 238
  2. 5000
  3. 4990
  4. 5100
Across many zeros

26. 6040 − 2795 = ?

  1. 4755
  2. 3255
  3. 3245
  4. 3345
Build the biggest sum

27. Use each of the digits 2, 4, 6, 8 once to make two 2-digit numbers with the largest possible sum.

  1. 86 + 42 = 128
  2. 84 + 62 = 146
  3. 86 + 24 = 110
  4. 68 + 24 = 92
Inverse check

28. Which check proves 3405 − 1879 = 1526?

  1. 3405 + 1879 = 5284
  2. 1879 − 1526 = 353
  3. 1526 + 1879 = 3405
  4. 1526 − 1879
How many regroups?

29. How many times do you regroup when adding 999 + 1?

  1. 1
  2. 2
  3. 3
  4. 0
Smallest difference

30. Using each of the digits 3, 5, 7 once, make a 3-digit number. Subtract it from 1000. What is the smallest possible answer?

  1. 100
  2. 243
  3. 357
  4. 247
Explain

31. Why does 1000 − 1 = 999 need three trades but 1000 − 1000 = 0 needs none?

Two methods

32. Find 4567 − 1892 two ways: with regrouping, and by adding up from 1892 (1892 → 1900 → 4500 → 4567). Do both give the same answer?

Answer key

  1. 1. 743
    8 + 5 = 13 ones: write 3, regroup 1 ten. 6 + 7 + 1 = 14 tens: write 4, regroup 1 hundred. 4 + 2 + 1 = 7 hundreds. 633 forgets both regrouped ones, 733 forgets the hundred, and 61313 writes 13 in two columns instead of trading.
  2. 2. 338
    4 − 6 needs a trade: 7 hundreds 2 tens 4 ones becomes 7 hundreds 1 ten 14 ones. 14 − 6 = 8. 1 − 8 needs a trade: 6 hundreds 11 tens. 11 − 8 = 3, 6 − 3 = 3. 462 takes the smaller digit from the bigger one in each column, 438 forgets that the hundreds went down to 6, and 348 forgets to reduce the tens after the first trade.
  3. 3. The ones and the tens
    2 + 9 = 11 ones (regroup), then 5 + 8 + 1 = 14 tens (regroup), then 3 + 4 + 1 = 8 hundreds. The sum is 841. The hundreds column does not reach 10.
  4. 4. 933
    6 + 7 = 13: write 3, carry 1. 8 + 4 + 1 = 13: write 3, carry 1. 5 + 3 + 1 = 9. 923 adds the tens wrong, 833 drops the regrouped hundred, and 81313 skips regrouping.
  5. 5. Trade 1 hundred for 10 tens, then 1 ten for 10 ones
    There are no tens to trade, so go to the hundreds first: 8 hundreds 0 tens becomes 7 hundreds 10 tens, then 7 hundreds 9 tens 16 ones. 6 − 7 is not 1, and 8 hundreds becomes 7, not 9.
  6. 6. 449
    After the trades: 16 − 7 = 9, 9 − 5 = 4, 7 − 3 = 4. 551 subtracts the smaller digits, 459 forgets that a ten moved to the ones, and 549 forgets the hundred that was traded.
  7. 7. 1
    9 + 4 = 13 ones, which is 1 ten and 3 ones. Write the 3 and put a small 1 above the tens. Writing 13 would put two digits in one column. The sum is 923.
  8. 8. 386 + 338
    Subtraction and addition undo each other: the part taken away plus the answer should give the starting number. 386 + 338 = 724, so she is right. The other choices do not rebuild 724.
  9. 9. 3843
    7 + 6 = 13 (carry 1), 5 + 8 + 1 = 14 (carry 1), 4 + 3 + 1 = 8, 2 + 1 = 3. 3733 forgets both carries, 3833 forgets the carry into the tens, and 3743 forgets the carry into the hundreds.
  10. 10. 636
    1000 is 9 hundreds 9 tens 10 ones after trading. 10 − 4 = 6, 9 − 6 = 3, 9 − 3 = 6. 1364 takes the smaller digit from the bigger one in each column, and 746 and 646 come from slips while trading.
  11. 11. 2738
    4005 becomes 3 thousands 9 hundreds 9 tens 15 ones. 15 − 7 = 8, 9 − 6 = 3, 9 − 2 = 7, 3 − 1 = 2. 3262 subtracts smaller digits, and 2848 and 2748 slip in the middle columns.
  12. 12. About 4500; exact 4524
    2700 + 1800 = 4500, and the exact sum is 4524, close to the estimate. 3424 is far from 4500, so it must be wrong, 4424 and 4514 each have a slip in a column.
  13. 13. 1549
    3008 becomes 2 thousands, 9 hundreds, 9 tens and 18 ones: 18 − 9 = 9, 9 − 5 = 4, 9 − 4 = 5, 2 − 1 = 1. 2451 subtracts smaller digits, and the others forget a trade.
  14. 14. Not reasonable: 900 − 500 = 400, and the exact answer is 447.
    553 is more than 100 away from 400, so check. Sam took 8 − 5 and 6 − 1 the wrong way round. 1400 is an addition estimate, and having three digits does not make an answer right.
  15. 15. 837
    Ones: 5 + 8 + 4 = 17 (write 7, carry 1). Tens: 6 + 7 + 9 + 1 = 23 (write 3, carry 2). Hundreds: 3 + 2 + 1 + 2 = 8. When three numbers are added, a column can carry 2. 737 carries only 1 into the hundreds, 827 slips in the tens, and 937 adds an extra hundred.
  16. 16. 628
    1206 − 578: 16 − 8 = 8, 9 − 7 = 2 after trading across the 0, 11 − 5 = 6, then 0 thousands. 1784 adds instead of subtracting, 728 forgets that a hundred was traded, and 1372 takes the smaller digit from the bigger one in each column.
  17. 17. In each column she took the smaller digit from the bigger one instead of regrouping.
    8 − 3 = 5, 7 − 0 = 7, 5 − 2 = 3 gives 375, but the top digit must come first. Regrouping across the 0 gives 225. Adding would give 781.
  18. 18. He wrote 13 and 13 in single columns instead of regrouping.
    Each column can hold only one digit. 13 ones is 1 ten and 3 ones, and 14 tens is 1 hundred and 4 tens. The correct sum is 743.
  19. 19. After trading for the ones, she forgot that the 10 tens became 9.
    She worked 10 − 5 = 5 in the tens instead of 9 − 5 = 4. The correct answer is 449.
  20. 20. Because there are 0 tens and 0 ones, so a hundred must become tens, and a ten must become ones.
    400 = 3 hundreds 9 tens 10 ones after the trades. Then 10 − 3 = 7, 9 − 6 = 3, 3 − 1 = 2, so 237. 163 is smaller than 400, and many subtractions need no trades.
  21. 21. 1623 + 2798 = 4421
    In all means add both months. 1175 is how many more in November, 3311 forgets the regroups, and 5596 adds November twice.
  22. 22. 843 − 397 = 446
    How far is left means take away the part already travelled. 554 subtracts smaller digits, adding gives the wrong idea, and 397 − 843 puts the numbers in the wrong order.
  23. 23. 352 + 489
    2050 − 1375 = 675 and 352 + 489 = 841, so the sum is greater. Estimating first (about 700 and about 850) already shows which is bigger.
  24. 24. 5
    805 − 247 = 558, so the missing tens digit is 5. Check: 558 + 247 = 805. The other digits give 795, 815 or 785.
  25. 25. 5000
    Undo the subtraction by adding: 2619 + 2381 = 5000. Check: 5000 − 2381 = 2619. 238 subtracts instead, and 4990 and 5100 have regrouping slips.
  26. 26. 3245
    Ones: 0 − 5 needs a trade from 4 tens: 10 − 5 = 5. Tens: 3 − 9 needs a trade from the 0 hundreds, so trade a thousand first: 5 thousands 9 hundreds 13 tens. 13 − 9 = 4, 9 − 7 = 2, 5 − 2 = 3. 4755 subtracts the smaller digit in each column; the others forget one trade.
  27. 27. 84 + 62 = 146
    Put the two largest digits in the tens places: 8 tens and 6 tens make 140, plus 4 + 2 = 6 ones, so 146. 86 + 42 and 86 + 24 put 4 or 2 in a tens place, and 68 + 24 puts the big digits in the ones places.
  28. 28. 1526 + 1879 = 3405
    Add the answer to the number taken away to get back the starting number. The other choices do not rebuild 3405.
  29. 29. 3
    Ones: 9 + 1 = 10, regroup. Tens: 9 + 1 = 10, regroup. Hundreds: 9 + 1 = 10, regroup into a thousand. The sum is 1000.
  30. 30. 247
    The biggest 3-digit number you can make is 753, and 1000 − 753 = 247. Subtracting the biggest number gives the smallest difference. 243 has a slip, 357 is the smallest number you can build, and 100 is not possible.
  31. 31. In 1000 − 1 the ones column has 0, so you trade a thousand into hundreds, a hundred into tens and a ten into ones. In 1000 − 1000 every column is subtracted from an equal digit, so no column needs more.
    Regrouping is needed only when a column on top is smaller than the one below.
  32. 32. Yes, 2675. Adding up: 8 + 2600 + 67 = 2675.
    Both methods find the same difference; adding up can avoid trades across zeros. Check: 2675 + 1892 = 4567.

Related practice

Estimate sums and differences: round first, then checkRounding to 10, 100 and 1000: then estimate and checkMental addition and subtraction: choose a shortcutMath hub