Subtract Within 1,000: Decompose Tens & Hundreds | Grade 2 Math
Subtract Within 1,000: Decompose Tens & Hundreds
Learn to subtract three-digit numbers when the ones or tens place does not have enough units. Use base-ten blocks, place-value charts, and drawings to decompose a ten into ten ones or decompose a hundred into ten tens without changing the starting amount.
Work through problems with one trade, two trades, and zeros, then practice choosing a strategy and checking each difference with addition. Includes 25 detailed worked examples, 30 independent practice problems, and a 15-question test with complete answer keys.
๐ฏ What You Will Learn
Rename 1 ten as 10 ones when there are not enough ones to subtract.
Rename 1 hundred as 10 tens when there are not enough tens to subtract.
Use a hundred to make tens, then a ten to make ones when the original tens place has zero.
Keep the starting value unchanged during trading, then verify by adding the difference and the amount removed.
๐ง Key Vocabulary
Rename the same value in smaller place-value units, such as 1 ten as 10 ones.
Exchange 1 hundred flat for 10 ten rods, or 1 ten rod for 10 one cubes.
A flat represents 100, a rod represents 10, and a cube represents 1.
The starting amount from which we subtract.
The amount we take away.
The amount remaining after subtraction, or how far apart two numbers are.
A chart that keeps hundreds, tens, and ones in their correct places.
An equation that checks the subtraction: difference + subtrahend = minuend.
๐ Why Do We Decompose?
๐ฆ Model Two Trades: 532 − 178
Build 532 with 5 hundred flats, 3 ten rods, and 2 one cubes. We need to remove 1 hundred, 7 tens, and 8 ones. There are not enough ones or tens in the original representation, so two exchanges will help.
Read every row: the first three rows are equal-value representations of 532. The last row shows the amount remaining after subtracting 178. These are not four separate amounts to be added.
↔ On a phone, swipe sideways inside the model; with a keyboard, focus it and use the arrow keys.
Remain: 354
๐ Follow the Same Trades on a Place-Value Chart
The numbers in the chart are counts of hundreds, tens, and ones, not separate three-digit numbers. An intermediate count may be greater than nine because the number has temporarily been renamed.
↔ Swipe the chart on a phone or focus it and use the keyboard arrow keys.
| Representation | Hundreds | Tens | Ones |
|---|---|---|---|
| Original 532 | 5 | 3 | 2 |
| After 1 ten → 10 ones | 5 | 2 | 12 |
| After 1 hundred → 10 tens | 4 | 12 | 12 |
| Take away 178 (1 H, 7 T, 8 O) | 4 − 1 | 12 − 7 | 12 − 8 |
| Difference 354 | 3 | 5 | 4 |
๐ Three Kinds of Subtraction with Decomposing
There are 2 ones, but we need to remove 6 ones. Trade 1 of the 4 tens. We then have 3 hundreds, 3 tens, and 12 ones. Remove 1 hundred, 1 ten, and 6 ones.
Four ones can lose one. But 2 tens cannot lose 8 tens. Trade one hundred for ten tens, leaving 5 hundreds and 12 tens. Then take away the matching units.
Start with 5 hundreds, 0 tens, and 3 ones. Because there are no tens available, trade 1 hundred for 10 tens. Next trade 1 of those tens for 10 ones. The same 503 is now 4 hundreds, 9 tens, and 13 ones. Subtract 1 hundred, 7 tens, and 8 ones to get 3 hundreds, 2 tens, and 5 ones.
๐งฉ Connect the Model to Expanded Form
Sometimes a trade can be shown without blocks. Rename 432 as 300 + 120 + 12. This is still 432 because one hundred was traded into ten tens and one ten was traded into ten ones.
Notice that the second expression is not obtained by changing digits at random. Each exchange preserves the starting value, and each subtracted part keeps its place-value meaning.
⬅️ Another Strategy: Backward Jumps
You can also subtract 178 from 532 by moving backward in helpful jumps: subtract 100, then 70, then 8. On a number line this gives 532 → 432 → 362 → 354. This uses the same place-value breakdown of 178, even though the model looks different.
✅ Always Check with Addition
If the difference is correct, the difference plus the number taken away gives the starting number:
To check the addition, combine 4 ones + 8 ones = 12 ones (compose a ten), then 5 tens + 7 tens + 1 new ten = 13 tens (compose a hundred). You get 5 hundreds, 3 tens, and 2 ones: 532.
๐งญ Choose the Right Trade
๐ 25 Worked Examples
Show solution
We have only 2 ones but need to subtract 6 ones. Decompose 1 ten into 10 ones. Now we have 3 hundreds, 3 tens, and 12 ones.
Subtract the ones: 12 ones − 6 ones = 6 ones.
We have enough tens to remove 1 ten, so we do not need another trade.
Subtract the tens: 3 tens − 1 ten = 2 tens.
Subtract the hundreds: 3 hundreds − 1 hundred = 2 hundreds.
Put the places together: 200 + 20 + 6 = 226. Check with addition: 226 + 116 = 342.
Show solution
We have only 1 one but need to subtract 6 ones. Decompose 1 ten into 10 ones. Now we have 5 hundreds, 6 tens, and 11 ones.
Subtract the ones: 11 ones − 6 ones = 5 ones.
We have enough tens to remove 3 tens, so we do not need another trade.
Subtract the tens: 6 tens − 3 tens = 3 tens.
Subtract the hundreds: 5 hundreds − 2 hundreds = 3 hundreds.
Put the places together: 300 + 30 + 5 = 335. Check with addition: 335 + 236 = 571.
Show solution
We have 4 ones and need to remove 1 one, so no trade is needed in the ones place.
We have only 2 tens but need to remove 8 tens. Trade 1 hundred for 10 tens. That gives 5 hundreds and 12 tens; the ones are still 4 ones before subtraction.
Subtract the ones: 4 ones − 1 one = 3 ones.
Subtract the tens: 12 tens − 8 tens = 4 tens.
Subtract the hundreds: 5 hundreds − 3 hundreds = 2 hundreds.
Put the places together: 200 + 40 + 3 = 243. Check with addition: 243 + 381 = 624.
Show solution
We have 6 ones and need to remove 2 ones, so no trade is needed in the ones place.
We have only 0 tens but need to remove 5 tens. Trade 1 hundred for 10 tens. That gives 7 hundreds and 10 tens; the ones are still 6 ones before subtraction.
Subtract the ones: 6 ones − 2 ones = 4 ones.
Subtract the tens: 10 tens − 5 tens = 5 tens.
Subtract the hundreds: 7 hundreds − 3 hundreds = 4 hundreds.
Put the places together: 400 + 50 + 4 = 454. Check with addition: 454 + 352 = 806.
Show solution
We have only 2 ones but need to subtract 8 ones. Decompose 1 ten into 10 ones. Now we have 5 hundreds, 2 tens, and 12 ones.
We have only 2 tens but need to remove 7 tens. Trade 1 hundred for 10 tens. That gives 4 hundreds and 12 tens; the ones are still 12 ones before subtraction.
Subtract the ones: 12 ones − 8 ones = 4 ones.
Subtract the tens: 12 tens − 7 tens = 5 tens.
Subtract the hundreds: 4 hundreds − 1 hundred = 3 hundreds.
Put the places together: 300 + 50 + 4 = 354. Check with addition: 354 + 178 = 532.
Show solution
There are no tens available to trade, so decompose 1 hundred into 10 tens. We now have 4 hundreds, 10 tens, and 3 ones.
We have only 3 ones but need to subtract 8 ones. Decompose 1 ten into 10 ones. Now we have 4 hundreds, 9 tens, and 13 ones.
Subtract the ones: 13 ones − 8 ones = 5 ones.
We have enough tens to remove 7 tens, so we do not need another trade.
Subtract the tens: 9 tens − 7 tens = 2 tens.
Subtract the hundreds: 4 hundreds − 1 hundred = 3 hundreds.
Put the places together: 300 + 20 + 5 = 325. Check with addition: 325 + 178 = 503.
Show solution
There are no tens available to trade, so decompose 1 hundred into 10 tens. We now have 6 hundreds, 10 tens, and 0 ones.
We have only 0 ones but need to subtract 6 ones. Decompose 1 ten into 10 ones. Now we have 6 hundreds, 9 tens, and 10 ones.
Subtract the ones: 10 ones − 6 ones = 4 ones.
We have enough tens to remove 5 tens, so we do not need another trade.
Subtract the tens: 9 tens − 5 tens = 4 tens.
Subtract the hundreds: 6 hundreds − 2 hundreds = 4 hundreds.
Put the places together: 400 + 40 + 4 = 444. Check with addition: 444 + 256 = 700.
Show solution
There are no tens available to trade, so decompose 1 hundred into 10 tens. We now have 8 hundreds, 10 tens, and 0 ones.
We have only 0 ones but need to subtract 7 ones. Decompose 1 ten into 10 ones. Now we have 8 hundreds, 9 tens, and 10 ones.
Subtract the ones: 10 ones − 7 ones = 3 ones.
We have enough tens to remove 5 tens, so we do not need another trade.
Subtract the tens: 9 tens − 5 tens = 4 tens.
Subtract the hundreds: 8 hundreds − 4 hundreds = 4 hundreds.
Put the places together: 400 + 40 + 3 = 443. Check with addition: 443 + 457 = 900.
Show solution
We have only 4 ones but need to subtract 7 ones. Decompose 1 ten into 10 ones. Now we have 8 hundreds, 0 tens, and 14 ones.
We have only 0 tens but need to remove 5 tens. Trade 1 hundred for 10 tens. That gives 7 hundreds and 10 tens; the ones are still 14 ones before subtraction.
Subtract the ones: 14 ones − 7 ones = 7 ones.
Subtract the tens: 10 tens − 5 tens = 5 tens.
Subtract the hundreds: 7 hundreds − 2 hundreds = 5 hundreds.
Put the places together: 500 + 50 + 7 = 557. Check with addition: 557 + 257 = 814.
Show solution
We have only 2 ones but need to subtract 5 ones. Decompose 1 ten into 10 ones. Now we have 4 hundreds, 2 tens, and 12 ones.
We have only 2 tens but need to remove 8 tens. Trade 1 hundred for 10 tens. That gives 3 hundreds and 12 tens; the ones are still 12 ones before subtraction.
Subtract the ones: 12 ones − 5 ones = 7 ones.
Subtract the tens: 12 tens − 8 tens = 4 tens.
Subtract the hundreds: 3 hundreds − 1 hundred = 2 hundreds.
Put the places together: 200 + 40 + 7 = 247. Check with addition: 247 + 185 = 432.
Show solution
There are no tens available to trade, so decompose 1 hundred into 10 tens. We now have 5 hundreds, 10 tens, and 7 ones.
We have only 7 ones but need to subtract 9 ones. Decompose 1 ten into 10 ones. Now we have 5 hundreds, 9 tens, and 17 ones.
Subtract the ones: 17 ones − 9 ones = 8 ones.
We have enough tens to remove 8 tens, so we do not need another trade.
Subtract the tens: 9 tens − 8 tens = 1 ten.
Subtract the hundreds: 5 hundreds − 2 hundreds = 3 hundreds.
Put the places together: 300 + 10 + 8 = 318. Check with addition: 318 + 289 = 607.
Show solution
We have only 0 ones but need to subtract 4 ones. Decompose 1 ten into 10 ones. Now we have 7 hundreds, 1 ten, and 10 ones.
We have only 1 ten but need to remove 8 tens. Trade 1 hundred for 10 tens. That gives 6 hundreds and 11 tens; the ones are still 10 ones before subtraction.
Subtract the ones: 10 ones − 4 ones = 6 ones.
Subtract the tens: 11 tens − 8 tens = 3 tens.
Subtract the hundreds: 6 hundreds − 1 hundred = 5 hundreds.
Put the places together: 500 + 30 + 6 = 536. Check with addition: 536 + 184 = 720.
Show solution
We have only 0 ones but need to subtract 1 one. Decompose 1 ten into 10 ones. Now we have 9 hundreds, 4 tens, and 10 ones.
We have only 4 tens but need to remove 7 tens. Trade 1 hundred for 10 tens. That gives 8 hundreds and 14 tens; the ones are still 10 ones before subtraction.
Subtract the ones: 10 ones − 1 one = 9 ones.
Subtract the tens: 14 tens − 7 tens = 7 tens.
Subtract the hundreds: 8 hundreds − 4 hundreds = 4 hundreds.
Put the places together: 400 + 70 + 9 = 479. Check with addition: 479 + 471 = 950.
Show solution
We have only 1 one but need to subtract 5 ones. Decompose 1 ten into 10 ones. Now we have 6 hundreds, 7 tens, and 11 ones.
We have only 7 tens but need to remove 9 tens. Trade 1 hundred for 10 tens. That gives 5 hundreds and 17 tens; the ones are still 11 ones before subtraction.
Subtract the ones: 11 ones − 5 ones = 6 ones.
Subtract the tens: 17 tens − 9 tens = 8 tens.
Subtract the hundreds: 5 hundreds − 3 hundreds = 2 hundreds.
Put the places together: 200 + 80 + 6 = 286. Check with addition: 286 + 395 = 681.
Show solution
We have only 1 one but need to subtract 8 ones. Decompose 1 ten into 10 ones. Now we have 7 hundreds, 7 tens, and 11 ones.
We have only 7 tens but need to remove 9 tens. Trade 1 hundred for 10 tens. That gives 6 hundreds and 17 tens; the ones are still 11 ones before subtraction.
Subtract the ones: 11 ones − 8 ones = 3 ones.
Subtract the tens: 17 tens − 9 tens = 8 tens.
Subtract the hundreds: 6 hundreds − 4 hundreds = 2 hundreds.
Put the places together: 200 + 80 + 3 = 283. Check with addition: 283 + 498 = 781.
Show solution
We have only 2 ones but need to subtract 8 ones. Decompose 1 ten into 10 ones. Now we have 4 hundreds, 5 tens, and 12 ones.
We have only 5 tens but need to remove 7 tens. Trade 1 hundred for 10 tens. That gives 3 hundreds and 15 tens; the ones are still 12 ones before subtraction.
Subtract the ones: 12 ones − 8 ones = 4 ones.
Subtract the tens: 15 tens − 7 tens = 8 tens.
Subtract the hundreds: 3 hundreds − 1 hundred = 2 hundreds.
Put the places together: 200 + 80 + 4 = 284. Check with addition: 284 + 178 = 462.
Show solution
We have only 4 ones but need to subtract 8 ones. Decompose 1 ten into 10 ones. Now we have 7 hundreds, 4 tens, and 14 ones.
We have only 4 tens but need to remove 6 tens. Trade 1 hundred for 10 tens. That gives 6 hundreds and 14 tens; the ones are still 14 ones before subtraction.
Subtract the ones: 14 ones − 8 ones = 6 ones.
Subtract the tens: 14 tens − 6 tens = 8 tens.
Subtract the hundreds: 6 hundreds − 2 hundreds = 4 hundreds.
Put the places together: 400 + 80 + 6 = 486. Check with addition: 486 + 268 = 754.
Show solution
We have only 0 ones but need to subtract 7 ones. Decompose 1 ten into 10 ones. Now we have 4 hundreds, 7 tens, and 10 ones.
Subtract the ones: 10 ones − 7 ones = 3 ones.
We have enough tens to remove 5 tens, so we do not need another trade.
Subtract the tens: 7 tens − 5 tens = 2 tens.
Subtract the hundreds: 4 hundreds − 2 hundreds = 2 hundreds.
Put the places together: 200 + 20 + 3 = 223. Check with addition: 223 + 257 = 480.
Show solution
There are no tens available to trade, so decompose 1 hundred into 10 tens. We now have 2 hundreds, 10 tens, and 4 ones.
We have only 4 ones but need to subtract 7 ones. Decompose 1 ten into 10 ones. Now we have 2 hundreds, 9 tens, and 14 ones.
Subtract the ones: 14 ones − 7 ones = 7 ones.
We have enough tens to remove 2 tens, so we do not need another trade.
Subtract the tens: 9 tens − 2 tens = 7 tens.
Subtract the hundreds: 2 hundreds − 1 hundred = 1 hundred.
Put the places together: 100 + 70 + 7 = 177. Check with addition: 177 + 127 = 304.
Show solution
We have only 0 ones but need to subtract 2 ones. Decompose 1 ten into 10 ones. Now we have 8 hundreds, 2 tens, and 10 ones.
We have only 2 tens but need to remove 7 tens. Trade 1 hundred for 10 tens. That gives 7 hundreds and 12 tens; the ones are still 10 ones before subtraction.
Subtract the ones: 10 ones − 2 ones = 8 ones.
Subtract the tens: 12 tens − 7 tens = 5 tens.
Subtract the hundreds: 7 hundreds − 4 hundreds = 3 hundreds.
Put the places together: 300 + 50 + 8 = 358. Check with addition: 358 + 472 = 830.
Show solution
The student subtracted the smaller digit from the larger digit in each column without considering the place-value units. That does not represent subtracting 286 from 643.
We have only 3 ones but need to subtract 6 ones. Decompose 1 ten into 10 ones. Now we have 6 hundreds, 3 tens, and 13 ones.
We have only 3 tens but need to remove 8 tens. Trade 1 hundred for 10 tens. That gives 5 hundreds and 13 tens; the ones are still 13 ones before subtraction.
Subtract the ones: 13 ones − 6 ones = 7 ones.
Subtract the tens: 13 tens − 8 tens = 5 tens.
Subtract the hundreds: 5 hundreds − 2 hundreds = 3 hundreds.
Put the places together: 300 + 50 + 7 = 357. Check with addition: 357 + 286 = 643.
Show solution
To find the unknown subtrahend, subtract the known difference from the starting amount: 704 − 368 = 336. So the missing amount taken away is 336. Verify it in the original equation: 704 − 336 = 368.
There are no tens available to trade, so decompose 1 hundred into 10 tens. We now have 6 hundreds, 10 tens, and 4 ones.
We have only 4 ones but need to subtract 6 ones. Decompose 1 ten into 10 ones. Now we have 6 hundreds, 9 tens, and 14 ones.
Subtract the ones: 14 ones − 6 ones = 8 ones.
We have enough tens to remove 3 tens, so we do not need another trade.
Subtract the tens: 9 tens − 3 tens = 6 tens.
Subtract the hundreds: 6 hundreds − 3 hundreds = 3 hundreds.
Put the places together: 300 + 60 + 8 = 368. Check with addition: 368 + 336 = 704.
Show solution
To find the starting number, add the subtrahend and the difference: 268 + 455 = 723. Check the subtraction by place value:
We have only 3 ones but need to subtract 8 ones. Decompose 1 ten into 10 ones. Now we have 7 hundreds, 1 ten, and 13 ones.
We have only 1 ten but need to remove 6 tens. Trade 1 hundred for 10 tens. That gives 6 hundreds and 11 tens; the ones are still 13 ones before subtraction.
Subtract the ones: 13 ones − 8 ones = 5 ones.
Subtract the tens: 11 tens − 6 tens = 5 tens.
Subtract the hundreds: 6 hundreds − 2 hundreds = 4 hundreds.
Put the places together: 400 + 50 + 5 = 455. Check with addition: 455 + 268 = 723.
Show solution
The remaining books equal the starting books minus the books loaned out. The subtraction equation is 813 − 457.
We have only 3 ones but need to subtract 7 ones. Decompose 1 ten into 10 ones. Now we have 8 hundreds, 0 tens, and 13 ones.
We have only 0 tens but need to remove 5 tens. Trade 1 hundred for 10 tens. That gives 7 hundreds and 10 tens; the ones are still 13 ones before subtraction.
Subtract the ones: 13 ones − 7 ones = 6 ones.
Subtract the tens: 10 tens − 5 tens = 5 tens.
Subtract the hundreds: 7 hundreds − 4 hundreds = 3 hundreds.
Put the places together: 300 + 50 + 6 = 356. Check with addition: 356 + 457 = 813.
Answer: 356 books remain in the school library.
Show solution
For this extension, represent 1,000 as 10 hundreds. It is still 1,000; we have not added any value. Trade one hundred into ten tens, and one of those tens into ten ones before removing 486.
There are no tens available to trade, so decompose 1 hundred into 10 tens. We now have 9 hundreds, 10 tens, and 0 ones.
We have only 0 ones but need to subtract 6 ones. Decompose 1 ten into 10 ones. Now we have 9 hundreds, 9 tens, and 10 ones.
Subtract the ones: 10 ones − 6 ones = 4 ones.
We have enough tens to remove 8 tens, so we do not need another trade.
Subtract the tens: 9 tens − 8 tens = 1 ten.
Subtract the hundreds: 9 hundreds − 4 hundreds = 5 hundreds.
Put the places together: 500 + 10 + 4 = 514. Check with addition: 514 + 486 = 1000.
๐ฎ Independent Practice — 30 Problems
Directions: Use a base-ten drawing, place-value chart, or clear written trade to solve each problem. Explain every exchange before subtracting. For the last six questions, show your reasoning. Keep the answer key closed until you have finished.
- 453 − 127 = ____.
- 684 − 258 = ____.
- 792 − 465 = ____.
- 563 − 248 = ____.
- 841 − 326 = ____.
- 926 − 419 = ____.
- 751 − 432 = ____.
- 975 − 648 = ____.
- 632 − 217 = ____.
- 584 − 376 = ____. Show each trade.
- 625 − 381 = ____.
- 713 − 462 = ____.
- 804 − 520 = ____.
- 905 − 641 = ____.
- 472 − 291 = ____.
- 643 − 382 = ____. Show each trade.
- 524 − 178 = ____.
- 731 − 256 = ____.
- 603 − 286 = ____.
- 800 − 347 = ____. Show each trade.
- 902 − 568 = ____.
- 650 − 284 = ____.
- 370 − 184 = ____.
- 921 − 574 = ____. Show each trade.
- Complete 704 − □ = 459. Find the missing subtrahend and check.
- Complete □ − 386 = 257. Find the starting number and check.
- A school had 824 admission tickets. It gave out 469 tickets. How many tickets remained?
- Maya started with 701 beads and used 258 beads for crafts. How many beads does she have left?
- A student writes 742 − 385 = 443 by subtracting the smaller digit from the larger in each column. Explain the error and correct the difference.
- Extension challenge: 1,000 − 678 = ____. Draw or explain the trades through the zeros.
๐ Lesson Test — 15 Questions
Directions: Work independently. For explanations and word problems, include a model or the sequence of trades. Check your work after completing the entire test.
- 572 − 346 = ____.
- 863 − 527 = ____.
- 941 − 618 = ____. Show the needed trade(s).
- 657 − 429 = ____.
- 786 − 359 = ____.
- 615 − 381 = ____. Show the needed trade(s).
- 704 − 230 = ____.
- 502 − 170 = ____.
- 842 − 578 = ____. Show the needed trade(s).
- 930 − 584 = ____.
- 601 − 287 = ____. Show the needed trade(s).
- 1,000 − 759 = ____. Show the needed trade(s).
- Complete □ − 487 = 256. Find the missing starting number and check.
- A school had 803 crayons. Students used 457 crayons for projects. How many crayons remained?
- A student says 621 − 385 = 364 because they subtract the smaller digit from the larger digit in each column. Explain the error and give the correct difference.
๐ Complete Answer Key
Show practice answer key
Practice Answers
Note for teachers and families: Accept other correct place-value strategies. The reasoning should show each unit trade accurately and verify the final result.
Show test answer key
Test Answers
Note: For reasoning questions, another valid model or written sequence of trades is acceptable when the subtraction and explanation are correct.
๐ก Common Mistakes to Watch For
When 1 ten becomes 10 ones, reduce the tens by 1 and increase the ones by 10. The total stays equal until subtraction.
After exchanging a ten for ones, there may still be too few tens to subtract. Check the tens place again.
For 503 − 178, first make 10 tens from 1 hundred; only then can 1 ten become 10 ones.
In 532 − 178, you cannot turn the ones step into 8 − 2. Decompose to make enough ones.
12 tens − 7 tens = 5 tens, meaning 50. This is not the same equation as 12 − 7 = 50.
To check 532 − 178 = 354, add the difference and subtrahend: 354 + 178 = 532.
๐ Related Lessons
๐จ️ Printable Companion
A future Subtract Within 1,000: Decompose Tens & Hundreds Practice Pack can include exchange mats, base-ten drawings, hundreds–tens–ones charts, missing-number problems, word problems, backward number lines, and complete answer keys.
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