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Adding and subtracting two- and three-digit numbers

Numbers · level 2, lesson 2 of 7

A question to start

Max has to work out 52 − 27. He thinks: "In the ones, 7 take away 2 is 5. In the tens, 5 take away 2 is 3. So the answer is 35."

Max is wrong. Can you see why? Here is a clue: 27 is a bit less than 30, and 52 is a bit more than 50. So the answer should be a bit more than 50 − 30 = 20. It cannot be 35.

Max made the most common mistake in all of subtraction. By the end of this lesson you will know what it is, how to fix it, and why the fix works. (The right answer is 25.)

The idea

Adding tens is easy

From level 1 you know that going one row down the hundred chart adds ten. The ones digit stays the same and the tens digit goes up by one. So adding tens is like counting by tens:

  • 47 + 10 = 57.
  • 47 + 30: count on three tens, "57, 67, 77". So 47 + 30 = 77.
  • 125 + 40: count on four tens, "135, 145, 155, 165". So 125 + 40 = 165.

Only the tens digit changes, as long as it does not go past 9.

Way 1: split into tens and ones

To add 47 + 25, split each number into its tens and its ones:

  • 47 is 40 and 7. 25 is 20 and 5.
  • Add the tens: 40 + 20 = 60.
  • Add the ones: 7 + 5 = 12.
  • Put them together: 60 + 12 = 72.

Why is this allowed? Adding just means putting everything together. It does not matter in what order you put the pieces together, so you may add all the tens first and all the ones second. You still get every piece exactly once.

Way 2: jump along a number line

Start at the first number and make jumps: first the tens, then the ones. A quick sketch of a number line, with only the numbers you need, is called an empty number line.

47677072 + 20+ 3+ 2

From 47, jump 20 to 67. Now there are 5 ones to add. Use "make ten" from level 1: 67 + 3 = 70, then 70 + 2 = 72. So 47 + 25 = 72. This is how most people add in their heads.

Way 3: in columns, with regrouping

For bigger numbers, we write one number under the other, with ones under ones and tens under tens. Then we add each column, starting from the ones on the right.

Here is 47 + 25 with blocks. 4 rods and 2 rods make 6 tens. 7 cubes and 5 cubes make 12 ones.

10 ones make 1 new ten

But 12 ones are too many to write in the ones place, because each place holds just one digit. So we bundle 10 of the ones into a new ten. That leaves 2 ones, and now there are 6 + 1 = 7 tens. The answer is 72.

On paper, it looks like this:

    1
    4 7
  + 2 5
  -----
    7 2
  1. Ones: 7 + 5 = 12. That is 1 ten and 2 ones. Write the 2 in the ones place. Write the 1 (it is one ten) at the top of the tens column, so you do not forget it.
  2. Tens: 1 + 4 + 2 = 7 tens. Write 7.

Moving a new ten into the tens column is called regrouping (people also say carrying). It is not a magic rule: it is bundling ten ones into a ten, just like 9 + 1 on the hundred chart moves to the next row.

Three-digit numbers

The same idea works for hundreds. If the tens add up to 10 or more, ten of them are bundled into a new hundred.

    1 1
    3 5 8
  + 2 7 6
  -------
    6 3 4
  1. Ones: 8 + 6 = 14. Write 4, carry 1 ten.
  2. Tens: 1 + 5 + 7 = 13 tens. 13 tens is 1 hundred and 3 tens. Write 3, carry 1 hundred.
  3. Hundreds: 1 + 3 + 2 = 6 hundreds. Write 6.

So 358 + 276 = 634. A quick check by rounding: 358 is about 360 and 276 is about 280; 360 + 280 = 640. The answer 634 is close to 640, so it is sensible.

Subtracting: when there are not enough ones

Now back to Max and 52 − 27. Make 52 with blocks: 5 rods and 2 cubes. We must take away 2 tens and 7 ones.

The problem is the ones. There are only 2 cubes, and we need to take away 7. We cannot. Max's mistake was to take the 2 away from the 7 instead, the wrong way round. That is not the question.

The fix: break one ten into 10 ones. Now there are 4 rods and 12 cubes. It is still 52 (40 + 12 = 52), just packed differently, like you saw in lesson 1.

4 tens and 12 ones; take away 2 tens and 7 ones

Now take away: 12 ones − 7 ones = 5 ones. 4 tens − 2 tens = 2 tens. What is left is 2 tens and 5 ones: 25.

On paper:

    4 12
    5  2
  − 2  7
  ------
    2  5
  1. Ones: 2 − 7 cannot be done. Break a ten: the 5 tens become 4, and the 2 ones become 12. Now 12 − 7 = 5.
  2. Tens: 4 − 2 = 2.

This is also called regrouping (people also say borrowing, though nothing is ever given back!). It works because breaking a ten into ten ones does not change the number.

Breaking a hundred, even across a zero

What about 403 − 158? The ones need 8 but there are only 3. We want to break a ten, but there are 0 tens! So first break a hundred into 10 tens, then break one of those tens into 10 ones.

  • 403 is 4 hundreds, 0 tens, 3 ones.
  • Break a hundred: 3 hundreds, 10 tens, 3 ones.
  • Break a ten: 3 hundreds, 9 tens, 13 ones.
  • Now subtract: 13 − 8 = 5 ones, 9 − 5 = 4 tens, 3 − 1 = 2 hundreds.

So 403 − 158 = 245.

Another way to subtract: count up

Subtraction can also mean "how far from one number to the other?" (level 1, lesson 4). To find 52 − 27, start at 27 and jump up to 52 in friendly steps:

  • 27 → 30 is 3.
  • 30 → 50 is 20.
  • 50 → 52 is 2.

The jumps add up to 3 + 20 + 2 = 25. Counting up is often the easiest way in your head, and it needs no regrouping. Try it for the opening puzzle: 307 → 310 is 3, and 310 → 370 is 60, so 370 − 307 = 63.

Checking with addition

Taking away and adding undo each other. So if 52 − 27 = 25, then 25 + 27 must give back 52. Check: 25 + 27 is 40 + 12 = 52. ✓ If you get back the number you started with, your subtraction is right. If not, look for the mistake.

Worked examples

1. Splitting. 36 + 48. Tens: 30 + 40 = 70. Ones: 6 + 8 = 14. Together: 70 + 14 = 84.

2. Columns with one regrouping. 165 + 227.

Ones: 5 + 7 = 12; write 2, carry 1 ten. Tens: 1 + 6 + 2 = 9; write 9. Hundreds: 1 + 2 = 3. Answer 392. Check by rounding: about 170 + 230 = 400. Close. ✓

3. Subtracting with regrouping. 81 − 46.

Ones: 1 − 6 cannot be done. Break a ten: 7 tens and 11 ones. 11 − 6 = 5. Tens: 7 − 4 = 3. Answer 35. Check: 35 + 46 = 70 + 11 = 81. ✓

4. Counting up. A rover is 600 metres from its target. It has already driven 380 metres. How far is left?

That is 600 − 380. Count up from 380: to 400 is 20, to 600 is 200. So 220 metres are left. Check: 380 + 220 = 600. ✓

5. Across a zero. 500 − 137.

500 is 5 hundreds, 0 tens, 0 ones. Break a hundred into 10 tens, and one ten into 10 ones: 4 hundreds, 9 tens, 10 ones. Then 10 − 7 = 3, 9 − 3 = 6, 4 − 1 = 3. Answer 363. Check: 363 + 137 = 500. ✓

Try it

  1. 54 + 30 and 54 + 35.
  2. 68 + 27, by splitting into tens and ones.
  3. 249 + 386, in columns.
  4. 73 − 38, by breaking a ten.
  5. 90 − 47, by counting up.
  6. 702 − 465, in columns.
  7. A bookshelf has 128 books. 59 more arrive. Then 74 are given away. How many are there now?
  8. Max's answer to 52 − 27 was 35. Show how checking with addition would have caught his mistake.
Answers
  1. 54 + 30: count on 3 tens, 64, 74, 84: 84. 54 + 35: 54 + 30 = 84, then + 5 = 89.
  2. Tens 60 + 20 = 80; ones 8 + 7 = 15; 80 + 15 = 95.
  3. Ones 9 + 6 = 15: write 5, carry 1. Tens 1 + 4 + 8 = 13: write 3, carry 1. Hundreds 1 + 2 + 3 = 6. 635. Check by rounding: 250 + 390 = 640, close to 635. ✓
  4. Ones 3 − 8 cannot be done. Break a ten: 6 tens and 13 ones. 13 − 8 = 5; 6 − 3 = 3. 35. Check: 35 + 38 = 60 + 13 = 73. ✓
  5. 47 → 50 is 3; 50 → 90 is 40. 3 + 40 = 43. Check: 43 + 47 = 90. ✓
  6. Ones 2 − 5 cannot be done, and there are 0 tens. Break a hundred: 6 hundreds, 10 tens, 2 ones. Break a ten: 6 hundreds, 9 tens, 12 ones. 12 − 5 = 7; 9 − 6 = 3; 6 − 4 = 2. 237. Check: 237 + 465 = 702. ✓
  7. 128 + 59 = 187 (ones 8 + 9 = 17, carry 1; tens 1 + 2 + 5 = 8; hundreds 1). Then 187 − 74 = 113 (7 − 4 = 3 ones, 8 − 7 = 1 ten, 1 hundred). Check: 113 + 74 = 187. ✓
  8. If 52 − 27 were 35, then 35 + 27 would be 52. But 35 + 27 = 50 + 12 = 62, not 52. So 35 is wrong. The right answer, 25, checks: 25 + 27 = 52. ✓

Common mistake

Taking the smaller digit from the bigger one. In 52 − 27, the ones column says 2 − 7. Swapping it to 7 − 2 gives the wrong answer every time. The top number is what you have; you can only take from it. If there are not enough, break a ten.

Forgetting the carried ten. In 47 + 25, if you write the 2 but forget the 1 ten, you get 62 instead of 72. Always write the carried 1 at the top of the next column, and add it in.

Where it leads