A three-digit number has a hundreds place, a tens place and a ones place, so 256 is 200 + 50 + 6.
When you add and a place adds up to 10 or more, you carry 1 to the next place; when you subtract and a place is too small, you borrow 10 from the next place.
That is why three-digit addition and subtraction are written in columns with the places lined up and worked from the ones place.
Choose addition or subtraction and change the two numbers to see where carrying and borrowing happen in the place-value blocks and in the column method.
What Is Place Value?
When you add or subtract two numbers, you only combine digits in the same place: hundreds with hundreds, tens with tens and ones with ones, so that their sizes never get mixed. If the ones add up to 10 or more, those 10 ones are the same as 1 ten, so they move to the tens place; if the ones are too few to subtract, 1 ten is traded for 10 ones and brought over. These trades always move from right to left, so the work also starts at the ones, then goes to the tens and finally the hundreds. Writing the numbers in columns with the places lined up makes it easy to keep that order.
🧱 Digits Have Places
①In 256, 2 is in the hundreds place, 5 is in the tens place and 6 is in the ones place
②The 2 means 200, the 5 means 50 and the 6 means 6
③So 256 = 200 + 50 + 6, and the blocks are stacked place by place
Working with Blocks
256
187
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Hundreds, tens and ones blocks with arrows showing carrying and borrowing
Pick addition or subtraction with the buttons and set two numbers from 100 to 499 with the sliders; the equation and its answer appear at the top of the first picture. Below it are three columns for hundreds, tens and ones: in each column the upper blocks are the first number, the lower blocks are the second number, and the digit under the line is that place in the answer. In addition, a red dashed arrow runs from ones to tens when the ones add up to 10 or more, and from tens to hundreds when the tens plus the carried 1 reach 10 or more. In subtraction, the arrow comes down from the place that lends to the place that cannot subtract. Subtraction always takes the smaller number from the larger, so if the first number is smaller the picture swaps them. The second picture writes the same problem in columns; small red digits above show the carried 1s in addition and the new digits after borrowing in subtraction.
💡 Carrying and Borrowing
①Addition: if a place adds up to 10 or more, carry 1 to the next place to the left
②Subtraction: if a place is too small to subtract, borrow 1 from the next place to the left and use it as 10
③Carrying and borrowing can happen between the ones and tens, and also between the tens and hundreds
The Column Method
It is tempting to start with the biggest place, the hundreds, but then a 1 coming up from a lower place forces you to go back and change a digit. Working 256 + 187 from the hundreds, you would write 3 first, but the tens are 5 + 8 plus the 1 carried from the ones, which makes 14, so another 1 goes up to the hundreds and the answer changes to 443. Starting at the ones, you jot each carried 1 above the next place and simply add it in, so nothing ever needs fixing. Subtraction works the same way: if the ones are short, you must borrow from the tens first, or the tens will come out wrong.
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The problem written in columns with the carrying and borrowing marks
✏️ Steps of the Column Method
①Write the numbers in columns with the places lined up
②Start with the ones place
③Write any carried or borrowed digit small above its place
④Finish with the tens, then the hundreds
⑤Check addition by subtracting one addend from the sum, and subtraction by adding the difference to the number subtracted
Summary
Order for Three-Digit Problems
Ones → Tens → Hundreds
Line up the places and work from the ones place
Carrying
Sum in a place ≥ 10 → carry 1 to the next place
A 1 carried to the tens is 10 ones, and a 1 carried to the hundreds is 10 tens
Borrowing
Can't subtract → take 1 from the next place as 10
The lending place goes down by 1 and the borrowing place goes up by 10
🎯 Summary
①Add and subtract three-digit numbers starting from the ones place
②If a place adds up to 10 or more, carry 1 to the next place
③If a place is too small to subtract, borrow 10 from the next place
④Writing in lined-up columns cuts down on mistakes
⑤Check addition with sum − addend and subtraction with difference + number subtracted
Examples and Unit Test
Example 1
Add 256 + 187 using the column method.
1
Ones: 6 + 7 = 13, so write 3 and carry 1 to the tens.
2
Tens: 5 + 8 + 1 = 14, so write 4 and carry 1 to the hundreds. Hundreds: 2 + 1 + 1 = 4. The answer is 443.
▸ 443
This problem carries in both the ones and the tens; the two red dashed arrows in the picture mark them.
Example 2
Subtract 423 − 158 using the column method.
1
Ones: you can't take 8 from 3, so borrow 10 from the tens: 13 − 8 = 5. The tens digit drops from 2 to 1.
2
Tens: you can't take 5 from 1, so borrow 10 from the hundreds: 11 − 5 = 6. Hundreds: 4 − 1 − 1 = 2. The answer is 265.
▸ 265
Write the new digit above each place that lends, so you remember it has gone down by 1.
Unit Test
What is 345 + 276?
①511
②611
③621
④521
▸ ③ 621
1
Ones: 5 + 6 = 11, so write 1 and carry 1. Tens: 4 + 7 + 1 = 12, so write 2 and carry 1.