Concept 4 of 7
The Column Stack: Carrying and Borrowing
Today's big idea
Borrow from the nearest room with something, and every empty room it passes keeps a 9.
Carrying pushes a spare ten left, and borrowing pulls one back the other way.
How it works
👀 What's going on?
A game scoreboard sits at 21,629. One big round adds 18,376 points and it climbs to 40,005. Then a penalty takes those same 18,376 points straight back off, so it drops to 21,629 again.
That is one addition and one subtraction with the same numbers, and the way to do both is the same: stack the numbers so the digits line up in columns, then work one column at a time from the right. Two things can trip you up. When you add and a column comes to more than 9, the spare ten jumps to the next column, which we call a carry. When you subtract and the top digit is too small to take the bottom one away, the column has to borrow from its neighbour. And in 40,005 every neighbour is a 0: the ones need to borrow, but next door is 0, and so is the room after that. So where does the borrow come from?
💡 Here's the trick
Look at the picture above. It runs one sum both ways. Adding 21,629 + 18,376, only the ones column spills on its own. Every column to its left is sitting on 9, so each tips over the moment the carry reaches it. That cascade is exactly what leaves the answer full of zeros. Taking the 18,376 back off, one shortage in the ones walks left along those same zeros, and every empty room it crosses keeps a 9.
🪄 How to do it
- 1.Stack the numbers so the columns line up, the way the picture above lines them up.
- 2.Work right to left, one column at a time. Adding, a column over 9 carries the spare ten left. Subtracting, a column that is too small borrows from the left.
- 3.A carry joins the next column along. A borrow takes 1 from the nearest room that has something, and every empty room it crosses keeps a 9.
✍️ Try it
First add 36,478 + 27,865. Then work out 60,003 − 24,517.
The sum carries in every column: 13, then 14, then 13, then 14, giving 64,343. For the difference, the ones need 3 take away 7, so they ask left. The tens, hundreds and thousands are all empty, so the borrow comes from the 6. The 6 becomes 5, those three rooms each keep a 9, and column by column that leaves 35,486.
⚠️ Watch out
You might be thinking each empty room turns into a 10. It does not. A room takes 10 from its left, then passes 10 on to its right, so it is left holding 9. Write 10 in the middle rooms and your answer comes out far too big.
🌟 Why this matters
Price tags, odometers and bank balances love round numbers, and round numbers are full of zeros. Taking 13,450 off a 20,000 budget is exactly this sum. Stacking is also what you fall back on when the numbers get too big to juggle in your head.
Exam tip: Zeros in the top number are the easiest place in a subtraction to drop a mark.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“Work out 50,004 − 27,158 with the column method.”
- 1
Stack and start at the right
Ones: 4 − 8. Too small. The ones column has to borrow.
- 2
Find a room that can lend
Tens is 0. Hundreds is 0. Thousands is 0. The nearest room holding anything is the ten-thousands 5.
- 3
Send the borrow right
The 5 drops to 4, and each empty room it crosses keeps a 9: 4 9 9 9 14
- 4
Now every column copes
14−8=6, 9−5=4, 9−1=8, 9−7=2, 4−2=2.
- 5
Read it off ✓
50,004 − 27,158 = 22,846
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
A column that cannot cope has to borrow. Find the room that can lend, then watch what the empty rooms in between keep.
PUZZLE 1 OF 2
The ones column needs 5 − 6, which it cannot do. Tap the nearest room that actually has something to lend.
Quick check
✅ VERIFICATION
In 70,004 − 25,318, what does the hundreds room become once the borrow has travelled through it?
⚡ Finish the activity above first to answer.
Keep going!
Up next: The Missing Piece
Find a hidden number by asking whether the start or a part is missing.