Written Column Subtraction

Column subtraction lines up numbers by place value and subtracts from right to left, borrowing from the next column when needed.

What Is Column Subtraction?

Column subtraction is the written method for finding the difference between two large numbers. You line them up by place value and subtract column by column from right to left. When a digit on top is smaller than the digit below, you borrow from the next column to the left.

Borrowing Rule

\text{if top digit} < \text{bottom digit} \Rightarrow \text{borrow 1 from the next column}

Borrowing adds 10 to the current column and reduces the left neighbour by 1.

Step-by-Step Method

1
Write the numbers in columns
Place the larger number on top and the smaller number below, aligning by place value.
2
Start at the ones column
Subtract the bottom digit from the top digit. If the top digit is smaller, borrow 1 from the tens column: add 10 to the top digit and reduce the tens digit above by 1.
3
Move to the tens column
Subtract, remembering that the tens digit may already have been reduced by a borrow. Borrow again from hundreds if needed.
4
Continue left
Repeat until all columns are done.
5
Check your answer
Add your answer to the number you subtracted. The result should equal the top number.

Worked Example

Calculate 83 042 − 27 658

  83 042
- 27 658
--------

Ones: 2 < 8 → borrow from tens. Tens digit (4) becomes 3; ones become 12. 12 − 8 = 4

Tens: 3 < 5 → borrow from hundreds. Hundreds digit (0) becomes −1, but 0 needs to borrow too — borrow from thousands. Hundreds become 9 (10 − 1), thousands digit (3) becomes 2; tens become 13. 13 − 5 = 8

Hundreds: 9 − 6 = 3

Thousands: 2 < 7 → borrow from ten-thousands. Ten-thousands digit (8) becomes 7; thousands become 12. 12 − 7 = 5

Ten-thousands: 7 − 2 = 5

Result: 55 384

Check: 55 384 + 27 658 = 83 042 ✓

Tip After you finish, always verify by adding your answer to the subtracted number. If you get the original number back, you know your answer is correct.

Real-World Applications

Real world: calculating change

A customer pays 50 € for a book that costs 33 €. How much change does she receive?

  50
- 33
----

Ones: 0 < 3 → borrow; 10 − 3 = 7. Tens: 4 − 3 = 1.

Change: 17 €

Real world: difference in distances

City A is 74 320 m from the capital; City B is 48 750 m. How much closer to the capital is City B?

  74 320
- 48 750
--------

Ones: 0 − 0 = 0

Tens: 2 < 5 → borrow; 12 − 5 = 7, hundreds reduced.

Hundreds: 2 (after borrow) < 7 → borrow; 12 − 7 = 5, thousands reduced.

Thousands: 3 (after borrow) < 8 → borrow; 13 − 8 = 5, ten-thousands reduced.

Ten-thousands: 6 − 4 = 2

Difference: 25 570 m

Real world: remaining budget

A school has a budget of 12 500 € for the year. After buying equipment for 7 840 €, how much remains?

  12 500
-  7 840
--------

Ones: 0 − 0 = 0

Tens: 0 < 4 → borrow; 10 − 4 = 6, hundreds reduced.

Hundreds: 4 (after borrow) < 8 → borrow; 14 − 8 = 6, thousands reduced.

Thousands: 1 (after borrow) < 7 → borrow; 11 − 7 = 4, ten-thousands reduced.

Ten-thousands: 1 − 0 = 1 (but 1 was borrowed, so 0... wait, 12 500 has ten-thousands digit 1 and thousands digit 2).

Corrected: 12 500 − 7 840. Ten-thousands: 1, thousands: 2. After borrowing at thousands: 11 − 7 = 4, ten-thousands: 1 − 0 = 1.

Remaining budget: 4 660 €

Common Mistakes

Forgetting to reduce the column you borrowed from
Trying to borrow from a zero (you must first borrow from further left to give the zero a value)
Subtracting top from bottom instead of bottom from top when borrowing was forgotten

Practice Problems

Calculate 62 400 − 38 756.

Show answer

Ones: 0<6, borrow → 10−6=4. Tens: 9−5=4 (tens was 9 after previous borrow chain). Hundreds: 3−7... borrow chain continues. Working through: 62 400 − 38 756 = 23 644.

A train route is 156 km long. A passenger has already travelled 89 km. How many kilometres remain?

Show answer

156 − 89 = 67 km remaining.