Money
A £27.68 purchase is paid for with £50. Written as 50.00 − 27.68, the exchange passes through the tenths and ones, giving £22.32 change.
When zero stands in the way
Before this page: setting a subtraction out in columns and exchanging one unit from the column on the left, both from column subtraction.
3,000 − 847 looks harmless until you start it. The ones need 7 and hold none, so they ask the tens for a ten. The tens hold none either. Neither do the hundreds.
1,000s
100s
10s
1s
3
0
0
0
−
0
8
4
7
?
Three columns in a row have nothing to give, and yet the number is three thousand. The value has not gone anywhere. It is all held in one place, in a unit far too large to spend in the ones column, and the job is to get it down there.
One thousand, broken up
A thousand comes apart without anything being lost:
1000 = 900 + 90 + 10
3000 = 2000 + 900 + 90 + 10
That second line is 3,000 written so that every column has something in it. The exchange does not leap over the empty columns; it stops in each one on the way. One thousand becomes ten hundreds, nine stay in the hundreds and one moves on. That hundred becomes ten tens, nine stay and one moves on. That ten becomes ten ones, and the ones finally have something to subtract from.
900 + 90 + 10 = 1000
In the columns
Written down, the 3, 0, 0, 0 becomes 2, 9, 9 and 10. A run of 9s written in above the top line is the mark of an exchange that had to travel. Here there is only one, taken from the thousands and spent in the ones.
Example: calculate 3,000 − 847
3,000 − 847
Align equal place values
Begin with equal place values directly beneath one another. The ones sit under the ones, the tens under the tens, and so on.
3,000 − 847 = 2,153
The check still works, and it is worth making here more than anywhere: 2,153 + 847 = 3,000.
Follow an exchange across zeroes
The second number is the larger one, so the answer falls below zero. Lower it, or read adding and subtracting negatives.
Align equal place values
Begin with equal place values directly beneath one another. The ones sit under the ones, the tens under the tens, and so on.
Zeros in a decimal
Padding a short decimal creates zeros of its own, and they behave exactly like the ones in 3,000. In 4.03 the first exchange is ordinary and the second has to travel.
Example: calculate 4.03 − 1.276
4.03 − 1.276
Align equal place values
Begin with the decimal points directly beneath one another. Write 4.03 as 4.030 so every occupied place has a digit to work with.
4.030 − 1.276 = 2.754
A whole number against a three-place decimal starts every decimal column empty, so the exchange travels the full width of the decimal.
Example: calculate 12 − 3.475
12 − 3.475
Align equal place values
Begin with the decimal points directly beneath one another. Write 12 as 12.000 so every occupied place has a digit to work with.
12.000 − 3.475 = 8.525
Exchanging across zeroes in context
Money and measures often create zero-filled columns when a whole quantity is written to match a decimal. The unit changes what the answer means, but it does not change the exchange.
Measure
A 1.276 m piece is cut from a 4.03 m length. Writing 4.03 as 4.030 exposes the thousandths column, and 4.030 − 1.276 = 2.754 m.
Common mistakes
The method is the one you already know. What goes wrong here is the bookkeeping of the run.
- Making the donor a 9 as wellOnly the columns the exchange passes through become 9. The column it finally reaches drops by one, so the 3 thousands become 2 — not 9.
- Stopping the run too earlyEvery empty column between the request and the donor becomes 9. Leaving one of them as 0 throws away ninety, or nine hundred, depending on where it sat.
- Reading the zero that is still writtenOnce a column has been rewritten, the digit above it is the one to use: 9 − 4, not 0 − 4.
- Reversing the digits to avoid the trouble0 − 7 is not 7. A column that cannot pay is the signal to exchange, and a row of zeros above the line is no exception.
Summary
- A column with nothing to give cannot stop an exchange. The request travels left until it reaches a column that has something.
- Every column passed on the way keeps 9 and sends the rest on; the column finally reached drops by one.
- 900 + 90 + 10 = 1000, which is why a whole run of 9s costs the donor exactly one unit.
- Padding a short decimal makes zeros of its own, and they travel in the same way.
- Subtract from the rewritten digits, never from the ones they replaced.