Joe Doe (joe_doe_zero) wrote in 0_arithmetic,
Joe Doe
joe_doe_zero
0_arithmetic

General subtraction - lesson 2

If we want to subtract 566 from 1707 how would we do it?

First, let's use the lessons we learned from adding multi-digit numbers. We would turn 566 into 0566.

1707 - 0566 = ?

1 - 0 = 1
7 - 5 = 2
0 - 6 = ???
7 - 6 = 1

You might have noticed that the tens digit in the minuend is smaller than the tens digit in the subtrahend and the problem seems incomputable. But we can always "borrow" from an upper row. That is, we subtract 1 from the row we are borrowing from and add ten to the row we are borrowing to.

Let's borrow a 10 from the hundreds column in this problem.

1 - 0 = 1
7 - 5 = 2 <- borrow 1 from this column (subtract 1 from the minuend)
0 - 6 = ??? <- borrow 10 to this row (add 10 to the minuend)

7 - 6 = 1

We get:

1 - 0 = 1
6 - 5 = 1
10 - 6 = 4

7 - 6 = 1

We get 1707 - 566 = 1141

Let's try another: 1200 - 975
1 - 0 = 1
2 - 9 = ???
0 - 7 = ???
0 - 5 = ???


First we borrow a 1 from the thousands row (in the minuend, of course). We add 10 to the hundreds row.
0 - 0 = 0
12 - 9 = 3
0 - 7 = ???
0 - 5 = ???

So far, we have no definite answer but at least the 3 in the hundreds row is settled. The problem of 1200 - 975 has been reduced to 300 - 75.

3 - 0 = 3
0 - 7 = ???
0 - 5 = ???

We "borrow" from the hundreds row again. We subtract 1 from the 3 and add 10 to the number beneath it.

2 - 0 = 2
10 - 7 = 3

0 - 5 = ???

Now we have a more definite answer: 230. The problem has been reduced from 300 - 75 to 230 - 5. But there's still more "borrowing" to do. One more "loan" ought to do it.

2 - 0 = 2
3 - 0 = 3 (borrow a 1 from here)
0 - 5 = ??? (add a 10 to here)


2 - 0 = 2
2 - 0 = 2
10 - 5 = 5


Done. And done. Our final answer is 225. Thus 1200 - 975 = 225

Another one to try: 9,000 - 5,678 = ?

9 - 5 = 4
0 - 6 = ???
0 - 7 = ???
0 - 8 = ???

A shortcut to take is to do multiple borrowing. Borrow a 1 from the 9 and put in a 10 into the 0 directly beneath it. Then borrow a 1 from that zero and put a 10 into the 0 directly beneath that. (The second zero beneath the 9) And finally, borrow a 1 from the tens row and put a 10 into the ones row (the bottom row). Here's what the problem looks like after the first step
8 - 5 = 3
10 - 6 = ?
0 - 7 = ???
0 - 8 = ???

After a second loan has been made:
8 - 5 = 3
9 - 6 = ?
10 - 7 = ?
0 - 8 = ???

And after the third loan:
8 - 5 = 3
9 - 6 = ?
9 - 7 = ?
10 - 8 = ?

And it's ready to be solved!
8 - 5 = 3
9 - 6 = 3
9 - 7 = 2
10 - 8 = 2

9,000 - 5,678 = 3,322.

Thus this stumper:

10,000,000,000 - 1,234,567,891

10 - 1 = 9
0 - 2 = ???
0 - 3 = ???
0 - 4 = ???
0 - 5 = ???
0 - 6 = ???
0 - 7 = ???
0 - 8 = ???
0 - 9 = ???
0 - 1 = ???

After a series of borrowing:
9 - 1 = 8
9 - 2 = 7
9 - 3 = 6
9 - 4 = 5
9 - 5 = 4
9 - 6 = 3
9 - 7 = 2
9 - 8 = 1
9 - 9 = 0
10 - 1 = 9


10,000,000,000 - 1,234,567,891 = 8,765,432,109

You might have noticed that the difference in that last problem has exactly one of each kind of digit in it.
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Some Life Help


If you must pay for something that costs $59.99 and you have a $100.00 bill, one way you can rapidly compute the difference is by doing "subtraction digit pairs". Pair each digit using this formula:

9 - (digit) = (its subtraction pair)
9 - 0 = 9
9 - 1 = 8
9 - 2 = 7
etc.

So 0 is paired with 9, 1 with 8, 2 with 7, 3 with 6, 4, with 5, and so on.

Then for every digit except for the ones digit, simple put in a pair. For the digits with the greatest value, simply subtract 1.

$100.00 - $59.99
10 - 5 = 4 (Think 9 - 5 = 4)
0 - 9 = 0 (0 is paired with 9)
0 - 9 = 0 (0 is paired with 9)
0 - 9 = 1 (1 + 9 = 10)
Your change is $40.01

Another: $800.00 - $567.98
8 - 5 = 2 (7 - 5 = 4)
0 - 6 = 3 (3 is paired with 6)
0 - 7 = 2 (2 is paired with 7)
0 - 9 = 0 (0 is paired with 9)
0 - 8 = 2 (2 + 8 = 10)
Your change is $$432.02
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