Associated Topics || Dr. Math Home || Search Dr. Math

### Borrowing in Subtraction

Date: 12/01/98 at 23:00:14
From: Christy
Subject: Subtraction with three numbers

When I work on the problem:

100
- 99
----

I don't know why the zero in the tens place becomes a 9 when I borrow
from it. I know that I am taking the 1 from the hundreds position to
borrow, but why does the zero in the tens become a nine?

Date: 12/02/98 at 12:03:30
From: Doctor Peterson
Subject: Re: Subtraction with three numbers

Hi, Christy. I like to teach "borrowing" using money. If you want, get
out some play money (\$100's, \$10's, and \$1's) and try acting out what
I say. It's fun, even if you don't need it.

When we write 100, it means 1 hundred, 0 tens, and 0 ones. So lay out
three piles (two of them will be empty!):

\$100's    \$10's    \$1's
------    -----    ----
1        0        0

I like to have fun with this and pretend these piles represent three
people, "Mr. Hundreds," "Mr. Tens," and "Mr. Ones." Now suppose we come
along and want to get \$99 from them (9 tens from Mr. Tens and 9 ones
from Mr. Ones). We first go to Mr. Ones and ask for the 9 ones, but he
doesn't have any ones today, so he says "I'll be right back," and goes
next door to Mr. Tens to ask for a ten dollar bill that he can change
into 10 ones. Mr. Tens doesn't have any, but he goes next door to Mr.
Hundreds and asks for a hundred dollar bill that he can change into 10
tens. He's in luck! He takes it, goes to the bank, and now he has 10
tens:

\$100's    \$10's    \$1's
------    -----    ----
0       10        0

Now he can give one of his tens to Mr. Ones, who changes it to 10 ones:

\$100's    \$10's    \$1's
------    -----    ----
0        9       10

Now Mr. Ones can give us 9 of his 10 ones:

\$100's    \$10's    \$1's
------    -----    ----
0        9       10
-9
--
1

Then we can go to Mr. Tens and ask for all of his 9 tens:

\$100's    \$10's    \$1's
------    -----    ----
0        9        1
-9
--
0

So they have nothing left but a single one.

Now let's write all this down the way we usually do:

1 0 0
-  9 9
------

We can't subtract 9 from 0, so we "go next door" to borrow a ten. Since
there are no tens to borrow, before we can do that we have to go yet
another place over to borrow a hundred, which turns into 10 tens:

0 10
/ /
1 0 0
-  9 9
------

Now we can take one from the tens (leaving 9 tens), which turns into
10 ones, from which we can subtract 9:

0 9 10
/ / /
1 0 0
-  9 9
------
1

Finally, we can subtract the 9 tens (and no hundreds):

0 9 10
/ / /
1 0 0
-  9 9
------
0 0 1

So where did that 9 come from in the tens place? It's the 10 tens we
borrowed (which was one hundred), minus the 1 ten that was borrowed
from it by the ones.

99
+  1
----

and you'll see a double carry just like the double borrow in your
subtraction.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/

Associated Topics:
Elementary Subtraction

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search