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

### Subtracting Decimals with Borrowing

```
Date: 10/09/2001 at 20:18:08
From: Mario
Subject: Subtraction with six numbers

I just don't understand how to borrow when there are five or six
numbers.

Example:

536.30
- 488.354
---------

```

```
Date: 10/10/2001 at 15:21:05
From: Doctor Ian
Subject: Re: Subtraction with six numbers

Hi Mario,

The first thing you need to know is that you can always add zeros
after the decimal point without changing anything:

1.3 = 1.30 = 1.300 = 1.3000 = ...

So you can add zeros to get both numbers to the same number of decimal
places:

536.300
- 488.354
----------

Okay, so now you can't subtract 4 from 0, so you have to 'borrow'.

Borrowing is sort of like converting dollars to dimes, or dimes to
pennies. For example, if you have 4 dollars and 2 dimes, that's the
same as having 3 dollars and 12 dimes:

4 dollars + 2 dimes = 3 dollars + 12 dimes

Make sure you understand why this is true. If you don't, nothing else
about borrowing is going to make any sense at all.

So if you have something like

35
- 18
----

This is really the same as

2   15
- 1    8
--------
1    7

^    ^
|    |
dimes   |
|
pennies

Sometimes you have to do it more than once:

Exchange           Exchange
one \$100           one \$10
for ten \$10's      for ten \$1's
_____________    _____________
/             \  /             \

3 0 4       2   10   4        2  9  14
-  2 9  ->  -     2   9  ->  -    2   9
------      -----------      ----------
2  7   5

But all this is kind of a drag to do, and it's easy to make mistakes.
Fortunately, there is a simpler way to do subtraction - by doing

Think about what it _means_ to subtract 18 from 35. It means there is
some number that you can add to 18 to get 35. You can try to do that
the standard way (lining up numbers and borrowing). But you can also
try to do it by building up the missing amount. For example, adding 2
to 18 gets you to 20:

18 + 2 = 20

Adding 10 to 20 gets you to 30:

18 + 2 + 10 = 30

Adding 5 more gets you to 35:

18 + 2 + 10 + 5 = 35
\________/

This must be
equal to 35 - 18         35 - 18 = 2 + 10 + 5 = 17

Let's try the second example:

304
- 29  add   1 to get to 30
----  add  70 to get to 100
add 200 to get to 300
add   4 to get to 304
---
275  <--  Add the differences to get the total difference.

You can even do it with your problem:

536.300
- 488.354   add    .006  to get to 488.360
----------   add    .04   to get to 488.40
add    .6    to get to 489
add   1      to get to 490
add  10      to get to 500
add  36.3    to get to 536.3
-------
47.946  <---  This is how much you have to add
to 488.354 to get to 536.300

There are teachers who would say that this is like counting on your
fingers to solve a problem. I say: Apart from the fact that it's slow,
there is nothing wrong with conting on your fingers!

I would rather come up with a 'dumb' solution that I _know_ is right
than follow a bunch of instructions that I don't understand to come up
with an answer that I _hope_ is correct.

At some point, when you're familiar enough with the 'dumb' method,
you'll actually begin to see why the 'standard' method makes sense, at
which point things like borrowing will seem perfectly obvious, rather
than mysterious.

more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Fractions
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