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```
Date: 08/26/2001 at 16:11:01
From: cassandra
Subject: Fractions

I am having problems with doing fractions on the calculator. Is there
a way to do it? I can't do long division, like 23/34, and how to get
the least common denominator.
```

```
Date: 09/04/2001 at 11:35:08
From: Doctor Ian
Subject: Re: Fractions

Hi Cassandra,

Let's look at the problem you mentioned, which is how to divide 23 by
34 (or, to put it another way, how to divide 23 into 34 parts):

____
34 ) 23

How many times does 34 go into 23?  Well, it doesn't go any times.
However, it goes into 230 some number of times:

?
_____
34 ) 230

How many times? Well, the only way to find out is to guess. If you've
memorized your times tables, you know that 3*8 is 24, so 30*8 would be
240, so 8 is probably too high. What about 7?

7 * 34 = 7 * (30 + 4)

= 7*30 + 7*4

= 210 + 28

= 238

which is still too big.  How about 6?

6 * 34 = 6*30 + 6*4

= 180 + 24

= 204

So 34 goes into 230 at least 6 times:

6
_____
34 ) 230
204
---
26        230/34 = 6 + 26/34

Now we're back in the same boat, trying to find out what 26/34 is.

34 doesn't go into 26, but it will go into 260.  We already saw that
7*34 is 238, so we know that it goes at least 7 times:

67
______
34 ) 2300
204
---
260
238
---
22

We can keep going on like this, and eventually, one of two things will
happen. Either we'll end up with no remainder,

375
______
8 ) 3000
24
--
60
56
--
40
40
--
0        Done!

or we'll end up with a remainder that we've already seen,

39
_______
33 ) 1300
99
---
310
297
---

in which case we can stop, because we know that we're just going to
keep repeating the same calculations over and over again:

39393...  (the sequence '39' repeats forever)
_________
33 ) 1300000
99
---
310
297
---
130
99
---
310
297
---
130
99

Okay, but what's really going on here? Why can we just keep adding
zeros? Don't we need a decimal point in here somewhere?

In fact, we do, but it certainly makes life easier if we forget about
it until the end.

Let's look at what happened with 3/8. Instead of taking 8 into 3, we
changed the 3 to 3000.  And we found that

3000 / 8 = 375

How does this help?  Well, we can multiply both sides of an equation
by the same thing without changing the truth of the equation, right?
Let's try multiplying both sides of _this_ equation by 1/10.  The easy
way to do that is to move the decimal point to the left by one place -
for example, 375 multiplied by 1/10 is 37.5.  So

3000
---- = 375
8

300.0
----- = 37.5             Multiply everything by 1/10.
8

30.00
----- = 3.75             Again...
8

3.000
----- = 0.375            Again...
8

Now we're back to our original problem, and we've found the answer.

Now, there is a quicker way to do this, which is to leave the decimal
point where it is, which gives us the answer including the decimal
point without having to go through this final step:

0.375
_______
8 ) 3.000
24
--
60
56
--
40
40
--
0

But if you don't see _why_ this 'adding more zeros' stuff works, then
this looks pretty much like some kind of magic spell.  And the
_reason_ that it works is that we're using the same multiplication
trick to try to change our original problem into one that we can solve
using only integers:

3   30    1
- = -- * --
8    8   10

300    1
= --- * ---
8    100

3000     1
= ---- * ----
8     1000

^      ^_____________
|                    |
We can solve            |
this problem
with integers... and then move the decimal point to

= 375 * (1/1000)

= 0.375

In other words, we use multiplication on both sides of the equation to
turn one messy problem into two simpler problems, whose solutions we
can easily combine to get the solution to the original problem.

As you take more mathematics, you'll see this pattern come up over and
over again, because mathematicians love to solve hard problems by
turning them into easy ones. In fact, when you get to algebra, _most_
of what you'll be learning is how to do this.

more, or if you have any other questions.

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

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