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

```Date: 01/06/2003 at 16:03:17
From: Sharon

My problem is 3/4 + 1/6. I have figured out how to get the same
denominators, but in the example 3/4 + 1/6 the denominator is 12 but
it says to build up my numerator, so then it is written 9/12 + 2/12 =
11/12. I don't understand how they get 9 and 2 for numerators.
```

```
Date: 01/06/2003 at 17:34:52
From: Doctor Ian
Subject: Re: Adding fractions

Hi Sharon,

Suppose you have a piece of cake, and you cut it into 4 pieces and
give one piece away. You now have 3/4 of the cake, right?

+-----------+
|           |
|   1/4     |
|           |
+-----------+-----------+  1/4 + 1/4 + 1/4 = 3/4
|           |           |
|   1/4     |   1/4     |
|           |           |
+-----------+-----------+

Now, suppose you cut each of these pieces into three smaller pieces:

+-----------+
|   .   .   |
|   .   .   |
|   .   .   |
+-----------+-----------+
|   .   .   |   .   .   |
|   .   .   |   .   .   |
|   .   .   |   .   .   |
+-----------+-----------+

How big is each of these pieces?  Well, if we'd cut the cake into
pieces of this size to begin with,

............+-----------+
.   .   .   |   .   .   |
.   .   .   |   .   .   |
.   .   .   |   .   .   |
+-----------+-----------+
|   .   .   |   .   .   |
|   .   .   |   .   .   |
|   .   .   |   .   .   |
+-----------+-----------+

we'd have had 12 pieces. (Count them to make sure.) So each of these
smaller pieces is 1/12 of the original cake, and when we have 3/4 of
the cake, that's the same as having 9/12 of the cake:

+-----------+                 +-----------+
|   .   .   |                 |           |
| 1 . 2 . 3 |                 |     1     |
|   .   .   |                 |           |
+-----------+-----------+  =  +-----------+-----------+
|   .   .   |   .   .   |     |           |           |
| 4 . 5 . 6 | 7 . 8 . 9 |     |     2     |     3     |
|   .   .   |   .   .   |     |           |           |
+-----------+-----------+     +-----------+-----------+

9/12                          3/4

Does that make sense?

Now, how do we get from 3/4 to 9/12 without drawing a picture? Well,
remember, we divided something into 4 pieces, and kept three of them.
If we divided it into 3 times as many pieces, we'd keep three times
as many of those, right?  So

keep 3 pieces                       keep 3*3 pieces
--------------- must be the same as -----------------
out of 4 pieces                     out of 3*4 pieces

Does _that_ make sense?  The way we write it in symbols is

3   3 * 3    9
- = ----- = --
4   3 * 4   12

Does this answer your question?  If so, you should be able to 'build
up' 1/6 in the same way that we 'built up' 3/4:

1   ? * 1    ?
- = ----- = --
6   ? * 6   12

If not, write back and let me know where I lost you.

I hope this helps!

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