Rounding Fractions to the Nearest Half or Whole

Date: 01/10/2003 at 01:04:01
From: Michelle
Subject: Rounding fractions to the nearest half or whole?

How do you round fractions to the nearest half or whole? I don't 
understand how you can tell what the fraction needs to be rounded to. 
What do you look for to know whether it goes up or down?

   9 1/4 

   6 2/15

    11/15

The example shown was 9/16 rounds to 1/2. Why?

1/14 rounds to 0. Why?

Date: 01/10/2003 at 08:55:07
From: Doctor Peterson
Subject: Re: Rounding fractions to the nearest half or whole?

Hi, Michelle.

Here is a number line containing 9/16:

                            |<-+------------------->|
                            |  |                    |
    0                      1/2 |                    1
    +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
    0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15  1
      -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
      16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 

I've indicated the fact that 8/16 is the same as 1/2, and I've 
located 9/16 on the line. Note that 9/16 is between 1/2 and 1, and is 
closer to 1/2 than to 1; so the nearest half is 1/2.

Now, how could you determine that without drawing the number line? 
One way is to first consider how many 16ths is 1/2; that is 1/2 of 16, 
or 8 16ths. Now compare 9 with 8, and see that it is greater; so 9/16 
is greater than 1/2. That puts it between 1/2 and the next whole, 
namely 1.

Which is closer? Just compare the difference 9-8 = 1 with the 
difference 16-9 = 7. That is, 9/16-8/16 = 1/16, while 16/16 - 9/16 = 
7/16. Since 1 < 7, 9/16 is closer to 1/2.

You can do the same with each of your problems. 1/14 is between 0 and 
7/14 = 1/2, and is closer to 0. When you have a mixed number, you can 
ignore the whole part while you are making this decision. When you 
have an odd denominator, 1/2 will not be a whole number of them; 1/2 
is equal to 7 1/2 15ths, for example. But you can still do all the 
same work: 2/15 is less than 7 1/2 15ths, so 6 2/15 is between 6 and 
6 1/2. Which is closer?

The hard case is 9 1/4. You will find that 1/4 is exactly between 0 
and 1/2, so it is a tossup whether to round up or down. I don't think 
there is a standard rule to decide this, as there is when rounding to 
a whole number, but you might as well follow a rule like the one 
usually taught in school, and always round up when you are exactly 
between. Then 9 1/4 will round up to 9 1/2.

If you have any further questions, feel free to write back.

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