Rounding to Nearest Multiple

Date: 07/31/2001 at 21:04:06
From: Pedro
Subject: Rounding numbers

Hello -

My name is Pedro and I'm in sixth grade. I can't figure out how to 
round to the nearest multiple of 10, 100, or 1,000. Can you please 
help me?  

For example: 

   27 + 56~ 30 + 60 = 90 

The teacher explained this example to me, but I don't understand.

Date: 07/31/2001 at 23:04:53
From: Doctor Peterson
Subject: Re: Rounding numbers

Hi, Pedro.

I'm not sure whether your problem is in rounding, or in using rounding 
to estimate a sum. I'll say a little about each, and you can write 
back and tell me what parts you have trouble with.

First, let's round some numbers to the nearest ten. What does "nearest 
ten" mean? We want to find the closest multiple of ten to a given 
number. For example, given the number 27, which of 10, 20, 30, and so 
on, is it closest to? Well, it's between 20 and 30, so the nearest 
must be one of those. And the number 25 is exactly between 20 and 30, 
so any number less than 25 is closer to 20, and any number bigger than 
25 is closer to 30. Since 27 is bigger than 25, the nearest multiple 
of ten must be 30. We call this rounding up, since the nearest 
multiple of ten is in that direction.

We can do that with a lot less thinking by making some rules. We first 
replace as many digits as we need at the right of our number, 27, with 
zeros to make it a multiple of ten: 27 --> 20. (If we were rounding to 
the nearest hundred, we would have replaced two digits with zero, 
rather than just one, like 273 --> 200.) That tells us what multiple 
of ten is LESS than the number. Now, we compare the first digit we 
dropped (the 7 in each example) with 5. If it's less, then we're done. 
If it's greater than 5, then we have to round up by adding on an extra 
ten or hundred, changing 20 to 30 or 200 to 300.

If the digit were exactly equal to 5, we would traditionally act as if 
it were greater than 5. That's because, if there are any digits beyond 
it (as in 251 rounded to the nearest hundred), our number will in fact 
be past the half-way point, and we do have to round up. If all the 
rest of the digits were zero (as in 250 rounded to the nearest 
hundred), then we would be exactly halfway, and we can choose either 
to round up or down; there is really no one nearest multiple. It's 
easiest just to always round up when we see a 5.

Now in your example of using rounding, you had to add 27 and 56. We 
rounded to the nearest ten (leaving only one nonzero digit to make 
things easy), and got 30 and 60 using the rules I just explained. 
These add up to 90. Now, since we rounded both numbers up, we expect 
90 to be too large; and in fact the correct sum is only 83, which 
would round down to 80, not up to 90. But 90 is close enough for many 
purposes.

Write back if you need any more help; give me some examples of your 
work, so I can see what mistakes you make, if any.

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