Rounding 3.445 to the Tenths Place

Date: 05/24/2001 at 22:34:34
From: Joseph Sciammarella
Subject: Rounding numbers

In my daughter's 6th grade math class, they are told to address the 
digit to the right of the place being rounded. For example, 3.445 
rounded to the tenths place, would be 3.4, since the number to the 
right of the tenth place is less than 5.

However, doesn't the presence of the 5 in the one-thousandths place 
round the 4 hundredths to 5 hundredths, which in turn would round the 
4 tenths to 5 tenths?

Thank you for your help. I have reviewed the archives, but haven't 
come across this level of rounding.

Date: 05/25/2001 at 08:46:57
From: Doctor Rick
Subject: Re: Rounding numbers

Hi, Joseph.

There is no need to consider digits beyond the digit to the right of 
the digit being rounded, except in one special case when using a 
different rounding algorithm.

Consider what rounding means. If I have a number, say 3.445, and I 
want to round it to the nearest tenth, this means I want to find the 
nearest number that is an even multiple of one tenth. The two 
candidates are the nearest such number below 3.445 (which is 3.4) and 
the nearest such number above 3.445 (which is 3.5). How close are 
these two candidates to 3.445?

     3.5 - 3.445 = 0.055

     3.445 - 3.4 = 0.045

The number 3.4 is therefore closer to 3.445 than is 3.5; so we should 
round down to 3.4.

This is what your daughter's algorithm does: seeing that the digit in 
the hundredths place is 4, she rounds down to 3.4.

When the digit to the right is 5, we may have a problem. Let's see if 
we do. When we round 3.452 to the nearest tenth, our candidates are 
3.4 and 3.5 again; the differences are:

     3.5 - 3.452 = 0.048

     3.452 - 3.4 = 0.052

The nearest candidate is 3.5, so this time we want to round up to 3.5. 
This, again, is what your daughter's algorithm does: seeing that the 
hundredths digit is 5, she rounds up.

But what about the number 3.45? Again the candidates are 3.4 and 3.5, 
and the differences are:

     3.5 - 3.45 = 0.05

     3.45 - 3.4 = 0.05

The differences are identical, so there is no obvious choice for which 
way we round! If I understand your daughter's algorithm correctly (the 
way most school children are taught these days), she would see the 5 
in the hundredths place and round up. This is a valid option.

You may be interested in the following item in our Archives, which 
addresses another rounding algorithm that many people were taught. You 
will see that, when properly applied, it only affects the last case I 
showed, in which it would be equally valid to round in either 
direction.

   Rounding Decimals: Even/Odd Issues
   http://mathforum.org/dr.math/problems/deborah.05.08.01.html   

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

Date: 05/25/2001 at 09:01:56
From: Doctor Peterson
Subject: Re: Rounding numbers

Hi, Joseph.

The problem here is that you have to round all at once, not one digit 
at a time. Rounding twice, to different digits, doesn't do what you 
would think it would.

Here's what happens: Since 3.445 is closer to 3.4 than to 3.5, it must 
round to 3.4; the border between 3.4 and 3.5 is at 3.45, and 3.445 is 
below that. But if you first round it to the nearest hundredth, it 
becomes 3.45, moving it from "below the border" to "right on the 
border" and allowing a second rounding to move it "over the border" to 
3.5.

It's as if the border patrol were to decree that anyone within ten 
feet of the boundary fence should be considered to be on the fence; 
and then said that anyone on the fence should be arrested for illegal 
entry. That wouldn't be right, since people ten feet outside of the 
country would be treated as if they were inside!

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