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/
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