Rounding to One Digit Accuracy

Date: 04/16/98 at 21:33:06
From: Lori
Subject: Rounding to 1-digit accuracy

Hello,

I can't figure out how do round to 1-digit accuracy. I received a 
paper and I only got a few right. I had to guess, and one of the 
problems was 3.82___________. I put 4 and I got it right. On all the 
other ones, like 14.98__________, I put 15.0 and that was wrong. 

Please explain this to me! I guess what I'm asking is please give some 
examples so I can see it too! Thank you so much.

LORI

Date: 04/21/98 at 14:11:09
From: Doctor Derrel
Subject: Re: Rounding to 1-digit accuracy

Hi Lori,

I'll try to help. I'm going to tell you how to do rounding and show 
you some examples; then I'll try to explain why you would want to 
do it.

When you round, only look at the digit to the right of the one that 
you are going to round. If that digit is 

     1) less than 5, round down 
     2) 5 or greater, round up

(Why does 5 always round up? Well, if the digit to the right of the 
one we are rounding is 0,1,2,3, or 4, we have five numbers that round 
down; then 5,6,7,8, or 9 are the five numbers that round up.)

Don't look more than one digit to the right of the one you are 
rounding to. For example, if you are given the number 564,982 and 
asked to round to two-digit accuracy, think of the number as 564,XXX 
where the "X" can be any digit. This is because the two digits that 
you will care about after rounding are "56" and the "4" tells you 
whether you have to round up or round down. In this case, you round 
down (the "6" stays a "6"), then all of the other digits besides the 
two you were asked to round to become zero. Your final answer is 
560,000.

Maybe you are confused that 560,000 has six digits, but we are talking 
about two-digit accuracy. The key word here is "accuracy." You have to 
know how big the number is; that is, is it in the hundreds? Thousands? 
Billions? Or on the other side of the decimal, in the ten-thousandths? 
Millionths? Trillionths? Otherwise you would end up with 56 as your 
answer, and you know that 56 is nowhere near 564,982!

Examples of rounding to one-digit accuracy (--> means "rounds to"):

     3.82    --> 4
     4.233   --> 4
     3.5     --> 4 (The 5 rounds up)
     4.5     --> 5 
     4.599   --> 5 (Only look at the digit to the right of the one you 
                    are rounding to - ignore the two 9's. It's the 5  
                    thatrounds up.)
     3.6837  -->   4
     10.962  -->  10 (Look only at the 0; ignore the .962)
     14.98   -->  10 (Look only at the 4; ignore the 98)
     153.68  --> 200

Did you notice that the "10," "10," and "200" are rounded to one-digit 
accuracy even though they have more than one total digits?

Try this one on your own: Round 428,379.62 to 1-digit accuracy and 
write down your answer. (The answer is at the bottom below my 
signature so that you won't be tempted to peek!) :-)

What if you write 4.0, 10.0, 10.0, or 200.0 for the last four numbers 
above? Well, those would be wrong because adding the digit to the 
right of the decimal point shows that you know something about what 
goes in the tenths place. In this case, you said it is zero tenths. 
Instead of one-digit accuracy, you would now have two digits for 4.0, 
three digits for 10.0, and a whopping four digits for 200.0 . Your 
15.0 was wrong for the 14.98 because you had three-digit accuracy.

Okay, I can see you scratching your head - how is it that putting in 
a decimal point and a zero in the tenths place, the way we did for the 
numbers in the previous paragraph, suddenly means we have more 
accuracy? The key here is the decimal point. By putting in a zero to 
the right of the decimal point, we said, "I know that the value in 
the tenths place is zero." And if we know the value in the tenths 
place, we know the values (and the accuracy) for all of the others. 

Now, let's try two-digit accuracy:

Rounding to two-digit accuracy:

     3.82    --> 3.8
     4.233   --> 4.2
     3.5     --> 3.5 (It already has two digits.)
     10.487  --> 10  (Look at the 4; ignore the 87.)

On the last one, we got "10," which is two-digit accuracy. But above, 
we also got "10" when we were rounding to one-digit accuracy. How do 
we know whether 10 is two-digit or one-digit accuracy? Well, we don't 
know just by looking at the number. You have to have some more 
information, like how many digits it was rounded to. (You will learn 
later that we do have a way of knowing how many digits of accuracy we 
have, but you need to know about something called "scientific 
notation.")

I hope this helped you with the rules of rounding when working only 
with numbers. I also suggest that you look up the term "significant 
digits" in a math dictionary (your teacher or library should have one) 
because it will go into more detail about rounding. 

However, rounding is something you do because it makes sense for the 
problem you are working on. When working on problems, you have to 
think about when to round, and sometimes you don't follow the rules. 

For example, if you are going on a school trip, each bus holds 50 
students, and you have 110 students going, would you say: "Well, 110 
divided by 50 is 2.2. Since we can only take whole buses, the rule 
says we round to one digit. Looks like we take only two buses." You 
would have 10 disappointed students if you followed that line of 
reasoning! 

In short, when doing tests where you are given numbers to round, just 
follow the rules. When you are working problems, think about whether 
it makes sense to round, and how you are going to round.

I did not give examples about rounding negative numbers and decimal 
numbers between 0 and 1. The rules for these really make sense if you 
think about why you are rounding. However, if you would like some 
examples of those, please write again. If you still have questions, 
please write again.

-Doctor Derrel,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/   

Answer to above question:
428,379.62 to one-digit accuracy is 400,000 -- can you explain why?