Checking When Rounding

Date: 12/10/2002 at 14:16:39
From: Stephanie
Subject: Checking Decimal Subtraction by Rounding


Hello,  

This is what I do understand:  

To check Decimal ADDITION:

    13.3 + 26.5 = 39.8  To check: Look at the tens, and round:
    13.  + 27.  = 40.  Then look at the addition answer and round it:
    39.8 is rounded to 40.  Answer is correct

What I DON'T understand:

To check Decimal SUBTRACTION:
  
    7.6 - 1.4 = 6.2  OR  86.8 - 43.9 = 42.9

   To round and then check, I don't understand what
   I need to round, and do I add to check?

Thank you very much for your time.
Stephanie

Date: 12/10/2002 at 22:19:49
From: Doctor Peterson
Subject: Re: Checking Decimal Subtraction by Rounding

Hi, Stephanie.

>    13.3 + 26.5 = 39.8  To Check: Look at the tens, and round:
>    13.  + 27.  = 40.  Then look at the addition answer and round it:
>    39.8 is rounded to 40.  Answer is correct

You aren't looking at the tens; you're rounding to the nearest whole 
number.

It's worth noting that this check does not tell you that the answer is 
correct; you just know that it makes sense - it is not too far off. 
But if the correct answer were 39.9, you would not know.

Also, the answer you get by rounding and adding will not always be 
the same as what you get by adding and then rounding. For example, 
12.5 + 23.6 = 36.1; but 13 + 24 = 37, and 36.1 does not round to 37. 
What has happened is that the errors introduced by rounding 
accumulated when they were added together, so that the result is more 
than .5 away from the exact answer. But since 36.1 and 37 are 
reasonably close, the answer is reasonable.

>    7.6 - 1.4 = 6.2  OR  86.8 - 43.9 = 42.9

To check subtraction by rounding, do the same thing: round and 
subtract, and see if the answer is close to the answer you got. The 
way the rounding test works is simply that you replace a detailed 
operation (adding decimal numbers) with the same operation on simpler 
numbers. So for the second problem, you would subtract 87 - 44 and 
compare that with 42.9.

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

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

Date: 12/12/2002 at 09:38:39
From: Stephanie
Subject: Checking Decimal Subtraction by Rounding

I understand your directions, and thank you. But I want to know if 
this is one of those weird Math things that isn't really needed?  
What's the use of checking, if you are only going to get a ballpark 
answer? That seems like a waste of time.  

Also, can you please show me how to check the first problem I gave 
you when I asked about subtraction?

I am wondering if my book has explained this wrong. This is what it 
says: "To round a decimal number to the nearest whole number, look at 
the tenths digit. If the digit is 0-4, the ones digit remains the same 
and all the digits to the right are dropped. If the tenths digit is 
5-9, the ones digit is raised one and all the digits to the right are 
dropped. This is called a CONVENTION."

Thank you again for your time.

Stephanie S.

Date: 12/12/2002 at 10:10:43
From: Doctor Peterson
Subject: Re: Checking Decimal Subtraction by Rounding

Hi, Stephanie. Thanks for writing back!

Checking is important, but each kind of check has a different value. 
Checking by estimation is especially important when you use a 
calculator, since you know it won't make mistakes on the details but 
if you fail to type in a decimal point or a digit, it can make big 
errors. So if you did a calculation that said you needed, say, 1.5 
tons of concrete to make a bridge strong enough, and your estimate 
said it should be about 150, you would go back and do the calculation 
again! Another kind of check, "casting out nines," is unaffected by 
the size of the answer, but would show if some one digit somewhere 
was wrong. And that check, in turn, is unaffected by the common error 
of transposing two digits, so you might prefer another check that 
would reveal that error.

Let's look at

    7.6 - 1.4 = 6.2

To estimate the answer, you can round 7.6 up to 8 (since 6 >= 5 ), and 
round 1.4 down to 1 (since 4 < 5 ); 8-1 = 7. That doesn't mean there's 
an error, because rounding can introduce an error this large. If the 
estimate were, say, 70, you would know something was wrong.

One way to improve the estimate when you do this is to think about how 
subtraction works. We added something to 7.6, and subtracted something 
from 1.4; both changes will have increased the answer. (Do you see 
why?) So we know the real answer is LESS than 7. That makes our check 
valid. Also, since I know about this problem, I prefer to round both 
numbers in the SAME DIRECTION when I subtract (and in opposite 
directions when I add). Here, the .6 wants to go up and the .4 wants 
to go down, and neither is more persuasive than the other (both are .4 
away from the nearest whole number). So I would arbitrarily choose, 
say, to round both numbers up:

    7.6 - 1.4 ~ 8 - 2 = 6

That gives a more accurate estimate. But even this way, it won't 
necessarily be exact.

You can find several discussions of rounding in our site (try the FAQ 
first); you'll see more about different conventions for rounding when 
you are exactly between two numbers. But the important thing here is 
to realize that rounding is only a tool, and considerations apart 
from the rule for rounding a single number can lead us to depart from 
that rule when the goal is to estimate the result of a calculation 
involving several numbers. The interactions among numbers can make a 
big difference.

I hope that helps.

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