Expert: Paul Klarreich - 4/15/2007

Question
"Round the number 43,678.426785 to the nearest ten thousand?

I'm confuse with the decimals. I know that the ten thousand place is the 4(the first four before the decimal. SO I thought that it would be 41,000. Since the number behind the four is 3 I should round down, and everything after becomes zero.  

Answer
Questioner:   Marie
Category:  Advanced Math
 
Subject:  Rounding
Question:  "Round the number 43,678.426785 to the nearest ten thousand?

I'm confused with the decimals. I know that the ten thousand place is the 4(the first four before the decimal.

>> Yes, and in that case, every digit after it will be zero.

SO I thought that it would be 41,000. Since the number behind the four is 3 I should round down, and everything after becomes zero.

>> Yes, but that means everything after the '4', so your rounding should give  40,000, not 41,000.

'Nearest ten thousand' means this:

Write down all the multiples of 10,000: [OK, OK, not all of them, just these:]

10,000
20,000
30,000
40,000
50,000
etc.

Now which of the numbers in that list is closest to your
43,678......... (never mind the fractional stuff)


The obvious candidates are 40,000 and 50,000 since 43,678 is between them.  Now find the differences:

50000 - 43678 = 6322;  43678 is 6322 away from 50000
43678 - 40000 = 3678;  43678 is 3678 away from 40000

Looks as if 40000 is closer to your number.  So 40000 is the 'nearest' ten thousand.

What about that 'next digit after' stuff?  Well, if the 'ten thousands' place is 'whatever', then you look at the next digit.  The rule is that if the next digit is LESS THAN 5, you keep the existing 'ten thousands' digit.  If it is 5 OR MORE, you round up and take the next higher digit.  

The principle is:

40xxx,41xxx,42xxx,43xxx,44xxx, are all definitely closer to 40000 than to 50000, so they all round to 40000.

46xxx,47xxx,48xxx,49xxx are all definitely closer to 50000 than to 40000, so they all round to 50000.

What about 45xxx?  If any of those x's, or any decimal parts are not zero, then this is definitely closer to 50000.

What about exactly 45000?  If you do those subtractions as I indicated above, you find a difference of 5000 for each one -- it is EXACTLY in between.  So which one do you pick?

A. If you are playing gin rummy you round down.
B. If you are playing bridge you round up.
C. If you want to make your teacher happy, round up anyway.

Sorry I can't do better than that.