Checking When RoundingDate: 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/ |
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