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Now that "everyone" has a calculator that will give a result to six or eight (or more) figures, it is important that we know how to round the answer off correctly. The typical rule taught is that you round up with five or more and round down with four or less.

THIS RULE IS WRONG!

However, please do not rush off to your elementary school teacher and read 'em the riot act!

The problem lies in rounding "up" (increasing) the number that is followed by a 5. For example, numbers like 3.65 or 3.75, where you are to round off to the nearest tenth.

OK, let's see if I can explain this. When you round off, you change the value of the number, except if you round off a zero. Following the old rules, you can round a number down in value four times (rounding with one, two, three, four) compared to rounding it upwards five times (five, six, seven, eight, nine). Remember that "rounding off" a zero does not change the value of the number being rounded off.

Suppose you had a very large sample of numbers to round off. On average you would be changing values in the sample downwards 4/9ths of the time, compared to changing values in the sample upward 5/9ths of the time.

This means the average of the values AFTER rounding off would be greater than the average of the values BEFORE rounding.

This is not acceptable.

We can correct for this problem by changing the rule for rounding 5 rounding "off" (keeping the number the same) in fifty percent of the roundings-even numbers followed by a 5. Then, on average, the roundings "off" will cancel out the roundings "up."

The following rules dictate the manner in which numbers are to be rounded to the number of figures indicated. The first two rules are more-or-less the old ones. Rule three is the change in the old way.

When rounding, you examine the digit following (i.e., to the right of) the digit that is to be the last digit in the rounded off number. The digit you are examining is the first digit to be dropped.

- If that first digit to be dropped is less than 5 (that is, 1, 2, 3 or 4), drop it and all the digits to the right of it.
- If that first digit to be dropped is more than 5 (that is, 6, 7, 8 or 9), increase by 1 the number to be rounded, that is, the preceeding figure (to the digit being dropped).
- If that first digit to be dropped is 5, round the digit that is to rounded off so that it will be even. Keep in mind that zero is considered to be even when rounding off.

The rules above are a bit technical, so here are some examples.

Example #1 - Suppose you wish to round 62.53__4__7 to four significant figures. Look at the fifth digit. It is a 4, a number less than 5. Therefore, you will simply drop every digit after the fourth, and the original number rounds off to 62.53. (rule #1 above)

Example #2 - Round 3.78__7__21 to three significant figures. Look at the fourth digit. It is 7, a number greater than 5, so you round the original number up to 3.79. (rule #2 above)

Example #3 - Round 726.83__5__ to five significant figures. To do this, you must look at the sixth digit. It is a 5, so now you must look at the fifth digit also. That is a 3, which is an odd number, so you round the original number up to 726.84. (rule #3 above)

Example #4 - Round 24.8__5__14 to three significant figures. Look at the fourth digit. It is a 5, so now you must also look at the third digit. It is 8, an even number, so you simply drop the 5 and the figures that follow it. The original number becomes 24.8. (rule #3 above)

Here are some more examples for rule #3.

Example #5 - Round 23.55 to the 0.1 place. To do this, you must look at the hundreths place (remember, we are going to keep the tenths place in our answer). It is a five, so now we look at the next digit inward (the tenth place) and see it is a five, an odd number. Since we are rounding off a 5 (in the hundreths place), we must round to an even number. The answer is 23.6.

Example #6 - Round 23.65 to the 0.1 place. To do this, you must look at the hundreths place (remember, we are going to keep the tenths place in our answer). It is a five, so now we look at the next digit inward (the tenth place) and see it is a six, an even number. Since we are rounding off a 5 (in the hundreths place), we must round to an even number. The answer is 23.6.

Notice the different phrasings of the problems. One says to round off to a specific number of significant figures and the other type says to round off to a specific decimal position. In both cases, you have to look at the digit just to the right of where you intend to wind up in your answer. For example, if you are to round to three sig figs, you have to look at the fourth significant figure. If you are to round off the the 0.01 place, you have to look at the 0.001 place as well. The digit in this place tells you to round up (if it is 6, 7, 8, or 9) or to round down (if it is a 1, 2, 3, or 4).

When the value you intend to round off is a five, you MUST look at the previous value ALSO. If it is even, you round down. If it is odd, you round up. A common question is "Is zero considered odd or even?" The answer is even.

Here are some more examples of the "five rule." Round off at the five. (Answers at the bottom of the file.)

3.075

3.85

22.73541

0.00565

2.0495

This last one is tricky (at least for high schoolers being exposed to this stuff for the first time!). The nine rounds off to a ten (not a zero), so the correct answer is 2.050, NOT 2.05.

Would your teacher be so mean as to include problems like this one on a test? In the ChemTeam classroom, the sufferers (oops, I mean students) have learned to shout "YES" in unison to such easy questions.

Lastly, before we get to the problems. Students, when they learn this rule, like to apply it across the board. For example, in 2.0495, let's say we want to round off to the nearest 0.01. Many times, a student will answer 2.04. When asked to explain, the rule concerning five will be cited. However, the important number in this problem is the nine, so the rule is to round up and the correct answer is 2.05.

Round the following numbers as indicated. To four figures: To the nearest 0.1: To nearest 0.01: To the nearest whole number: 1) 2.16347 x 10Answers^{5}13) 3.64 25) 6.675 37) 56.912 2) 4.000574 x 10^{6}14) 4.55 26) 0.4203 38) 3.4125 3) 3.682417 15) 7.250 27) 0.03062 39) 251.7817 4) 7.2518 16) 0.0865 28) 4.500 40) 112.511 5) 375.6523 17) 0.5182 29) 2.473 41) 63.541 6) 21.860051 18) 2.473 30) 7.555 42) 7.555 To two figures: To one decimal place: To the nearest 0.001: Round off the farthest right digit 7) 3.512 19) 54.7421 31) 5.687524 43) 2.473 8) 25.631 20) 100.0925 32) 39.861214 44) 5.396 9) 40.523 21) 1.3511 33) 104.97055 45) 8.235 10) 2.751 x 10^{8}22) 79.2588 34) 41.86632 46) 3.05 11) 3.9814 x 10^{5}23) 0.9114 35) 0.03765 47) 8.25 12) 22.494 24) 0.2056 36) 0.0045 48) 8.65

Answers to "rounding off a 5" rule

3.08

3.8

22.74

0.0056