Making Averaging EasierAuthor(s): Michael S. SmithSource: The Arithmetic Teacher, Vol. 29, No. 4 (December 1981), pp. 40-41Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41190030 .

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Making Averaging Easier

By Michael S. Smith

The average of two numbers is the number that is half way between them. Normally we compute averages by adding the two numbers and dividing the sum by two. Alternatively, we could take half of each number and then add the halves.

Example 1.

46 Hi «57 2

68 114

or

(1/2 X 46) + (1/2 X 68) 23+34 57

Example 2.

78 111 = 56 2

34 112

or

(1/2 X 78) + (1/2 X 34) 39 + 17 56

If we work from left to right and think of each number in terms of the sums of

Michael Smith is an assistant professor of neurol- ogy at the University of Arizona Health Sciences Center in Tucson. He is a graduate of Dartmouth College and of the University of Colorado medical school. Following his internship, he served two years in the U. S. Navy aboard the USS St. Louis, both as medical officer to her and to am- phibious squadron seven, of which she was a part. The method of averaging that he describes in this article was first tested on some sailors somewhere in the western Pacific. Dr. Smith is interested in neuro-ophthalmology - and arithmetic.

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columns of thousands, hundreds, tens, and ones, then example 1 looks as fol- lows:

40 + 6 60 + 8

50 + 7 = 57

We can easily average 40 and 60 (to get 50) and can easily average 6 and 8 (to get 7). Adding the two results, 50 + 7 = 57, which is the same as obtained be- fore. Similarly, example 2 would look as follows:

70 + 8 30 + 4

50 + 6 = 56

What about the case in which one of the numbers is odd?

Example 3.

61 38

We would rewrite the example as follows

60 + 1 30 + 8

The average of 60 and 30 is 45. The aver- age of 1 and 8 is 4 1 /2. Adding the two, the average is 49 1 /2.

Larger numbers are averaged in the same fashion.

Example 3.

The average of 276 and 1 92 becomes

200 + 70 + 6 100 + 90 + 2 150 + 80 + 4 = 234

Example 4.

The average of 1 934 and 872 becomes

1000 + 900 + 30 + 4 0 + 800 + 70 + 2

500 + 850 + 50 + 3 = 1403

Notice that if we are to average the thousands column in the fourth ex- ample, then we must place a zero in the thousands column for 872. This zero is averaged with the thousand to make 500. Similarly, if we were finding the average of 12 and 8, the example could be written as follows:

10 + 2 0 + 8

5 + 5 = 10

The rewriting step can be omitted as long as we remember whether we are working with thousands, hundreds, tens, or ones.

Example 5.

To find the average of 42 and 86:

42 First average the 4 and 86 8 in the tens column.

6

42 Then average the 2 and 6 86 in the ones column.

64

Half the time one of the numbers will be odd.

Example 6. To find the average of 79 and 47:

7 9 Average the 7 and 4 in the 4 7 tens column. This is 5 1 /2,

5 1 /2 which means 5 1 /2 tens, or 55.

Arithmetic Teacher

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7 9 Next average the 9 and 7 4 7 in the ones column. This is

51/2 8 8. So, 55 + 8 = 63.

Example 7. To find the average of 276 and 529:

2 7 6 Average the 2 and 5 to 5 2 9 get 3 1/2 hundreds or 350; 3 5 0 the 7 and 2 to get 4 1 /2

4 5 tens or 45; and the 6 and 9 to 7 1/2 get 7 1 /2. The average is the

4 0 0 2 1 /2 sum of 350, 45, and 7 1 /2.

If one is in doubt, the numbers could be re- written:

200 + 70 + 6 500 + 20 + 9

350 + 45 + 7 1/2 = 402 1/2

Further Shortcuts So far we have identified two ways of finding the average of two numbers, (1)

adding the two numbers and finding half the sum, and (2) taking the aver- age of each of the columns. There is yet another method. We can find the difference between the two numbers, take half of the difference, and then ei- ther add it to the smaller number or subtract it from the larger number in order to arrive at the average. Al- though this "difference splitting" method may appear cumbersome, in cases where the difference is easily cal- culated, the method may be the fastest of the three.

Example 8. To find the average of 46 and 96:

The difference is 50. Half of the difference is 25. Add the 25 to the smaller number,

46 + 25 = 71, or subtract it from the larger number,

96 - 25 = 71.

This method of "difference splitting" also enables one to check results. The smaller number plus half the difference should equal the larger number minus half the difference.

Which method should one use to av- erage two numbers? The answer is, use the one that you are most comfortable with. In some cases, column averaging is faster; in other cases "difference splitting" is faster. Ideally, one should be familiar with all methods and able to recognize which method is best suit- ed for a given problem. Obviously, such facility is gained only with thoughtful practice.

Since there are alternative ways to approach many arithmetic problems, no one method should be used blindly; a quick inspection of the problem may reveal alternative routes for solu- tion, w

RavlacDlng and Uiaœing

Etcetera Edited by Roslynn Seidenstein Illinois State University Normal, Illinois

Disney's Basic Math Program. Six color filmstrips, six records or cassettes, six spirit masters, and a teacher's guide. $142. Walt Disney Educational Media Company, 500 South Buena Vista Street, Burbank, С А 91521.

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Prices on books and materials are included with reviews for general information only. Prices change and actual prices may vary from those listed. Please check with the individual publish- ers and producers.

December 1981

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