SUPPLEMENTARY DRILL EXERCISES

132. Some computers check addition, subtraction, multiplication, and division by casting out nines. First you must learn how to cast out nines. Example 1. Consider the number 3786. 3+7+8+6=24, and 24 = (2×9)+6.

18 is the largest multiple of 9 which is less than 24.
6 is the remainder when 18 is subtracted from 24.
6 is called the excess of 3786 after casting out nines.

Example 2. - Find the excess of 285.

2+8+5=15=9+6.

.. the excess of 285 is 6.

Example 3. - Find the excess of 232.

2+3+2=7.

Since 7 is less than 9, 7 is the excess of 232.

Example 4.

Find the excess of 882.

8+8+2=18= (2×9)+0.
.. the excess of 882 is 0.

- Checking addition by casting out nines.

α.

Since the excess of 14 equals the excess of 1958, the sum is probably correct.

That is, find the excess of each addend. Add these excesses and find the excess of the result. If this final excess equals the excess of the sum of the addends, the addition is probably correct.

Since the excess of 17 equals the excess of 19,349, the addition is probably correct.

b.-Checking subtraction by casting out nines.

The excess of 156, the same as the excess of the minuend.

That is, add the excesses of the subtrahend and the remainder and find the excess of the result. If this equals the excess of the minuend, the result is probably correct.

The excess of 9=0, the same as the excess of the minuend.
Therefore the subtraction is probably correct.

The excess of 6 = 6. Since this equals the excess of 1,636,755, the multiplication is probably correct.

That is, find the excesses of the multiplier and multiplicand, and find their product. Obtain the excess of this result. If this final excess equals the excess of the product of the multiplicand and multiplier, the multiplication is probably correct.

the multiplication is probably correct.

d.-Checking division by casting out nines.

That is, multiply the excess of the divisor by the excess of the quotient. To the result, add the excess of the remainder. Find the excess of the sum. If this equals the excess of the dividend, the solution is probably correct.

133. The following exercises should be used according to the teacher's judgment of the needs of the class and

of individuals in the class. Many teachers spend a few minutes (e.g. five minutes) daily (or frequently) on such drill work for the whole class, and then assign special practice for the pupils who are found to be slow or inaccurate. For convenience in making assignments, the exercises are grouped according to the various processes. If a miscellaneous exercise is wanted, select one part from each of a number of exercises; e.g. one from each exercise.

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375

554

SUGGESTED EXERCISES

(Do not do two consecutive ones on the same day.)

1. Hold a slip of paper below Row 2. Write upon it the sum of the two numbers in each column that appear above the paper.

2. Hold a slip of paper below Row 3. Write upon it the sum of the three numbers in each column that appear above the paper.

3. Similarly, obtain the sum of the numbers in each column that are in other sets of consecutive rows. (For example, the numbers in the third to the seventh rows

inclusive. Hold a slip of paper below the seventh row and above the third row in this case.)

4. Speed and accuracy tests.

a. Hold a slip of paper below Row 10. Upon it write the sum of the numbers in each column that appear above the paper. Obtain the median time as described in Book I, page 212.

b. Courtis type tests and standard. Any who wish to compare with the Courtis Standard scores may proceed as follows:

Ask the pupils to find the sum of the numbers in each column that are in the first nine rows, allowing them only eight minutes in which to add. At the end of the allotted time, stop them.

Median speed. - Have each pupil count the number of examples tried. Direct all who have tried only 1, only 2, only 3, etc., to stand until one half the class is standing. The number of examples tried by the last pupil to rise is the median speed of the class on this test, for the number of minutes allowed for the test.

Median accuracy. Have each pupil note the number of examples tried and the number solved correctly. Have each pupil find what per cent of the number tried by him or her were solved correctly; this is the accuracy mark of that pupil. Direct all pupils whose mark is 100, 99, 98, 97, etc., to stand until one half the class is standing. The mark of the last pupil to rise is the median accuracy mark of the class.

The median speed for grade eight is twelve such examples in eight minutes.

The median accuracy in one school system is about