20. It is helpful also to know combinations, or groups 1 2 3 4 5 that form certain numbers. Thus, 5 4 and 9 8 7 6 5 4 3 3, etc., are groups that form 10, and 541I 2 3 8 1 2 2 3 2 3 4 5 4 5 7 are groups that form 20. 6 6 21. Such groups should be carefully studied and practiced until the student readily recognizes them in his work. He should also familiarize himself with other groups. The nine-groups and the eleven-groups are easy to add, since adding nine to any number diminishes the units' figure by one, and adding eleven increases the units' and the tens' digits each by one. EXERCISE 3 1. Begin with 8 and add 7's till the result is 50. 2. Begin with 3 and add 8's till the result is 67. Form the following sums till the result exceeds 100: 3. Begin with 3 and add 7's. 4. Begin with 7 and add 8's. 5. Begin with 5 and add 9's. 6. Begin with 8 and add 5's. Add the following columns, beginning at the bottom, and check the results by adding downward. Form such groups as are convenient and add them as a single number. In the first two exercises groups are indicated. In commercial operations it is sometimes convenient to add numbers written in a line across the page. If totals are required at the right-hand side of the page, add from left to right and check by adding from right to left. Add: 17. 23, 42, 31, 76, 94, 11, 13, 27, 83, 62, 93. 18. 728, 936, 342, 529, 638, 577, 123, 328, 654. 4 55 679 Find the sum of the following numbers by adding the columns and then adding the results horizontally. Check by adding the rows horizontally and then adding the columns of results. Exercises for further practice in addition can be readily supplied by the teacher. The student should be drilled till he can add accurately and rapidly. Accuracy, however, should never be sacrificed to attain rapidity. Expert accountants, by systems of grouping and much practice, acquire facility in adding two or even three columns of figures at a time. Elaborate calculating machines have also been invented, and are much used in banks and counting offices. By means of these machines, columns of numbers can be tabulated and the sum printed by simply turning a lever. SUBTRACTION 22. In subtraction it is important that the student should be able to see at once what number added to the smaller of two numbers of one figure each will produce the larger. Thus, if the difference between 5 and 9 is desired, the student should at once think of 4, the number which added to 5 produces 9. 23. Again, if the second number is the smaller, as in 7 from 5, the student should think of 8, the number which added to 7 produces 15, the next number greater than 7 which ends in 5. 24. The complete process of subtraction is shown in the following exercise : 8534 5627 2907 7 and 7 are 14, carry 1. (Why carry 1?) 6 and 9 are 15, carry 1. 6 and 2 are 8. 25. The student should think "What number added to 5627 will produce 8534?" After a little practice, it is unnecessary to say more than 7 and 7, 3 and 0, 6 and 9, 6 and 2, writing down the underscored digit just as it is named. 26. Check. To check, add the remainder and the subtrahend upward, since in working the exercise the numbers were added downward. 27. The above method of subtraction is important not only because it can be performed rapidly, but because it is very useful in long division. It is also the method of "making change" used in stores. 28. There are two other methods of subtraction in The processes are shown in the following common use. exercises: 1. Define the terms subtrahend, minuend, difference. 2. How should the terms be arranged in subtraction? Where do we begin to subtract? Why? 3. Is the difference affected by adding the same number to both subtrahend and minuend? Is this principle used in either (1) or (2)? 4. If a digit in the minuend is less than a digit of the corresponding order in the subtrahend, explain how the subtraction is performed in both (1) and (2). 29. Arithmetical Complement. The arithmetical complement of a number is the difference between the number and the next higher power of 10. Thus, the arithmetical complement of 642 is 358, since 358 +642=1000. The arithmetical complement of 0.34 is 0.66, since 0.66 +0.34 = 1. |