18. 10, and 9, and 8, and 7, and 6, and 5, and 4, and 3, and 2, are how many? 19. 11, and 10, and 9, and 8, and 7, and 6, and 5, and 4, and 3, are how many? 20. 12, and 11, and 10, and 9, and 8, and 7, and 6, and 5, and 4, and 3, and 2, and 1, are how many? 21. Add each of these rows of figures, first from the bottom to the top, then from the top to the bottom, then from right to left, and then from left to right. 1 2 3 4 5 6 7 8 9 [This addition should be made aloud, as should every other calculation in the mental exercises.] 22. How many are 9 and 2? 19 and 2? 29 and 2? 39 and 2? 49 and 2? 59 and 2? 69 and 2? 79 and 2? 89 and 2? 99 and 2? [The teacher should here take 9 and 3, 29 and 3, and so on; after which, he should begin 8 and 2, 18 and 2, &c.; 8 and 3, 18 and 3, &c.; 7 and 2, &c.; 7 and 3, &c., as far as 2 and 2, 12 and 2, 22 and 2, &c.; 1 and 2, &c., up to 91 and 9. This is intended for inexperienced pupils. A few questions are given below, for those in every stage of improvement.] 23. How many are 6 and 6? 16 and 6? and 6? 36 and 7? 46 and 7? 56 and 8? and 8? 86 and 9? 96 and 9? 26 and 6? 36 76 and 8? 96 24. How many are 25 and 9? 35 and 9? 45 and 8 ? 55 and 8? 65 and 7? 75 and 7? 85 and 4? 95 and 4? 25. How many are 3 and 4? 13 and 4? 23 and 4? 33 and 4? 43 and 5? 53 and 5? 63 and 6? and 8? 93 and 9? 104 and 10? 26. How many are 4 and 2? 34 and 2? 44 and 3? 54 and 4? 84 and 7? 94 and 8? 104 and 9? 27. How many are 7 and 4? 37 and 5? 47 and 6? 87 and 9? 97 and 9? 28. How many are 38 and 4? 48 and 5? 88 and 9? 98 and 9? 73 and 7? 83 14 and 2? 24 and 2? 64 and 5? 74 and 6? 17 and 4? 27 and 5? 57 and 6? 67 and 7? 77 and 8 ? 18 and 3? 28 and 4 ? 68 and 7? 78 and 7 ? 118 and 8? 8 and 2? 58 and 5? 108 and 8? 29. Add these rows of figures, from left to right, and write down the amount of each row. 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 55 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 7 7-7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 999999999999 2 2 3 3 4 4 5 5 6 6 7 7 § VIII. 1. Copy these rows of figures on your slate, add them up, and set down the amount in figures. 8 3 4 5 4 6 4 5 3 1 6 8 2. In the last example you added rows of units, and set down the amount. Add, now, these rows of tens, and hundreds, and thousands, &c., in the same manner. 3000 90000 500000 7000000 10 - 100 3. Here are some simple examples, containing different orders. 70 4. The following examples are similar. 400 70 8000 70000 6 60 4 700 3046 60000 700000 3040 5. The following are larger, but are to be added in the [The learner will readily understand and perform the above examples. If however, he should hesitate, let him study, or let the teacher repeat to him, the following explanation.] In the third row of examples above, 1 ten is added to 1 unit. The ten is a unit of the second order, and should therefore stand in the second place, and the unit is a unit of the first order, and ought to be written in the first place. We will take away, therefore, the cypher on the right of the ten, and in its place will put the unit. All of the first row of examples are performed in this way, and the same explanation will apply to them. In the fourth row of examples, 70 or 7 tens are added to 6 units, and 1 unit. The 6 units, and 1 unit, make 7 units, which are written, as before, in place of the 0 in 70. The others are all similar. [The teacher will perceive, that, thus far in the examples containing several columns, no one column added up, has exceeded 9. He should require his pupils to explain the addition of every column, in the manner given above, using the name of the order he is adding. Thus, in the last example, he should begin 1 unit and 8 units are 9 units, which are to be written in the units' place, in the answer. 1 ten and 1 ten are two tens; 2 tens and 7 tens are 9 tens, to be written in the tens' place in the answer, &c.] It is plainly of no consequence on which side you begin to add the examples above, since each column is added separately. It is better, however, to begin on the right hand, as will presently be seen. It is, also, very manifest, that as all the units are to be added together, all the tens together, &c., it is most convenient to write the numbers so that units shall stand under units, and tens under tens, &c. The pupil should be very careful never to add the figures of different orders together; for 3 units and 2 tens, are neither 5 units, nor 5 tens. But 3 units and 2 units are 5 units; and 3 tens and 2 tens are 5 tens. 6. Write down 15 and 6, and add them. 15 .6. Set this down. Bring it down 1 ten and 1 ten First, add the units together. 5 units and 6 units are 11 units; that is, 1 ten and I unit, written 11. 11 There is one ten in the upper number. under the tens' place. Now add the tens. are 2 tens, or twenty. Join this to the 1 unit, and it is 21. 1 Ans. 21 7. Write 17 and 26, and add them. the tens. First, add the units. 7 units and 6 units are 13 units; equal to 1 ten and 3 units. Set this down. Now add 1 ten and 2 tens are 3 tens. Set these under Now add again. 3 tens and 1 ten are 4 tens ; which, joined with the 3 units, make 43. the tens. 8. Let the following examples be performed and explained in the The learner will see, that, when his units are more than 9, it is impossible to write them in the units' place; because there are no characters to express them. Every even ten, therefore, he is obliged to add with the tens. And if there are any units over, he sets them down in the units' place. Of course, after he has added his units column, a part of the amount goes to the units' place, and a part to the tens' place. In the examples above, these two parts are both set down. But this is inconvenient, because it renders, at least, two additions necessary. The following process is better, 10. Add 54 and 39. 54 9 units and 4 units are 13 units; that is, 1 ten and 3 units. 39 The 1, is the part that belongs to the tens' place. Therefore, in mind, add it with the tens' column. Thus, 1 ten and 3 tens 93 are four tens; 4 tens and 5 tens are 9 tens. The answer is 93. In the same manner, we add 1 to the hundreds' column, for every ten tens, 1 to the thousands' column for every ten hundreds, and 1 to any column, or order of figüres, for every ten of the next lower order. THIS ADDITION OF ONE FOR EVERY TEN TO THE NEXT HIGHER ORDER IS CALLED CARRYING. If we were to write down the whole amount of any column, whose sum is more than 9, as in the examples above, it would take at least two figures, and sometimes more. Of these, it is plain, that the one on the right hand is the one to be set down, and the other, or others, the number to be carried, Thus: 11. Add 375, and 463, and 999, and 888. 375 463 999 888 2725 The amount of the units is 25; of course, we set down the 5, and carry the 2. There are 32 tens. Therefore, we set down 2, and carry 3. There are 27 hundreds. We therefore set down 7, and carry 2. As there is nothing to add it to, we put it in the next place, by itself. [The pupil should, frequently, be exercised in explaining operations in addition, in the manner of the preceding illustrations.]. The following are to be performed, in the manner of the preceding. 12. In an orchard, 19 trees bear cherries, 28 bear peaches, 8 bear plums, and 58 apples. How many trees in all ? Ans. 113. 13. Four men purchased a field. The first gave 74 dollars; the second, 67; the third, 41; and the fourth, 27. How much did all give? Ans. 209. 14. How many times does the hammer of a clock strike in 24 hours, if it strikes regularly from 1 to 12, and then from 1 to 12 again? Ans. 156. 15. How many days in the Spring months, containing March, 31 days; April, 30; and May 31? Ans. 92. 16 How many in the Winter months, containing December, 31; January, 31; February, 28 ? Ans. 90. 17. How many in the Summer and Autumn months, containing June, 30; July, 31; August, 31; September, 30; November, 30? 18. Then, how many days in the year ? October, 31; Ans. 365. 19. If it require 650 men to man a 74 gun ship; 475, to man a 44; 350, to man a 36; 275, to man a 32; 200, to man a 20; and 180, to man an 18; how many men will it require to man the whole ? Ans. 2,130. 20. A man spent 30 years in the United States, 5 years in France, 12 years in Italy, 7 years in Germany, 4 years in the Netherlands, and afterwards returned to the United States, where he lived 26 years. How old was he at his death? Ans. 84. The pupil is now prepared to understand the following rule. I. WRITE DOWN THE NUMbers to be aDDED, SO THAT THE SAME ORDERS IN EACH, MAY STAND IN THE SAME COLUMN, UNDER EACH OTHER. |