48 tained. 16 0 1 Sum of partial sums first answer obThus showing the work to be correct. NOTE. When long columns are to be added, it may sometimes be convenient to divide them in this way in performing the first addition. The student should, however, accustom himself to adding the longest columns without any separation into parts. Fourth Method. Beginning either at the right or at the left hand to add, write the sum of each denomination separately, and then add these sums together. Fifth Method.. ceed as follows: Begin with the left hand column, and pro PROOF OF THE EXAMPLE IN 50. By adding the ten-thousands column we find that its sum is 28; but as there are 31 ten-thousands in the answer first obtained, we infer that 3 ten-thousands were brought from the lower denominations. 3 ten-thousands = 30 thousands, which, added to the 7 written in the thousands' place of the answer, gives 37 thousands to be accounted for. The sum of the thousands column is 34, which, taken from 37, leaves 3; thus showing, that if the work is correct, 3 thousands must have been brought from the lower denominations. 3 thousands = 30 hun dreds, which, added to the 7 written in the hundreds' place of the answer, gives 37 hundreds to be accounted for. The sum of the hundreds column is 33, which, taken from 37, leaves 4; thus showing, that if the work is correct, 4 hundreds must have been brought from the lowe denomination. 4 hundreds = 40 tens, which, added to the 9 tens writ ten in the tens' place of the answer, gives 49 tens to be accounted for The sum of the tens column is 45, which, taken from 49, leaves 4; thus showing, that if the work is correct, 4 tens must have been brought from the units column. 4 tens 40 units, which, added to the 1 unit written in the units' place of the answer, gives 41 units to be accounted for. As the sum of the units column is 41, we infer that the work is correct. L PROOF OF THE EXAMPLE IN 51. By adding the pounds column, we find its sum is £43, which, taken from the £48 written in the answer, leaves £5; thus showing, that if the answer is correct, £5 must have been brought from the lower denominations. £5100 s., which, added to the 16 s. written in the answer, gives 116 s. to be accounted for. The sum of the shillings column is 111 s., which, subtracted from the 116 s., leaves 5 s.; thus showing, that if the answer is correct, 5 s. must have been brought from the lower denominations. 5 s. = 60 d., which, as there are no pence written in the answer, gives 60 d. to be accounted for. The sum of the pence column is 57 d., which, taken from 60 d., leaves 3 d.; thus showing, that if the answer is correct, 3 d. must have been brought from the column of farthings; 3 d. = 12 qr., which, added to the 1 qr. written in the answer, gives 13 qr. to be accounted for. As the sum of the farthings column is 13, we infer that the answer is correct. (b.) The first, second, and third methods of proof are the most practical, but as the fourth and fifth furnish valuable illustrations of the nature of the various changes and reductions, and call the reasoning faculties into healthful exercise, they should not be omitted by the student. (c.) If, by any of these methods, we obtain a different result from the one we first obtained, we may be sure there is an error in one or both operations, and should examine both carefully to find it. (d.) Some method of proof should always be resorted to, until the pupil acquires sufficient skill to be sure of the accuracy of his work without it. 54. Importance of Accuracy and Certainty. (a.) No person who is willing to allow an error to pass undetecte can be a good arithmetician. Accuracy, absolute accuracy, should be aimed at in every operation; and no labor is too great which is necessary to secure it. Not only should the results be accurate, but the computer should know for himself that they are so. If he has any doubt concerning a result, he should examine each and every step of his work, to see, First. That it was a proper one to take. Second. That it was taken at the right time. Third. That it was taken correctly. (b.) One problem thus solved and proved by a learner is of more real value to him than ten solved by him and proved by another, or tested by comparison with a printed answer. The accountant does not hesitate to spend hours, and even days, in looking over long and complicated accounts, to discover the cause of an error of a few cents in a trial balance sheet,* and surely the student ought not to shrink from the task of proving the correctness of his solutions of the much more simple problems contained in a school text book. (c.) Rapidity in the performance of numerical operations is scarcely of secondary importance to accuracy and certainty. The most accurate computers are usually the most rapid in their work. * The trial balance sheet is used in keeping books by double entry, as a means of determining whether any errors exist in the entries which have been made in some given time, as a month, a quarter, (i. e., three months,) six months, or a year. By its aid, the existence of an error may be ascertained; but the error itself cannot be discovered without examining the separate entries and accounts. An intelligent and highly accomplished accountant, who has charge of the books of a large manufacturing establishment, employing three hundred men, once spent nearly a week in examining his accounts, to discover the cause of an error of a few cents; and said he, "I never spent the same amount of time more profitably." Another gentleman, bearing also a high reputation, and receiving a good salary as an accountant, spent, to use his own language, "the greater part of four days in searching out the cause of an error of ten cents." Both these gentlemen say, that if they should adopt any other principle than that of absolute accuracy, they could not retain their situations. Every accountant, business man, and practical man bears similar testimony, and confirms these views. Indeed, most of them say, that the knowledge of arithmetic acquired in the school room has been of little practical value to them, because they did not learn to be accurate and rapid in performing their work, and to know for themselves that they had been accurate. 13. What is the sum of 679487 +386754 +329687 + 435429 +276834 + 579487 ? 14. What is the sum of 324067 +235143 +543345 + 425341 +876583 +947869 ? 15 What is the sum of $473.87 + $526.94 + $857.93 +$297.16+ $87.43 +$528.60 + $35.29 ? 16. What is the sum of 857436.57 + 25986.483 + 295463.867297484.253 +80672.005? 17. What is the sum of 258.647943 +547.685329 + 27.843729765.4837 + 736.852066 + 542.063794? 18. What is the sum of 984137.612 + 257.00684 + 43687.5792 + 574869.23757 +2068439.14238 + 1748.2 + 13.37 ? 19. What is the sum of 1864 + 437.29 + 58.697 + 12.86 +7527.385 + 167.97 + 848.396 + 4.584? 20. What is the sum of 389.40067 +2768.4372 + 5894.2761385.7281 ? 21. What is the sum of 728 + 436 + 549 +278 +367 + 825 ? 22. What is the sum of 426764572681 +894737629437 +179428630006 + 576428670639 +584967245876 ? 23. What is the sum of 3798643 +5978642 + 5489379 +6759863768543+27864+ 37987428957387 + 95837968395989 +3865372 ? 24. What is the sum of 83679 +54873 + 72352 + 9587387563590687506 +29764 + 38756 + 35742 ? 25. What is the sum of 57386 + 2864.3 + 379.86 + 28.6975.4738+.97986 +7.5983 +86.794 + 886.79 +2937.670003 + 9764.2 +859.86 +48.375 ? 26. What is the sum of $8.69 + $13.48 + $4.48 + $8.64 $37.15 + $47.13 + $.86 + $.25 + $9.37 + $6.08 + $3.54 + $7.06 + $2.37 + $4.68 + $20.08 +$7.57 + $7.48 ? 27. What is the sum of $4.175 + $3.867 + $5.384 -+$9.375 $5.78 +$8.378+ $2.635+ $.875+ $1.25 + |