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them; add first the units' column, and mark below it the righthand figure of the amount; then carry to the tens' column the other figure or figures, if any (if there are none, then nothing is carried), and add it as before, including what has been brought from the units' column; put down below the tens' column the righthand figure of the amount, and carry the other figures, if any, to the hundreds' column; add the hundreds' column, including what has been brought from the tens' column; and so on till all the columns have been added. In adding the last column, put down below it all the figures of the amount, as the operation is now completed.
The meaning of this operation is, that the units are first added; the units are marked down, and the tens contained in the sum carried to the next or tens' column; the tens' column is then added, the tens marked down, and the hundreds contained in the sum carried to the hundreds' column; and so on. Example 1.-Add together 5, 3, 4, and 7.
In this example the numbers to be added together are all single numbers or units, and, being placed under each other, they form but one column; beginning at the lowest figure of which, we add it to the one above it, the sum of the two to the next above, and so on to the top;
the amount of them all is placed under the line. For 19 Sum example, in the present account, we begin by saying
7 and 4 are 11, 1] and 3 are 14, 14 and 5 are 19, which
is then placed under the line. Example 2.-Add together 27, 5, 536, 352, and 275.
In this example, beginning at the lowest figure of the right-hand column, we say 5 and 2 are 7, and 6 are 13, and 5 are 18, and 7 are 25—that is, 2 tens and 5 units ;
but since there are other columns to be added, we put 536
down the 5 only, under the units' column, and carry or 352 add the 2 tens to the lowest figure of the next column, 275 saying 2 and 7 are 9, and 5 are 14, and 3 are 17, and 2 are 1195
19. Here, as before, the 9 only is put down under the second column, and the 1 carried to the next; thus
1 and 2 are 3, and 3 are 6, and 5 are 11. No more figures remaining to be added, both these figures are now put down, and the amount or sum of them all is found to be 1195.
If the amount of any column be in three figures, still only the last or right-hand figure of the three is to be put down, and the other two carried to the next column. For example, if the amount of a column be 127, put down the 7 and carry the other two-viz. 12; if it be 234, put down the 4 and carry 23. Thus, all the figures except the last or right-hand one are to be carried to the succeeding column.
PROOF.—To prove that any question in addition is correctly wrought, we may add from top to bottom of the columns, and see if the sum be the same as by adding from bottom to top. Another method consists in striking off the upper row of figures, making
the addition without them, and then adding the top row to the sum. For example, treat the above question as follows:
Here the top row, 27, is struck off, and being after27
wards added to the product, the result is the same as if
all had been added up together; this proves that the 536 summation was properly performed. 352 Advice to Learners.—The art of adding up quickly is 275 considered a great accomplishment, and we strongly 1168
recommend young persons to acquire it with all reason
able diligence. It will save much time if they learn to 27
sum up the columns by a glance of the eye, without 1195 naming the numbers ; for instance, taking the above
question, instead of saying 5 and 2 are 7, 7 and 6 are 13, 13 and 5 are 18, 18 and 7 are 25, acquire the knack of summing up the figures in the mind ; thus—5, 7, 13, 18, 25.
Exercises. 1. 2. 3. 4. 5. 6. 7. 8. 9. 3 6 16 27 55 317 450 412 653 2 7 13 30 61 68 73 6 43 705 9 10 43 18 40 307 507 690 24
618 2 5 32
10. George has 13 marbles, John has 19, James 16, William 20, and Thomas 27. How many have they in all ? . . Ans. 95
11. Charles got several bundles of pens to count: he found in one bundle 24, in another 37, in another 53, in another 40, and in the last 17. How many were there in all ? . . Ans. 171
12. Add together 86, 5, 41, 7, 26, and 357, . . 1 522 13. Add 45, 60, 764, 37, and 78, . . .
984 14. Add 375, 460, 85, 67, and 43, . ,
1030 15. Add 4763, 2768, 437, and 8326, .
16294 16. How many do 735, 4628, 39, and 57 come to? ., 5459 17. How many are 85, 79, 632, and 781 ?
65. What is the amount of seventeen thousand three hundred and ten, five hundred and seven, two thousand four hundred and fifty, and fifty thousand one hundred and twelve ?. Ans. 70379
66. Add together four thousand two hundred and eleven, two thousand and forty, six hundred and twenty-seven, ninety-eight, and seven thousand nine hundred and three, . . Ans. 14879
54. 38499 55. 34643 56. 383034
63. 4157698 64. 5301997
SUBTRACTION is the taking or deducting of a smaller number from a greater, to find what remains, or what is the difference between them. The number left after deducting the one from the other, is called the remainder.
We subtract when we say, 3 from 5, and 2 remains. Here 2 is the difference between 3 and 5. If John has 5 marbles, and he gives James 3 of them, he will have 2 remaining.
Subtraction is denoted by a small horizontal line, thus [-] between two figures ; as, for example, 9 -- 5=4, means, 5 subtracted from 9, and 4 remains.
RULE FOR SUBTRACTING. 1. Place the smaller number below the greater, writing units under units, tens under tens, and so on, as in Addition.
2. Draw a line under them, and beginning at the right hand, deduct in succession each figure in the lower line from the figure immediately above it in the upper line, marking down the remainder below each figure.
3. If any figure in the lower line be greater than the figure above it, add 10 to the upper figure, and then go on with the subtraction. The 10 thus added is said to be borrowed from the next upper figure, and, as an equivalent for having borrowed it, the next under figure requires to be considered as 1 more before subtracting.
The reason that the 10 borrowed from the next figure is counted as 1 in making an equivalent allowance for it, is, that the figure from which 10 is borrowed is of a higher order or rank than that to which it is carried; and, consequently, 1 of the former is equal to 10 of the latter. The 10 borrowed is allowed for by making the next under figure 1 more, instead of making the next upper figure I less (as, strictly speaking, should be done), because it is more convenient in practice, and produces the same result. Example 1.-Take 325 from 537.
Here, 325 being the smaller number, it is placed 537
under 537, the greater, and commencing at 5, we 325
take 5 from 7, and the remainder is 2, which we
place below the line, directly under the 5; then 212 Remainder.
proceeding to the next figure, we say 2 from 3,
and I remains, which is also placed under the line; then going on to the next figure, we say 3 from 5, and 2 remains; this being placed below the line, the whole remainder or difference between the two numbers is found to be 212.