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ART. 49. Ex. 1. A man bought three lots of land; the first contained 23 acres, the second 9 acres, and the third 15 acres : how many acres did he buy?
Solution.-23 acres and 9 acres are 32 acres, and 15 are 47 acres. Ans. 47 acres.
OBS. It will be seen, that the solution of this example consists in finding a single number, which will exactly express the value of the several given numbers united together.
50. The process of uniting two or more numbers together, so as to form a single number, is called ADDITION.
The answer, or the number thus found, is called the Sum or Amount.
OBS. When the numbers to be added are all of the same denomination, as all dollars, all pounds, &c., the operation is called Simple Addition.
Ex. 2. A miller bought 7864 bushels of wheat of one man, 4952 bushels of another, and 3273 bushels of another: how many bushels did he buy of all ?
Ans. 16089 bu.
Write the numbers under each other, so that units may stand under units, tens under tens, &c., and draw a line beneath them. Then beginning at the right hand or units, add each column separately. Thus, 3 units and 2 units are 5 units, and 4 are 9 units. Write the 9 in units' place under the column added. Next 7 and 5 are 12, and 6 are 18 tens. But 18 requires two figures to express it; (Art. 34;) consequently it cannot all be written under its own column. We therefore write the 8 or right hand figure in tens' place under the column added, and reserving the 1 or left hand figure, add it with the hundreds. Thus, 1 which was reserved, and 2 are 3, and 9 are 12, and 8 are 20 hundreds. Set the 0 or right hand figure under the column
QUEST.-50. What is Addition? What is the answer called? Obs. When the num bers to be added are all of the same denomination, what is the operation called? 51. What orders of figures do you add together?
added, and reserving the 2 or left hand figure, add it to the next column as before. Thus, 2 which were reserved and 3 are 5, and 4 are 9, and 7 are 16 thousands. Set the 6 under the column added; and since there is no other column to be added, write the 1 in the next place on the left.
51. It will be perceived in this example, that units are added to units, tens to tens, &c.; that is, figures of the same order are added to each other. All numbers must be added in the same manner. For, figures standing in different orders or columns express different values; (Art. 35;) consequently, they cannot be united together directly in a single sum. Thus, 3 units and 5 tens will neither make eight units, nor eight tens, any more than 3 oranges and 5 apples will make 8 apples, or 8 oranges. In like manner it is plain that 7 tens and 2 hundreds will neither make 9 tens, nor 9 hundreds.
OBS. The object of writing units under units, tens under tens, &c., is to prevent mistakes which might occur from adding different orders to each other
52. When the sum of a column does not exceed 9, it will be noticed, we set it under the column added but if it exceeds 9, we set the units or right hand figure under the column added, and reserving the tens or left hand figure, add it to the next column. In adding the last column on the left, we set down the whole sum.
OBS. The process of reserving the tens, or left hand figure, and adding it to the next column, is called carrying tens.
53. The principle of carrying may be illustrated in the following manner.
Take, for instance, the last example, and adding as before, write the sum of each column in a separate line. Thus, the sum of the units' column is 9 units; the sum of the tens' column is 18 tens, or 1 hundred and 8 tens; the sum of the hundreds' column is 19 hundred, or 1 thousand 9 hundred; the sum of the
9 sum of units
QUEST.-Why not add figures of different orders together?
thousands' column is 14 thousand. Now, adding these results together as they stand, units to units, tens to tens, &c., the amount is 16089 bushels, which is the same as in the solution above.
Thus, it is evident, when the sum of a column exceeds 9, the right hand figure denotes units of the same order as the column added, and the tens or left hand figure denotes units of the next higher order. Hence,
The reason we carry the tens or left hand figure to the next column, is because it is of the same order as the next column, and figures of the same order must always be added together. (Art. 51.)
OBS. 1. The reason for setting down the whole sum of the last or left hand column, is because there are no figures in the next order to which the left hand figure can be added. It is, in fact, carrying it to the next column.
2. From the preceding illustration it will also be seen, that the object of beginning to add at the right hand is, that we may carry the tens, as we proceed in the operation.
54. From the preceding illustrations and principles we deive the following
GENERAL RULE FOR ADDITION.
I. Write the numbers to be added, under each other; so that units may stand under units, tens under tens, &c. (Art. 51. Obs.)
II. Begin at the right hand, and add each column separately. When the sum of a colu.nn does not exceed 9, write it under the column; but if the sum of a column exceeds 9, write the units' figure under the column added, and carry the tens to the next column. (Arts. 52, 53.)
III. Proceed in this manner through all the orders, and set down the whole sum of the last or left hand column. (Art. 53. Obs.)
55. PROOF.-Beginning at the top, add each column downwards, and if the second result is the same as the first, the work is supposed to be right.
QUEST.-54 How do you write numbers to be added? Why place units under units, &c.? Where do you begin to add? When the sum of a column does not exceed 9, what do you do with it? When it exceeds 9, how proceed? What is meant by carrying the tens? Why carry the tens to the next column? Why begin to add at the right hand? What do you do with the sum of the last column? 55. How is addition proved?
Note.-The object of beginning at the top and adding downwards, is that the figures may be taken in a different order from that in which they were added before. The order being reversed, the presumption is, that any mistake which may have been made will thus be detected; for it can hardly be supposed that two mistakes exactly equal will occur.
56. Second Method.-Cut off the bottom line, and find the sum of the rest of the numbers; then add this sum and the bottom line together, and if the second result is the same as the first, the work is supposed to be right.
Note.-1. This method of proof depends on the axiom, that the whole of a quantity is equal to the sum of all its parts. (Ax. 11.)
2. The method of cutting off the top line, and afterwards adding it to the sum of the others, is objectionable on account of adding the numbers in the same order as they were added in the solution. (Art. 55. Note.)
57. Third Method.-From the amount, subtract all the given numbers but one, and if the remainder is equal to the number not subtracted, the work may be supposed to be right.
Note.—This method supposes the pupil to be acquainted with subtraction, before he commences this work. It is placed here on account of the convenience of having all the methods of proving the rule together.
58. Fourth Method.*-Cast the 9s out of each of the given numbers separately, and place each excess at the right of the number. Then cast the 9s out of the sum of these excesses; also cast the 9s out of the amount; and if these two excesses are equal, the work may be supposed to be right.
Note.-1. This mode of proof is based on a peculiar property of the number 9. For its illustration and demonstration, see Art. 161. Prop. 14.
2. To cast the 9s out of a number, begin at the left hand, add the digits together, and, as soon as the sum is 9 or over, drop the 9, and add the remainder to the next digit, and so on. For example, to cast the 9s out of 4626357, we proceed thus: 4 and 6 are 10; drop the 9 and add the 1 to the next figure. 1 and 2 are 3, and 6 are 9; drop the 9 as above. 3 and 5 are 8, and 7 are 15; dropping the 9, we have 6 remainder.
EXAMPLES FOR PRACTICE.
59. Ex. 1. A man paid 2468 dollars for his farm, 1645 dollars for a house, 865 dollars for stock, and 467 dollars for tools: how much did he pay for the whole?
2. A produce merchant bought 5 cargoes of corn; the first con
QUEST.-Note. Why add the columns downwards, instead of upr¬rds? Cai, addition be proved by any other methods?
"Wallis' Arithmetic, Oxford, 1657.
tained 6725 bushels, the second 7208, the third 5047, the fourth 12386, and the fifth 10391 bushels: how many bushels did he buy?
3. A tavern-keeper bought six loads of hay which weighed as follows: 1725 pounds, 2163 pounds, 1581 pounds, 1908 pounds, 2340 pounds, and 1879 pounds: what was the weight of the whole?
4. A man gave 5460 dollars to his oldest son, to the next 4065, to the next 6750, to the next 8000, and to the youngest 7276 dollars: how much did he give to all?
5. A merchant, on settling up his business, found he owed one creditor 176 dollars, another 841 dollars, another 1356 dollars, another 2370 dollars, another 840 dollars: what was the amount of his debts?
6. The state of Maine contains 32400 square miles; New Hampshire, 9500; Vermont, 9700; Massachusetts, 7800; Rhode Island, 1251; and Connecticut, 4789: how many square miles are there in the New England States?
7. The state of New York contains 46220 square miles; New Jersey, 7948; Pennsylvania, 46215; and Delaware, 2068: how many square miles are there in the Middle States?
8. The state of Maryland contains 10755 square miles; Virginia, 65700; North Carolina, 51632; South Carolina, 31565; Georgia, 61683; Florida, 56386; Alabama, 54084; Mississippi, 49356; Louisiana, 47413; and Texas, 100000: how many square miles are there in the Southern States?
9. The state of Tennessee contains 41752; Kentucky, 40023; Ohio, 40500; Michigan, 60537; Indiana, 35626; Illinois, 56506; Missouri, 70050; Arkansas, 54617; Iowa, 173786; and Wisconsin, 92930: how many square miles are there in the Western States?
10. What is the whole number of square miles in the United States?
11. What is the sum of 75234+41015+19075+176+88350 +10040?
12. What is the sum of 250120+30402+7850+465000+ 10046+65045 ?