REMARKS. Addition and Subtraction are the bases of all operations in Arithmetick. This is evident from the following fact, that we cannot alter a number, unless we make it either larger or smaller. Neither can we alter a quantity, as it respects the amount of quantity, for Arithmetick considers quantity in no other respect, without making it more or less. But if we increase a number or a quantity we add to it; and if we lessen a number or quantity, we subtract from it. ADDITION OF SIMPLE NUMBERS. A number is called simple when all the places of figures represent the same kind of quantity, or when ten in any one place is equal to one in the next left hand place; thus, 651 dollars, 723 years, are simple numbers. A given number is a number mentioned with which something is to be done. The answer in addition is called the sum or amount. The Addition of Simple Numbers is putting together two or more simple numbers, so that all of them may be expressed by one number. Small numbers can be added in the mind. Thus, we say, 2 put with 4, make 6; 1 put with 7, make 8; 7 put with 3, make 10; 9 put with 4, make 13; 11 put with 5, make 16; 5 put with 13, make 18. Let the pupil answer the following questions : 6 added to 3, make how many ? 6 added to 2, make how many ? 7 added to 2? 7 added to 4 ? 7 added to 5? 6 added to 7 ? 6 added to 9 ? 6 added to 4? 10 added to 1 ? 10 added to 4 ? 5 added to 10? 5 added to 11 ? 5 added to 12 ? 6 added to 12? What is the sum of 6 and 14 ? of 7 and 14 ? of 8 and 13 ? of 3 and 14 ? of 2 and 15 ? of 4 and 16? of 15 and 5 ? of 7 and 14 ? of 6 and 15? of 8 and 16 ?. of 7 and 17 ? of 9 and 15 ? of 19 and 5 ? RULE For stating Questions in Addition of Simple Numbers. Set down the given numbers so that every figure may stand directly under one of its own local name; that is, units under units, tens under tens, hundreds under hundreds, &c. State the following numbers for adding: 16'709, 87'634, 10'245, 1'175'302. They will stand thus, 1 6 7 09 1 0 2 4 5 RULE I, For adding simple numbers. Add up each column by itself, commencing at the right hand, and set the sums underneath. EXAMPLES. 3 3.2 1 Explanation of the operation. Given 2 1 3 3 numbers, In the first place, we addup the 1 3 1 2 column of units, saying, 2 and 3 are 5, and I are 6. We then Sum 6 7 6.6 1 add the column of tens, saying, and 3 are 4, and 2 are 6; next, the column of hundreds, 3 and I are 4, and 3 are 7; lastly, the column of thousands, 1 and 2 are 3, and 3 are 6. (2.) 1 0 1 2 3 1 Explanation. As ciphers do not express 32'45 3 any number, in adding we have nothing to 26°114 do with them when they stand in a lumn with other figures. 6 877 08 The pupil is required to state and work the following questions. 6. A man has two casks of grain ; one contains 12 bushels, the other 21: How many bushels in both ? Ans. 33. 7. A mán bought a yoke of oxen for 56 dollars, and a horse for 43 dollars : How many dollars did he give for both ? Ans. 99. 8. The eldest of three brothers has 102 cents : the second 75: and the third 22: How jaany cents have the three? Ans. 199. 9. A boy goes to a store and buys a yard of cloth for 45 cents : a book for 21 : and a quart of molasses for 12: How many cents did he give for all the articles ? Ans. 78, 10. What number of dollars are there in three bags : the first containing 1201 dollars : the second 3486: the third 2210 ? Operation. 1 2 0 1 added together : for by adding up Sum 6 8 9 7 the several columns, we set a figure in the sum expressing as many units, tens or hundreds, &c. as there are units, tens or hundreds, &c. in all the given numbers. In the operation of the last question, 7 expresses the same number of units as 6 and 1 in the given numbers; the 9 tens, as many tens as there are tens in the given numbers ; 8 hundred, as many hundreds as are represented by all the given numbers : and the 6 thousand denotes as many thousands as the thousands contained in the given numbers, therefore 6 thousand, 8 hundred, 9 tens and 7 units, or 6897, express a number equal to 1201, 3486 and 2210. If it were required to add the quantities which these given numbers express, we should put the dollars contained in the three bags into one bag, and the 6897 represents the number of dollars which the one bag would contain. an ex When the sum of any column is ten, or act number of tens, set a cipher under that column, and add one for each ten to the next column ; but if the sum be more than an exact number of tens, set down what there is over the tens and add the tens as before. (11.) 6 7 2 Explanation. In this example, because 9 3 3 there is just ten in the right hand column, 8 7 3 we set down a cipher, and add the 1 ten 9 6 2 to the column of tens. The sum of the tens in the second column, with the one 3 4 4 0 ten brought from the units' place, is 24. There being 2 tens in 20, we set down the 4 and add the two tens to the next column. The third column consists of 32, and the 2 brought from the tens' place make 34 ; we set down the 4, what there is over 3 tens, and as there is no other column, we set the 3 tens in the next place. (12.) (13.) (14.) 6 7 6 9 1 2 ng OO 1 2 9 O O O O 1 2 4 5 3 6 1 2 3 4 5 8 7 0 0 0 7 6 8 9 0 2 9 8 7 6 5 7 6 5 0 0 1 2 3 4 5 6 8 7 6 5 4 6 5 4 3 0 5 4 3 2 1 1 6 9 3 8 0 6 2 6 8 7 7 6 3 7 3 2 5 1 PROOF. Begin at the top and add together all the columns of figures: and if the sums are the same as those produced when added from the bottom, the work is supposed to be correct. Remark. No method of proving the operation of a question, will demonstrate the work to be correct, for the same error may be made in both operations ; but there are methods of rendering the accuracy of all operations in Arithmetick more probable. The scholar is required to prove the following questions. (15.) (16.) (17.) 6 7 6 9 1 2 2 0 6 7 6 1 3 4 3 7 1 2 4 4 98 6 7 0 2 8 39 1 7 8 9 8 18. What is the sum of the following numbers; 67601, 87654, 1206 and 10101? Ans. 166562. 19. A certain farm is divided into 4 lots; the first containing 97 acres; the second, 108; the third, 70; and the fourth, 39. How many acres in the farm ? Ans. 314. 20. What is the amount of twenty-five, two thousand five hundred, twenty-five thousand, and two hundred fifty thousand ? Ans. 277525. 21. A merchant imported a bale of broadcloths containing 4 pieces, which measured as follows; the first, 31 yards; the second, 29; the third, 27; and the fourth, 32. How many yards of broadcloth were there in the bales. Ans. 119. 22. A butcher killed an ox, which weighed as follows; the two hind quarters, two hundred and eighty. pounds; one of the fore quarters, one hundred and thirty-five pounds; the other, one hundred and forty-one pounds ; the hide, eighty-nine ; and the tallow, ninety ; what was the whole weight of the ox? Ans. 735. 23. A gentleman owns 4 farms; the first is worth 965 dollars; the second, 3579; the third, 4018; the fourth, 5617; how much are they all worth? |