FIRST OPERATION. SECOND OPERATION. In the first operaBushels. Bushels. tion, the several subMinuend 6 57 Minuend 657 trahends are added for a single subtrahend to 141 141 be taken from the 2 44 2 44 minuend. In the sec134 134 ond, the subtrahends Subtrahend 519 Remainder 138 are subtracted as they are added, at one operRemainder 138 ation, thus: beginning with units, 4 and 4 and 1 9, which from 17 units leaves 8 units; passing to tens, 1 (carried) and 3 and 4 and 4 : = 12 tens; reserving the left-hand figure to add in with the figures of the subtrahends in the next column, the right-hand figure, 2 tens, which we subtract from the 5 tens of the minuend, and have left 3 tens; and, passing to hundreds, we add in the left-hand figure 1, reserved from the 12 tens, which with the other figures 1 and 2 and 1 = 5 hundreds, which, taken from 6 hundreds, leaves 1 hundred; and 138 is the answer sought. 2. John Drew has a yearly income of 2,500 dollars ; his family expenses are 1,300 dollars, his expenditures in improving his estate 450 dollars, and his contributions to several worthy objects 225 dollars. What remains to lay up or invest? 3. A speculator bought four village lots; for the first he paid 620 dollars; for the second, 416 dollars; for the third, 350 dollars; for the fourth, 225 dollars; and sold the whole for 2,000 dollars. What did he gain ? 4. Daniel White, dying, left property to the amount of 27,563 dollars, of which his wife received 9,188 dollars, each of his two daughters, 4,594 dollars, and his only son the bal What did his son receive ? Ans. 9,187 dollars. 5. The United States contain 2,983,153 square miles, of which the Atlantic slope includes 967,576, the Pacific slope 778,266, and the Mississippi Valley the remainder. How many square miles does the Mississippi Valley contain ? Ans. 1,237,311. 6. The British North American Provinces contain 3,125,401 square miles; of which 147,832 square miles belong to Canada West; 201,989 to Canada East; 27,700 to New Brunswick ; 18,746 to Nova Scotia ; 2,134 to Prince Edward's Island; 57,000 to Newfoundland ; 170,000 to Labrador ; and the re ance. mainder to the Hudson's Bay Territory. What number of square miles belong to the Hudson's Bay Territory? Ans. 2,500,000. 7. James Howe has property to the amount of 63,450 dollars, and owes in all three debts; one of 1000 dollars, another of 350 dollars, and another of 12,468 dollars. How much has he after paying his debts? 8. The entire coinage of the mint of the United States, including the coinage of its branches, from 1792 to 1856, amounted in value to $ 498,197,382, of which $ 396,895,574 was gold, $ 100,729,602 silver, and the remainder of the amount copper. What was the value of the copper coinage ? Ans. $ 572,206. MULTIPLICATION. 54. MULTIPLICATION is the process of taking one number as many times as there are units in another number. In multiplication three terms are employed, called the Multiplicand, the Multiplier, and the Product. The multiplicand is the number to be multiplied or taken. The multiplier is the number by which we multiply, and denotes the number of times the multiplicand is to be taken. The product is the result, or number produced by the multiplication. The multiplicand and multiplier together are called FACTORS, from the product being made or produced by them. When the multiplicand consists of a simple number, the process is termed Multiplication of Simple Numbers. In the following table, the invention of Pythagoras, may be found all the elementary products necessary in performing any operation in multiplication, since the multiplication of numbers, however large, depends upon the product of one digit by another. The products, therefore, of each digit by any other, should be thoroughly committed to mem emory. Considerable more of the table, even, may be memorized with fully compensating results. find, where the lines intersect, the same result. may look for the 7 at the left hand, and the 5 at the top, and and where the lines intersect is 35, the number sought ; or, we we look for 7 at the top of the table, and for 5 at the left hand, For example, suppose we wish to find the product of 7 by 5; 11_21 31 41 51 61 71 81 91 101 11 12 13 14 15 16 17 18! 191 201 21 22 23 24 25 21 41 61 81 101 121 141 161 181 20 22 24 26 28 30 32 34 361 38| 40| 42| 441 46 48 50 361 91 12| 15| 181 21| 24 | 271 301 33| 36| 39| 42| 45 | 481 511 541 571 601 63/ 66|69| 72 | 75 41 81 121 16| 201 241 28 32 36| 40| 44! 48 52 56 601 611 68| 72?6 801 84 88 92 96|100, 5/ 101 15 20 25 30 35 40 45 50 55 60 65 70 75 801 851 901 95/100/105110|11511201125' 61 12| 18|24| 30/ 36| 42| 48 54 601 661 72 781 8+1 901 96|102|108/11411201126/132/138|14+1150 7| 14| 21| 28|35| 42| 49 56 631 701 771 8+1 911 98/105|112|1191126/133/140147 154/161|168|175 8| 16| 24/ 32| 40| 481 56 641 721 801 881 961101111211201128, 136/1441152|160|1681761184|192|200 9| 181 27| 36| 45| 54 | 631 72 81| 901 99|108|117|126|135|144|153162|171|180/189|198|207|216225 10| 201 301 401 501 601 701 80 90/100/110/120/130/140|150/160/1701180|190/200/210/220/230|240|250 11| 22| 33| 44| 55| 661 771 88| 99|110|121|132|143|15+1165|176|187|198/209/220/231|242253|264/275 12| 21| 361 48 601 721 841 96|108|120|132|14+|156|16811801192/20+)216|228|240252|264/276/288|300 13 26| 39| 52/ 65/ 78| 911104|117|130/143|156|169|182|195/208/221|234|247|260|273/286/299|312|325 14 28 42 56 70 841 38|112126 || 40|156|16|1 82|196/2101224 238 1252 | 266 260|294 308|322 336|350 MULTIPLICATION TABLIC 15| 30| 45 | 60|75| 90/105|120(135|150|165|1801195|210/225! 240/255|2701285|300|315|330|345\360|375 16| 32481 648096|112|128|144|160|176|192/208/224/240/256/272|288|304|3201336|352|368|3841400 171 341 51 681 85 102|119/136|153|17011871204/221/238/255/272289|306 (323340\357|374/391|408|425 18| 361 541 72|90|108/12611441162|180|198|216234252 2701288|306|324\342\360\378|396|4141432|450 191 381 571 76|95|1141133|152|171|190209|228|247|266|285|304/323|342\3613801399|418|437|456|475 20| 401 601 80/100/120/140/160/180/200/220/24012601280|300\320\340\360|380/400/420|440/460|4801500 21| 42| 631 84|105|126/147|168|189|210/231|252|273|294/315|3:36|357|378|399|420|441|462|483|5041525 22| 44| 66 88|110|132|154|176|198/220/242|264/286|30813301352|374|396|418|4401462/484|506|5281550 23| 46 691 92|115|138|161|1841207|230|253|276/299|322|345|368|391/4141437 460/483|5061529|552|575 241 481 72; 9611201144|1681192|216/240/264|288|312|336|360|384|40814:32|456|180)50115281552|5761600 25/ 501 75/100|125|150|175/200/225/250/275/300|325|350|375|400/425/450/4751500|525/550/57516001625 x 3 6 x 55. The repeated addition of a number to itself is equivalent to a multiplication of that number. Thus, 7 +7+7+ 7 is equivalent to 7 X 4, the sum of the former and the product of the latter being the same. Hence multiplication has sometimes been called a concise method of addition. 56. The product must be of the same kind or denomination as the multiplicand, since the taking of a quantity any number of times does not alter its nature. Thus: 5, an abstract number, 15, an abstract number; and 9 yards x 7 63 yards. 57. The multiplier must always be considered as an abstract number. Thus, in finding the cost of 4 books at 9 dollars each, we cannot multiply books and dollars together, which would be absurd, but we can, by regarding the 4 as an abstract number, take the 9 dollars, or cost of 1 book 4 times, and the product, 36 dollars, will be the result required. 58. The product of two factors will be the same, whichever is taken as the multiplier. Thus, 8 x 6 18 48; and the cost of 5 hats at 2 dollars each gives the same product as 2 hats at 5 dollars each. Also, the product of any number of factors is the same, in whatever order they are multiplied. Thus, 2 X 3 X 5 = 3 X 5 X 2 = 5 X 2 X 3 = 30. 59. A COMPOSITE number is a number produced by multiplying together two or more numbers greater than 1. Thus, 10 is a composite number, since it is the product of 2 x 5; and 18 is a composite number, since it is the product of 2 X 3 X 3. 65. To multiply simple numbers. Ans. 13842. Having written the multiplier, 9, unMultiplicand 1538 der the unit figure of the multiplicand, Multiplier 9 we multiply the 8 units by the 9, ob taining 72 units 7 tens and 2 units. Product 138 4 2 We write down the 2 units in the units' place, and reserve the 7 tens to add to the product of the tens. We then multiply the 3 tens by 9, obtaining 27 tens, and, adding the 7 tens which were reserved, we have 34 tens 3 hundreds and 4 tens. We write down the 4 tens in the tens' place, and reserve the 3 hundreds to add to the product of the hundreds. We next multiply the 5 hundreds by 9, obtaining 45 hundreds, and, adding the 3 hundreds which were reserved, we have 48 hun OPERATION. OPERATION. dreds 4 thousands and 8 hundreds. We write down the 8 lundreds in the hundreds' place, and reserve the 4 thousands to add to the product of the thousands. By multiplying the 1 thousand by 9 we obtain 9 thousands, and, adding the 4 thousands reserved, we have 13 thousands, which we write down in full; - and the product is 13842. 2. Let it be required to multiply 2156 by 423. Ans. 911988. In this example the multiplicand is Multiplicand 215 6 to be taken 423 times 3 units times Multiplier 4 2 3 + 2 tens times + 4 hundreds times. 3 units times 2156 = 6468 units ; 2 6 4 6 8 Partial tens times 2156 4312 tens; and 4 4 3 1 2 Products hundreds times 2156 : 8624 hundreds; 8 6 2 4 the sum of which partial products 911988, or the total product required. Product 911988 In the operation the right-hand figure of each partial product is written directly under its multiplier, that units of the same order may stand in the same column, for convenience in adding. RULE. — Write the multiplier under the multiplicand, arranging units under units, tens under tens, &c. Multiply each figure of the multiplicand by each figure of the multiplier, beginning with the right-hand figure, writing the right-hand figure of each product underneath, and adding the left-hand figure or figures, if any, to the next succeeding product. If the multiplier consists of more than one figure, the right-hand figure of each partial product must be placed directly under the figure of the multiplier that produces it. The sum of the partial products will be the whole product required. NOTE. — When there are ciphers between the significant figures of the multiplier, pass over them in the operation, and multiply by the significant figures only, remembering to set the first figure of the product directly under the figure of the multiplier that produces it. 61, First Method of Proof. – Multiply the multiplier by the multiplicand, and, if the result is like the first product, the work is supposed to be right. (Art. 58.) 62. Second Method of Proof.— Divide the product by the multiplier, and, if the work is right, the quotient will be like the multiplicand. · NOTE. — This is the common mode of proof in business; but, as it anticipates the principles of division, it cannot be employed without a previous knowledge of that process. 63. Third Method of Proof. — Begin at the left hand of the |