4 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 ; il 21 31 41 51 61 71 81 91 101 111 12 13 14 15 16 17 18 19 20 21 22 23 24 25 21 61 81 10 12 14 16 18 20 22 24 26 281 301 32| 341 361 381 401 421 44 46 48 50 3. 61 9| 12| 15| 181 21 241 27| 3033| 36 39 42 45 48 51| 54 571 601 63 66 69| 72| 75 41 81 121 161 201 24| 28 32 36| 40| 44! 48 52 56 60 641 68 72 76 80 84 88 92 96|100 51 10 151 20 25 30 35 401 45 50 55 60 65 701 75 80 85| 901 95/100/105/110|115 120 125 6 12 181 24 30 361 42| 481 541 601 66| 72 78 84) 901 961102, 108/11411201126/132|138|144|150 71 14| 21| 28| 35| 42| 49 561 631 701 77 841 911 984105|112|119|126 133/140|147 | 154|161|168|175 8| 161 24| 32| 40 48 56 641 721 801 88/ 961101|1121120)128, 136/144|152|1601681176/184/192|200 9| 18| 27361 450 541 631 72 81| 901 99|108|117|126|135|144|1531162|171|180|189|1981207|216|225 10 20 30 40 50 60 70 80 90|1001110|120/130/140|150160|170|180|190/200/210/220/230/240/250 11 22 33 44 55| 661 771 88|99|110|121|132|143|154/165|1761187|198/209/220/231242|253 264 275 12| 21| 36/ 481 60 721 841 96|108|120/132|144|156|16811801192|204 216/228/240/252|264276288|300 13 26| 39| 52 65 78| 91110411171130|143|156|169/182|195/208221|234|247|260/273/286/299|312|325 14| 28 42 56 70 811 98/112|126|140|154/168|182|196/21012241238|252|266/280/294/308|322|336|350 15 30 45| 60|75| 90|105|120|135|150|165|180|195|210/225! 240|255 270/285|300|315 330|345\360\375 16| 32 48 64 801 961112|128|144|160|176|192208|224/240/256/272|288|304|320|336|352|368|384400 171 341 51 68 85 102|119|136|153|17011871204|221/238/255/272/289|306|323|340\357|374|391|408|425 18 36 541 72|90|108|126/144|162|180/198/216234/25212701288|306|324\342\360\378|396|414|432|450 191 38 571 76 95/1141133|152|171|190/209/2281247|266|285|304/323/342\361\380 399|418|437|456|475 201 40 60 80/100/120/140/160/180/200/220/240/2601280\300\320\340\360\380|4001420/440/460/480|500 21| 42| 631 84|105|126|147|168/1891210231|252|273|294/315/336357|378399420441|462483504|525 22| 441 661 881110|132|154|1761198/220/2421264|286|308|330|352|374|396|41844014624841506|528|550 23 46 691 92|115/138|161|184/2071230/253/276/299|322|345|368|391/414/437|4601483|506|5291552|575 24! 481 72; 96/120|144|168|192|216240264|288|312|336|360|384|408|432|456|480504|528|552|576|600 25/ 501 75/100/125/150/175/200|225/250/275|300\325|350|375400/425/450/47515001525/550/575160016251 MULTIPLICATION TABLE 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, X 3 = 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 6 x 8 = 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. 60. 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 1 3 8 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 Irundreds 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 68 Partial tens times 2156 4312 tens; and 4 431 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 multiplicand, and add together its successive figures toward the right, till the sum obtained equals or exceeds the number nine. If it equals it, drop the nine, and begin to add again at this point, and proceed till you obtain a sum equal to, or greater than, nine. If it exceeds nine, drop the nine as before, and carry the excess to the next figure, and then continue the addition as before. Proceed in this way, till you have added all the figures in the multiplicand and rejected all the nines contained in it, and write the final excess at the right hand of the multiplicand. Proceed in the same manner with the multiplier, and write the final excess under that of the multiplicand. Multiply these excesses together, and place the excess of nines in their product at the right. Then proceed to find the excess of nines in the product obtained by the original operation ; and, if the work is right, the excess thus found will be equal to the excess contained in the product of the above excesses of the multiplicand and multiplier. NotE. — This method of proof, though perhaps sufficiently sure for common purposes, is not always a test of the correctness of an operation. If two or more figures in the work should be transposed, or the value of one figure be just as much too great as another is too small, or if a nine be set down in the place of a cipher, or the contrary, the excess of nines will be the same, and still the work may not be correct. Such a balance of errors will not, however, be likely to occur. EXAMPLES. 3. Multiply 7325 by 3612. Ans. 26457900. OPERATION PROOF BY MULTIPLICATION. Multiplicand 7 3 2 5 Multiplicand 3 6 1 2 Multiplier 3 6 1 2 Multiplier 7 3 2 5 1 4 6 5 0 180 6 0 7 3 2 5 7 2 2 4 4 3 9 50 10 8 3 6 2 1 9 7 5 2 5 2 8 4 Product 2 6 4 5 7 9 0 0 Product 2 6 4 5 7 9 0 0 4. Required the product of 82967 by 652. Ans. 54094484. 17. What will 365 acres of land cost at 73 dollars per acre ? Ans. $ 26645. 18. What will 97 tons of iron cost at 57 dollars a ton ? Ans. $ 5529. 19. What will 397 yards of cloth cost at 7 dollars per yard? Ans. $ 2779. 20. What will 569 hogsheads of molasses cost at 37 dollars per hogshead? Ans. $ 21053. 21. If a man travel 37 miles in one day, how far will he travel in 365 days? Ans. 13505 miles. 22. If a vessel sails 169 miles in one day, how far will she sail in 144 days? 23. What will 698 barrels of flour cost at 7 dollars a barrel? 24. What will 376 lbs. of sugar cost at 13 cents a pound? Ans. 4888 cts. 25. What will 97 lbs. of tea cost at 93 cents a pound? Ans. 9021 cts. |