are 20. B4, 8. MULTIPLICATION. Four booke will evidently cost four times Addition. as much as one book; and to answer the Multipucation 5 question by Addition, we should write 5 down 4 fives, and add them, as at the left 5 hand. By Multiplication we should pro5 ceed as at the right hand, thus, 4 times 5 Ans. 20 cto. Now these two operations differ Ans. 20 cts. only in the form of expression; for we can arrive at the amount of 4 times 5 only by a mental process similar to that at the left hand. Hence, in order to derive any advantage from the use of Multiplication over that of Addition, it is necessary that the several results arising from the multiplication of the numbers below ten, should be perfectly committed to memory. They may be learned from the Mub tiplication table, page 19. (16) 2. If 1 pound of raisins cost 9 cents, what will 7 pounds cost? 84. 3. There are 24 hours in a day; how many hours are there in 3 days! Three days will evidently contain Addition. three times as many 'hours as 1 day, Multiplication Ist day 24 hours. or 3 times 24 huurs; we may there- 24 hours. 24 hours. fore write down 24 three times, and 3 hours. 24 hours. add them together, as at the left hand, or we may write 24 with 3, the num- Ans. 72 Ans. 72 hours. ber of times it is to be repeated, un:ler it, as at the right hand, and say 3 times 4 are 12, (the same as 3 fours added together) which are 1 ten and 2 units. We there fore write down the 2 units in the place of units, and reserving the 1 ten to be joined with the tens, we say, 3 times 2 tens are 6 tens, to which we add the 1 ten reserved, making 7 tens. We the efore write 7 at the left hand of the 2, in the place of tens, and we have 72 hours, the same as by Addition. In Multiplication the two numbers which produce the resuli, as 24 and 3 in this example, are called fuctors. The factor which is repeated, as the 24, is called the muliiplicand; the number which shows bow many times the multiplicand is repeated, as the 3, is called the mubo tiplier; and the result of the operatioa, as the 72, is called the product. 4. There are 320 rods in a mile; how many rods in 8 miles ? 85. 5. A certain orchard consists of 26 rows of trees, and in each roo are 26 trees; how many trees are there in the orchard? Here we find it impracticable to multiply by the whole 28 Operation. at once; but as 26 is made up of 2 tens and 6 units, we may 26 separate them, and multiply first by thc units, and then by the 26 tens; thus, 6 times 6 are 36, of which we write down the 6 un.is, and reserving the 3 tens, we say 6 times 2 are 12, and 156 8, which was reserved, are 15, which we write down, the 5 52 in the place of tens, and the ! in the place of hundreds, and thus find that 6 of the rows contain 156 trees. We now pro676 ceed to the 2, and say 2 times 6 are 12; the 2 by which we multiply being 2.tens, it is evident that the 12 are so many Lens; but 12 tens are 1 hundred and 2 tens; we therefore write the 2 under the place of tens, which is done by putting it directly under the 2 in the multiplier, and reserve the 1 to be united with the hundreds. We then say 2 times 2 are 4; both these 2's being in the tens' places, their product 4 is nundreds, with which we unite the 1 hun: ed reserved, making 5 hundreds. The 5 being written at the left hand of the 2 tens, we have 5 hundreds and 2 tens, or 520 for the number o: trees in 20 rows. These being added to 156, the number in 6 rows, we have 676 for the number of trees in 26 rows, or in the whole orchard. 86. 6. There are in a gentleman's garden 3 rows of trees, and 5 trees in cach row; how many trees are there in the whole? We will represent the 3 rows by 3 lines of l's, :nd the 1, 1, 1, 1, 1, 5 trees in each row by 5 l’s in nach line. Here it is 1, 1, 1, 1, 1, evident that the whole number of l's are as many times 5 1, 1, 1, 1, 1, as there are lines, or 3 times 5=15, and as many tiines 3 as there are columns, or 5 times 3=15... This proves that 5 multiplied by 3 gives the same product as 3 multiplieel isy 5; and the same may be shown of any other two factors. Hence either of tie two factors may be made the multiplicand, or the multiplier, and the pr duct will still be the same. We may therefore prove multiplication by changing the places of the factors, and repeating the operation. SIMPLE MULTIPLICATION. 87. Simple Multiplication is the method of finding the amount of a given number by repeating it a proposed number of times. There must be two or more numbers given in order to perform the operation. The given numbers, spoken of together, are called factors. Spoken of separately, the number which is repeated, or multiplied, is called the mulliplicand; the number by which the multiplicand is repeated, or multiplied, is called the multiplier; and the number produced by the operation is called the product. RULE. 88. Write the multiplier under the multiplicand, and araw a line below them. If the multiplier consist of a single figure only, begin at the right hand and multiply each figure of the multiplicand by the multiplier, -setting down the excesses and carrying the tens as in Addition. (84) If the multiplier consists of two or more figures, begin at the right hand and multiply all the figures of the multiplicand successively by cach figure of the multiplier, remembering to set the first figure of each product directly under the figure by which you are multiplying, and the sum of these several products will be the total product, or answer required.(85) PROOF. 89. Make the former multiplicand the multiplier, and the former multiplier the multiplicand, and proceed as before ; if it be right, the product will be the same as the former. (86) A QUESTIONS FOR PRACTICE. 7. In the division of a prize 14. If a man's income be i among 207 men, each man's dollar a day, what will be the share was 534 dollars ; what amount of his income in 45 was the value of the prize? years, allowing 365 days to 534 dolls. each year? Ans. 16425 dolls. 207 men 15. A certain brigade con sists of 32 companies, and 37 38 each company of 86 soldiers; 1068 how many soldiers in the bric Ans. 2752. 16. A man sold 742 thou8. If a man earn 3 dolls. a sand feet of boards at 18 dol week, how much will he earn lars a thousand; what did in a year, or 52 weeks? they come to? Ans. 156 dolls. Ans. 19956 dolls. 9. If a man thrash 9 bush- 17. If a man spend 6 cents els of wheat a day, how much a day for cigars, how much will he thrash. in 29 days? will he spend in a year of 365 Ans. 261 bush. days? Ans. 2190 cts.=$21.90. 10. In a certain orchard 18. If a man drink-a glass there are 27 rows of trees, of spirits 3 times a day, and and 15 trees in each row, I will be the cost for a year? each glass cost 6 cents, what how many trees are there? Ans. 405. Ans. 6570 cts.= $65.70. 11. If a person count 180 in 19. Says Tom to Dick, you a minute, how many will he have 7 times 11 chesnuts, but count in an hour? I have 7 times as many as you, how have I? Ans. 539. 12. A man had 2 farms, on 20. In a prize 47 men shared one he raised 360 bushels of equally, and received 25 dob wheat, and on the other 5 lars each ; how large was the times as much ; how much prize? Ans. 1175 dolls. did he raise on both? 21. What is the product, Ans. 2160 bush. t. 808879 by twarty, thousand 13. In dividing a certain i fre hundred pe ihree? fum of money among 352, Ans 0532946137. each man received 17 dollars, :: 22. What will be the cost of what was the sum divided? 924 tons of potash at 95 dolls. Ans. 5994 dolls. Ans. 87780 dolls. 23. Multiply 848329 by 4009. Product, 3400950961 24. Multiply 64+7001+-103-83 by 18 H6. Prod. 170040 25. 49X15X17X12X100=bow many Ans. 14994000 y a tond answer. CONTRACTIONS OF MULTIPLICATION. 90. 1. A man. bought 17 cows for 15. dollars apiece; what did they all cost? If we muliiply 17 by 5, we find the cost at 5 dollars apiere, Operation, and since 15 is 3 times 5, the cost, at 15 dollars apicce, will 17 manifestly be 3 times as much as the cost at 5 dollars apiece. 5 If then we multiply the cost at 5 dollars by 3, the produci must be the cost at 15 dollars apiece. 85 A number (as 15). which is produced by the multiplicaliun 3 of two, or more, other numbers, is called a composite num'e'. 'The factors which produce a coinposite nuinber (as 5 ani 3) Aps.$255 are called the component parts. 1. To multiply by a composite number. RULE. --Multiply first by one component part, and that product by the other, and so on, if there be more than two; the last product will be the 2. What is the weight of 82 boxes, 3. Multiply 2478 by 36. each weighing 42 pounds? Product 89208. 42 6X7 Ans. 3444 lbs. 4. Multiply 8462 by 56. Product 473872. 91. 5. What will 16 tons of hay cost at 10 dollars a ton? It has been shown (73) that each removal of a figure one place towards the left increases its value ten times. Hence to multiply by 10, we save only to annex a cipher to the multiplicand, because all the significant figures are thereby removed one place to the left. In the present &le we add a cipher to 16, making 160 dollars for the answer. 6. A certain army is made up of 125 companies, consisting of 100 men each; how many men are there in the whole? For the reasons given under example 5, a number is multiplied by 100 by placing two ciphers on the right of it, for the first cipher multiplies it by 10, and the second multiplies this product by 10, and thus makes it 10 umes 10, or 100, times greater; and the same reasoning may be extended to 1 with any number of ciphers annexed. Henee 2. To mulliply by 10, 100, 1000, or 1 with any number of ciphers annexed. RULE.-Annex as many ciphers to the multiplicand as there are ciphers in the multiplier, and the number thus produced will be the product. 7. Multiply 3579 by 1000. 8. Multiply 789101 by 100000. Prod. 78910100000. 92. 9. What is the weight of 250 casks of sugar, each weighing 300 Here 300 may be regarded as a composite number, 28 2 whose component parts are 100 and 3; hence in 3 multiply by 300, we have only to multiply by 8. and join two ciphers to the product; and as the operation Ans. 75000 lbs. must always commence with the first significant figo are, when the multiplicand is terminated by ciphers, the cipher in that may be omitted in multiplying, and be joined afterwards to the product. Hence 8. When there are ciphers on the right of one or both the factors: RULE-Neglecting the ciphers, multiply the significant figuren by the general rule, and place on the right of the product as many ciphan * were neglected in both factors Prod. 9699000.1 93, 94, 95. MULTIPLICATION. 13 10. Multiply 3700 by 200. 11. Multiply 7830 by 97000. Prod. 740000. Prod. 759510000. 33. 12. Peter has 17 chesnuts, and John 9 times as many; how many has John? 170 Here we annex a cipher to 17, which multiplies it by 10. 17 If now we subtract 17 from this product, we have the 17 nina times repeated, or multiplied by 9. Ans. 153 13. A certain cornfield contains 228 rows, which are 99 hills long; buw many hills are there? 22800 Annexing two ciphers to 228, multiplies it 100; 228 we thon subtract 228 from this product, which leaves 99 times 223; and in general, Ans. 22572 4. When the multiplier is 9, 99, or any number of nines. RULE.—Annex as many ciphers to the multiplicand as there are nines in the multiplier, and from the sum chus produced, suburact the multiplicand, the remainder will be the answer. 14. Multiply 99 by 9. | 15 Multiply 6473 by 999. 3. SUBTRACTION, ANALYSIS. 94. 1. A boy having 18 cents, lost 6 of them; how many had he left? Here is a collection of 18 cents, and we wish to know how many there will be after 6 cents are taken out. The most natural way of doing this, would be to begin with 18, and take out une cent at a time till we have taken 6 cents; thus, 1 froin 18 leaves 17, 1 from 17 leaves 16, 1 from 16 leaves 15, 1 from 15 leaves 14, 1 from 14 leaves 13, 1 from 13 leaves 12. We have now taken away 6 unes, or 6 cents, from 18, and have arrived, in the descending series of numbers, at 12; thus discovering that if 6 be taken from 18, there will remain 12, or tha: 12 is the difference between 6 and 18. Hence Subtraction is the reverse of Addition. When the nurnbers are small, as in the preceding example, the operation may be performed wholly in the mind ;(102) but if they are large, the work is facilitated by writing them down. 95. 2. A person owed 75 dollars, of which he paid 43 dollars; how much remains to be paid? Now to find the difference between 75 and 43, we Operation. write down the 75, calling it the minuend, or numFrom 75 minuend. ber to be diminished, and write under it the 43, Take 43 subtrahend. calling it the subtrahend, with the units under units and the tens under tens, and draw a line below, as 82 remainder. at the left hand. As 75 is made up of 7 tens and 5 units, and 43 of 4 tens and 3 units, we take the 3 75 proof. units of the lower from the 7 units of the upper line, and find the remainder to be 2, which we write be. low the line in the place of units. We then take the 4 tens of the lower from the 7 tens of the upper line, and find the remainder to ke 3, which we write below the line in the ten's place, and thus we find 32 to be the 3 |