1. If a book cost 5 cents, what will 4 such books cost? | Four books will evidently cost four | Multiplication, Addition. | times as much as one book; and to answer | the question by Adlition, we should write down 4 fives, and add them, as at the left i hand. By Multiplication we should pro- i Ans. 20 cts. | ceed as at the right hand, thus, 4 times 5 are 20. Now | these two operations differ only in the form of expression; Ans. 20 cts. | 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 Multiplication Table, page 19. (16) 2. If one 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? Addition. I Three days will evidently contain | Multiplication. Ist day 24 hours. i 3 times as many hours as one day, or i 24 hours. 20 - 24 hours, 3 times 24 hours; we may therefore 3 times. 3d - 24 hours. | write down 24 three times, and add i | them together, as at the left hand, or | Ans. 72 hours. Ans. 72 hours. I we may write 24 with 3, the number of times it is to be. repeated, under it, as at the right, and say 3 times 4 are 12, (the same as 3 Sours added together) which are 1 ten and 2 units. We therefore 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 I ten reserved, making 7 tens. We therefore 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 on the two numbers which produce the result, as 24 and 3 in this example, are called factors. The factor which is repeated,as the 24, is called the multiplicand; the number which shows how many times the multiplicand is repeated, as the 3, is called the multiplier; and the result of the operation, 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 row are 26 trees; how many trees are there in the orchard ? Operation. I Here we find it impracticable to multiply by the whole 26 1 26 at once; but as 26 is made up of 2 tens and 6 units, | we may separate them and multiply first hy the units and then by the tens; thus, 6 times 6 are 36, of which 156 we write down the 6 units, and reserving the 3 tens, we | say 6 times 2 are 12, and 3, which was reserved, are 15, | which we write down, the 5 in the place of tens, and the 676 1 1 in the place of hundreds, and thus find that 6 of the rows contain 156 trees. We now proceed 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 teng; 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 hun. dreds. We then say 2 times 2 are 4; both these 2's being in the tens' places, their produci 4 is bundreds, with which we unite ihe 1 hundred reserved, making 5 hundreds. The 5 being written at the left hand of the Multi 2 tens, we have 5 hundred and 2 tens, or 520 for the number of 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 each row; how many trees are there in the whole ? 1, 1, 1, 1, 1, 1 We will represent the 3 rows by 3 lines of 1s, and 1, 1, 1, 1, 1, the 5 trees in each row by 5 ls in each line. Here it 1, 1, 1, 1, 1, 1 is evident that the whole number of Is are as many times 5 as there are lines, or 3 times 5=15, and as many times 3 as there are columns, or 5 times 3=15. This proves that 5 multiplted by 3 gives the same produci as 3 multiplied by 5; and the same may be shown of any other two factors. Hence either of the two factors may be made the multiplicand, or the multiplier, and the product 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 multiplicand; 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 draw 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 consist of two or more figures, begin at the right hand and multiply all the figures of the multiplicand successively by each figure of the multiplier, remembering to set the first figure of each product directly under the figure by which you are mul. tiplying, 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 What is Simple Multiplication? How must the numbers be written tiplier is a single figure ? What are they called spoken of to. How when the multiplier consists gether? of two or inore figures ? What, spoken of separately? What is the method of proof? What is the result called ? former multiplier the multiplicand, and proceed as before; if it be right, the product will be the same as the former. (86) QUESTIONS FOR PRACTICE. 7. In the division of a prize 1 14. A certain city is divided among 207 men, each man's into 12 wards, each ward conshare was 534 dollars; what tains 2000 families, and each was the value of the prize ? family 5 persons; what is the 534 dolls. whole population ? 207 men. Ans. 120000. 15. If a man's income be one 3738 dollar a day, what will be the 1068 amount of his income in 45 years, allowing 365 days to Ans. 11053 8 dolls. each year? Aos. 16425 dolls. 8. If a man earn 3 dolls. a 16. A certain brigade conweek, how much will he earn sists of 32 companies, and each in a year, or 52 weeks? company of 86 soldiers ; how Aas. 156 dolls. many soldiers in the brigade? 9. If a man thrash 9 bushels Ans. 2752. of wheat a day, how much 17. A man sold 742 thousand will be thrash in 29 days? I feet of boards at 18 dollars a Ans. 261 bush. I thousand; what did they come 10. In a certain orchard | to ? Ans. 13356 dolls. there are 27 rows of trees, 18. If a man spend 6 cents and 15 trees in each row; how many trees are there? | a day for cigars, how much Ans. 405. will he spend in a year of 365 days? Ans. 2190 cts.=$21,90. 11. If a person count 180 in 19. If a man drink a glass a minute, how many will he count in an hour? Ans. 10800. of spirits 3 times a day, and į each glass costs 6 cents, what 12. A man had? farms, on will be the cost for a year? one he raised 360 bushels of Ans. 6570 cts.=$65,70. wheat, and on the other 5 times as much ; how much did! 20. Says Tom to Dick, your he raise on both? have 7 times 11 chesauts, but Ans. 2160 bush. I have 7 times as many as you, 13. In dividing a certain | how many have 1 ? Ans. 539. sum of money among 352, each 21. In a prize 47 men shared man received 17 dollars, what | equally, and received 25 dol. was the sum divided. ? lars each ; how large was the Ans. 5984 dolls. I prize ? Aas. 1175 dolls. Upon what principle does it de. | What is the sign of mukiplication ? 22. What is the product, 23. What will be the cost of 308879 by twenir thousand 924 tons of potash at 95 dolls. five hundred and three ? a ton ? Aps, 87780 dolls, pend? Ans. 6332946137. 24. Multiply 37934 by 2. Product 75868 25. Multiply 357 by 56. Prod. 19992 26. Multiply 46891 by 325. Prod. 15239575 27. Multiply 848329 by 4009. Prod. 3400950961 28. Multiply 64+7001 + 103—83 by 18-+-6. Prod. 170040 29. 49 x 15 x 17 x 12 x 100=how inany ? Ans. 14994000 CONTRACTIONS OF MULTIPLICATION. 90. 1. A man bought 17 cows for 15 dollars apiece; what did they all cost? Operation. If we multiply 17 by 5, we find the cost at 5 dollars 17 | apiece, and since 15 is 3 times 5, the cost at 15 dollars i apiece will manifestly be 3 times as much as the cost at 15 dollars a piece. If then we multiply the cost at 5 dol. 85 flars by 3, the produet must be the cost at 15 dollars a piece. ī A number (as 15:) which is produced by the multiplicaAns. 255 doils. I tiun of two, or more, other numbers, is called a composite number. The factors which produce a composite number (as 5 and 3) are called tlie component parts. 1. To multiply by a composite number. RULE.-Multiply first by one component part and that product by the other, aid so on, if there be more than two, the last product will be the answer. 2. What is the weight of 82 boxes 3. Multiply 2478 by 36. each weighing 42 pounds ? Product 69208, 42=6X7 Ans. 5444 lbs. 1 4. Multiply 8402 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 have only to annex a cipher to the multiplicand, because all the significant figures are thereby removed one place to the left. In the present example we adu 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 ? Fur the reasons given under example 5, a number is multiplied by 100 by placing two ciphers on the right of ii, for the first ciplier multiplies it by 10, and the second multiplies this product by 10, and thus makes it 10 tinies 10, or 100 tinies greater; and the same reasoning may be extended to 1 with any number of ciphers annexed. Hence What is a composite number? What is the rule for multiplying by What is meant by the component i a composite number? parts of a number? 92, 93. CONTRACTIONS OF MULTIPLICATION. · 73 -2. T'o multiply by 10, 100, 1000.or 1 with any number of ciphers annered. ROLE.-Annex as many ciphers to the multiplicand as there are ciò phers in the multiplier, and the number thus produced will be the produci. 7. Multiply 3579 by 1000. 1 8. Multiply 789101 by 100000. Prod. 3579000. 1 Prod. 78910100000.92. 9. What is the weight of 250 casks of sugar, each weighing 300 Here 310 máy be regarded as a compositë number: 25 whose component parts are 100 and 3; hence to multiply by 300, we have only to multiply by 3 and join two ciphers' to the product; and as the operation Ans. 75000 lbs. must always commence with the first significant figo ure, when the multiplicand is terminated by ciphers, the cipher in that may be omitted in multiplying, and be joined afterwards to the product. Hence* : 3. When there are ciphers on the right of one or both the factors : RULE.-Neglecting the ciphers, multiply the significant figures by the general rule, and place on yhe right of the product as many ciphers as were neglected in both factors. 10. Multiply 3700 by 200. 11. Multiply 7830 by 97000. Prod. 740000. Prod. 759510000. 93. 12. Peter has 17 chesnuts, and John 9 times as many; how many has Joho? 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 - 9 times repeated, or multiplied by 9. Ans. 153 · 13. A certain cornfield contains 228 rows, which are 99 hills long, how many hills are there? 22800' Annexing two ciphers to 228. multiplies it 100 ; we then 228 subtract 228 from this product, which leaves 99 times 228';' and in general Ans. 22572" 4. When the multiplier is 9, 99, or any number of nines : RULE. —Andex as many ciphers to the multiplicand as there are nines in the multiplier, and from the sum thus produced, subtract the multiplicand, the remainder will be the answer.. 14. Multiply 99 by 9. | 15. Multiply 6473 by 999. How do you proceed when the mol- | How do you proceed when there tiplier is 10, 100, &c.? Explain are ciphers on the right of both the reason. factors ? Explain by an example |