2. The process of finding the factors of which a given number is composed, is called resolving it into factors. 56. Some numbers may be resolved into more than two factors ; and also into different sets of factors. Thus the factors of 24 are 3, 2, 2 and 2 ; or 4, 3 and 2; or 6, 2 and 2, or 8 and 3, or 6 and 4, or 12 and 2. Obs. We have seen that the product of any two numbers is the same, whichever factor is taken for the multiplier. (Art. 47.) In like manner, the product of any three or more factors will be the same, in whatever order they are multiplied. For, the product of two factors may be considered as one number, and this may be taken either for the multiplicand or the multiplier. Again, the product of three factors may be considered as one number, and be taken for the multiplicand or the multiplier, &c. Thus 2433X2X2X2=6X2X 2=12X2=6X4=4X2X3=8X3. 3. What will 24 hogsheads of molasses cost, at 37 dolapiece ? Ans. 888 dollars. Suggestion. Resolve 24 into any two or more factors and proceed as before. Hence, 57. To multiply by a composite number. Resolve the multiplier into two or more factors ; multiply the multiplicand by one of these factors, and this product by another factor, and so on till you have multiplied by all the factors. The last product will be the product required. Obs. The factors into which a number may be resolved, must not be confounded with the parts into which it may be separated. (Art. 26.) The former have reference to multiplication, the latter to addition ; that is, factors must be multiplied together, but parts must be added together to produce the given number. Thus 56 may be resolved into two factors, 8 and 7; it may be separated into two parts, 5 tens or 50, and 6. Now 8X7=56, and 50+6=56. 4. What will 36 cows cost, at 19 dollars a head? Quest.- Obs. What is a composite number? What are the factors which produce it, sometimes called ? What is meant by resolving a number into factors? 56. Are numbers ever composed of more than two factors ? What are the factors of 24? 32 ? 36 ? 40 ? 42 ? 60 ? 64? 72 ? 108? Obs. When three or more factors are to be multiplied together, does it make any difference in what order they are taken? 57. When the multiplier is a composite number, how do you proceed? Obs. What is the difference between the factors into which a number may be resolved, and the parts into which it may be separated ? 5. What cost 45 acres of land, at 110 dollars per acre ? 6. At 36 shillings per week, how much will it cost a person to board 52 weeks ? 7. If a man travels at the rate of 42 miles a day, how far will he travel in 205 days ? 8. At the rate of 56 bushels per acre, how much corn can be raised on 460 acres of land ? 9. What cost 672 yards of broadcloth, at 24 shillings per yard ? 10. What cost 1265 yoke of oxen, at 72 dollars per yoke ? CASE II.- When the multiplier is 1 with ciphers annexed to it. 58. It is a fundamental principle of notation, that each removal of a figure one place towards the left, increases its value ten times ; (Art. 9 ;) consequently, annexing a cipher to a number will increase its value ten times, or multiply it by 10; annexing two ciphers will increase its value hundred times, or multiply it by 100; annexing three ciphers will increase it a thousand times, or multiply it by 1000, &c.; for each cipher annexed, removes each figure in the number one place towards the left. Thus 12 with a cipher annexed, becomes 120, and is the same as 12x10 ; 12 with two ciphers annexed, becomes 1200, and is the same 12 x 100; 12 with three ciphers annexed, becomes 12000, and is the same as 12 x 1000, &c. Hence, 59. To multiply by 10, 100, 1000, &c. Annex as many ciphers to the multiplicand as there are ciphers in the multiplier, and the number thus formed will be the product required. OBs. To annex means to place after or at the right hand. 11. What will 10 drums of figs weigh, at 28 pounds a drum? Ans. 280 pounds. as Quest.-58. What effect vloes it have to remove a figure one place towards the left hand ? Two places ? 59. How do you proceed when the multiplier is 10, 100, 1000 ?' Obs. What is the meaning of the term an12. How many pages are there in 100 books, each book having 352 pages? nex. 13. Multiply 476 by 1000. 14. Multiply 53486 by 10000. 15. Multiply 12046708 by 100000. 16. Multiply 26900785 by 1000000. 17. Multiply 89063457 by 10000000. 18. Multiply 9460305068 by 100000. 19. Multiply 78312065073 by 10000. Case III.--When the multiplier has ciphers on the right. 20. What will 32 acres of land cost, at 20 dollars per acre ? Note.—Any number with ciphers on its right hand, is obviously a composite number ; the significant figure or figures being one factor, and 1 with the given ciphers annexed to it, the other factor. Thus 20 may be resolved into the factors 2 and 10. We may therefore first multiply by 2 and then by 10, by annexing a cipher as above. Thus 32 X2=64, and 64 X10=640 dolls. Ans. 21. If the expenses of an army are 2000 dollars per day, what will it cost to support the same army 365 days? Operation. 2000 may be resolved into the factors 2 and 365 1000. Then 2 times 365 are 730; now 2 adding three ciphers to this product, multiplies it by 1000, and we have 730000 dollars for 730000 the answer. Hence, 60. When there are ciphers on the right hand of the multiplier. Multiply the multiplicand by the significant figures of the multiplier, and to this product annex as many ciphers as there are on the right of the multiplier. Obs. It will be perceived that this case is a variety of Case I. and of course depends upon the same principle. Quest.–60. When there are ciphers on the right of the multiplier, how do you proceed? Obs. Upon what principle does this case depend ? 22. How many days are there in 36 months, reckoning 30 days to a month ? 23. If 1 barrel of flour weighs 192 pounds, how much will 200 barrels weigh? 24. Multiply 4376 by 2500. CASE IV.--When the multiplicand has ciphers on the right. Suggestion. 12000 is a composite Operation. number, the factors of which are 12 and 12000 1000. Now the product of two or more 31 numbers is the same in whatever order 12 they are multiplied ; (Art. 47 ;) conse 36 quently multiplying the factor 12 by 31, and this product by 1000, will give the Ans, 372000 same result as 12000 x 31. Thus 31 times 12 are 372; then annexing three ciphers, we have 372000, which is the same as 12000 x 37. Hence, 61. When there are ciphers on the right of the multiplicand. Multiply the significant figures of the multiplicand by the multiplier, and to the product annex as many ciphers as there are on the right of the multiplicand. OBs. When the multiplier and multiplicand both have ciphers on the right, multiply the significant figures together, and to their product annex as many ciphers as there are on the right of both factors. 28. Multiply 370000 by 32. Ans. 37800000. QUEST.-61. When there are ciphers on the right of the multiplicand, how proceed? Obs. How, when there are ciphers on the right both of the multiplier and multiplicand ? 33. Multiply 4300 by 600. Ans. 2580000. Art. 63. Ex. 1. How many oranges, at 3 cents apiece, can you buy for 12 cents ? Suggestion. If 3 cents buy one orange, 12 cents will buy as many oranges as there are 3 cents in 12 cents ; that is, as many as 3 is contained times in 12. Now 3 is contained in 12, 4 times. Ans. 4 oranges. 2. How many lemons, at 4 cents apiece, can you buy for 20 cents? Suggestion. To find how many times 4 cents are contained in 20 cents, think how many times 4 make 20, or what number multiplied by 4, produces 20. 3. At 3 dollars per yard, how many yards of cloth can be bought for 15 dollars ? 4. How many hats, at 5 dollars apiece, can you buy for 30 dollars ? 5. How many barrels of flour will 36 bushels of wheat make, allowing 4 bushels to one barrel ? 6. If you pay 6 cents a mile for riding in a stage, how far can you ride for 48 cents ? 7. If a pound of sugar costs 7 cents, how many pounds can you buy for 56 cents ? |