ILLUSTRATION.-22 pounds 12 pounds. If 12 cost 60 = will cost of 60 cents = cents, 5 cents; and, or a pound, 50. How many times will 60 contain 2? Ans. $0.25. 51. Paid $54 for 71⁄2 barrels of oil; what cost 1 barrel ? 52. How many times is 7 contained in 54? 53. How many cords of wood, at $5 for $66? Ans. $7. per cord, can be bought 54. How many times will 66 contain 51 ? 55. Gave $40 for 63 yards of broadcloth; what cost 1 yard? 56. How many times is 63 contained in 40? 57. The distance between two places is 110 rods. I wish to divide this distance into spaces of 5 rods each. Required the number of spaces. We multiply by 100, by annexing two ciphers to the multiplicand; and since 25, the multiplier, is only one fourth of 100, we divide by 4 to obtain the true product. RULE. Annex two ciphers to the multiplicand, and divide it by 4. *If the principles on which these contractions depend are considered too difficult for the young pupil to understand at this stage of his progress, they may be omitted for the present, and attended to when he is further advanced. QUESTIONS. What is the rule for dividing when there are ciphers on the right of the divisor?— Art. 61. What is the rule for multiplying by 25? What is the reason for the rule? RULE. Annex two ciphers to the multiplicand, and divide it by 3. EXAMPLES FOR PRACTICE. 2. Multiply 356789541 by 33. 3. Multiply 871132182 by 33. 4. Multiply 583647912 by 33. ART. 63. To multiply by 125. Ex. 1. Multiply 7896538 by 125. OPERATION. 8)78965 38000 987067250 Product. Ans. 11892984700. Ans. 987067250. We multiply by 1000, by annexing three ciphers to the multiplicand; and since 125, the multiplier, is only one eighth of 1000, we divide by 8 to obtain the true product. Annex three ciphers to the multiplicand, and divide it by 8. QUESTIONS. Art. 62. What is the rule for multiplying by 33? What is the reason for this rule?-Art. 63. What is the rule for multiplying by 125? Give the reason for the rule. ART. 64. To multiply by any number of 9's. Ex. 1. Multiply 4789653 by 99999. OPERATION. 4789 6 5 3 0 0 0 0 0 4789653 478960510347 Product. Ans. 478960510347. By adding 1 to any number composed of nines, we obtain a number expressed by 1 with as many ciphers annexed as there are nines in the number to which 1 is added. Thus, 999 + 1 = 1000. Therefore, annexing to the multiplicand as many ciphers as there are nines in the multiplier is the same thing as multiplying the number by a multiplier too large by 1, and subtracting the number to be multiplied from this enlarged product will give the true product. RULE. — Annex as many ciphers to the multiplicand as there are 9's in the multiplier, and from this number subtract the number to be multiplied. EAXMPLES FOR PRACTICE. 2. Multiply 1234567 by 999. Ans. 1233332433. Ans. 876542123457. Ans. 999998000001. CONTRACTIONS IN DIVISION. ART. 65. To divide by 25. OPERATION. 1234567 4 49 3 8 2/6 8 Quotient. Ans. 49382 68 Multiplying the dividend by 4 makes it four times too great; therefore, to obtain the true quotient, we must divide by 100, a divisor four times greater This we do by cutthan the true one. ting off two figures on the right. RULE. - Multiply the dividend by 4, and divide the product by 100. QUESTIONS. Art. 64. What is the rule for multiplying by any number of 9's? What is the reason for the rule? Art. 65. What is the rule for dividing by 25? Give the reason for the rule. ART. 66. To divide by 334. Ex. 1. Divide 6789543 by 33. OPERATION. 6789543 3. Ans. 203686,2. Multiplying the dividend by 3 makes it three times too great; therefore, to obtain the true quotient, we must divide by 100, a divisor three times greater This is done by cutting off two figures on the right. 2036 8 6/29 Quotient. than the true one. Multiply the dividend by 3, and divide the product by 100 EXAMPLES FOR PRACTICE. 2. Divide 987654321 by 331⁄2. ART. 67. To divide by 125. Ex. 1. Divide 9874725 by 125. OPERATION. 9874725 8 789971800 Quotient. Multiplying the dividend by 8 makes it eight times too great; therefore, to obtain the true quotient, we must divide by 1000, a divisor eight times greater than the true one. We do this by cut ting off three figures on the right. RULE. Multiply the dividend by 8, and divide the product by 1000. QUESTIONS. Art. 66. What is the rule for dividing by 33? Give the reason for the rule. Art. 67. What is the rule for dividing by 125? What is the reason for the rule? § VIII. MISCELLANEOUS EXAMPLES, INVOLVING THE FOREGOING RULES. 1. A BOUGHT 73 hogsheads of molasses at 29 dollars per hogshead, and sold it at 37 dollars per hogshead; what did he gain? Ans. 584 dollars. 2. B bought 896 acres of wild land at 15 dollars per acre, and sold it at 43 dollars per acre; what did he gain? Ans. 25088 dollars. 3. N. Gage sold 47 bushels of corn at 57 cents per bushel, which cost him only 37 cents per bushel; how many cents did he gain? Ans. 940 cents. 4. A butcher bought a lot of beef weighing 765 pounds at 11 cents per pound, and sold it at 9 cents per pound; how many cents did he lose? Ans. 1530 cents. 5. A taverner bought 29 loads of hay at 17 dollars per load, and 76 cords of wood at 5 dollars a cord; what was the amount of the hay and the wood? Ans. 873 dollars. 6. Bought 17 yards of cotton at 15 cents per yard, 46 gallons of molasses at 28 cents per gallon, 16 pounds of tea at 76 cents a pound, and 107 pounds of coffee at 14 cents a pound; what was the amount of my bill? Ans. 4257 cents. 7. A man travelled 78 days, and each day he walked 27 miles; what was the length of his journey? Ans. 2106 miles. 8. A man sets out from Boston to travel to New York, the distance being 223 miles, and walks 27 miles a day for 6 days in succession; what distance remains to be travelled? Ans. 61 miles. 9. What cost a farm of 365 acres at 97 dollars per acre? Ans. 35405 dollars. 10. Bought 376 oxen at 36 dollars per ox, 169 cows at 27 dollars each, 765 sheep at 4 dollars per head, and 79 elegant horses at 275 dollars each; what was paid for all? Ans. 42884 dollars. 11. J. Barker has a fine orchard, consisting of 365 trees, and each tree produces 7 barrels of apples, and these apples will bring him in market 3 dollars per barrel; what is the income of the orchard? Ans. 7665 dollars. 12. J. Peabody bought of E. Ames 7 yards of his best broadcloth at 9 dollars per yard, and in payment he gave Ames a |