EXAMPLES FOR PRACTICE. 2. Multiply 76589658 by 25. 3. Multiply 567898717 by 25. 4. Multiply 123456789 by 25. Ans. 1914741450. Ans. 14197467925. Ans. 3086419725. e 62. To multiply by 334. OPERATION. Ex 1. Multiply 87678963 by 33). Ans. 2922632100. We multiply by 100, as be3 ) 8 7 6 7 8 9 6 3 0 0 fore; and since 33}, the multi plier, is only one third of 100, 2 9 2 2 6 3 2 1 0 0 Product. we divide by 3 to obtain the true product. RULE. — Annex two ciphers to the multiplicand, and divide it by 3. EXAMPLES FOR PRACTICE. 2. Multiply 356789541 by 337 3. Multiply 871132182 by 331. 4. Multiply 583647912 by 334. Ans. 11892984700. Ans. 29037739400. Aus. 19454930400. 63. To multiply by 125. OPERATION. Ex. 1. Multiply 7896538 by 125. Ans. 987067250. We multiply by 1000, by an8) 78 9 6 5 3 8 0 0 0 nexing, three ciphers to the 9 8 7 0 6 7 2 5 0 Product. multiplier, is only one eighth of multiplicand; and since 125, the 1000, we dividę by 8 to obtain the true product. RULE. Annex three ciphers to the multiplicand, and divide it by 8. EXAMPLES FOR PRACTICE. 2. Multiply 7965325 by 125. 3. Multiply 1234567 by 125. 4. Multiply 3049862 by 125. Ans. 995665625. 62. The rule for multiplying by 33}? The reason for this rule ? - 63. The rule for multiplying by 125 ? The reason for the rule ? OPERATION. 64. To multiply by any number of 9's. Ex. 1. Multiply 4789653 by 99999. Ans. 478960510347. By adding 1 to any number 47 8 9 6 5 3 0 0 0 0 0 composed of nines, we obtain a 47 8 9 6 5 3 number expressed by 1 with as many ciphers annexed as there 47 8 9 6 0.5 1 0 3 4 7 Product. are nines in the number to which the 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 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. EXAMPLES FOR PRACTICE. 2. Multiply 1234567 by 999. Ans. 1233332433. Ans. 876542123457. Ans. 999998000001. CONTRACTIONS IN DIVISION. OPERATION. 65. To divide by 25. Ans. 49382148 Multiplying the dividend by 4 makes 1 2 3 4 5 6 7 it four times as great; therefore, to ob4 tain the true quotient, we must divide by 100, a divisor four times as great as the 49 3 8 26 8 Quotient. true one. This we do by cutting off two figures on the right. RULE. Multiply the dividend by 4, and divide the product by 100. EXAMPLES FOR PRACTICE. 2. Divide 9876525 by 25. 3. Divide 1378925 by 25. 4. Divide 899999 by 25. Ans. 395061. Ans. 55157. Ans. 3599986 64. The rule for multiplying by any number of g's? The reason for the rule? - 65. The rule for dividing by 25? The reason for the rule ? 66. To divide by 33). OPERATION. Ex. 1. Divide 6789543 by 333. Ans. 203686100 Multiplying the dividend by 3 makes 6 7 8 9 5 4 3 it three times as great; therefore, to 3 obtain the true quotient, we must divide by 100, a divisor three times as great 20 3 6 8 6 2 9 Quotient. as the true one. This is done by cut ting off two figures on the right. RULE. — Multiply the dividend by 3, and divide the product by 100. EXAMPLES FOR PRACTICE. 2. Divide 987654321 by 33. Ans. 2962962963 Ans. 261371,34 62 67. To divide by 125. OPERATION. Ex. 1. Divide 9874725 by 125. Ans. 789978 Multiplying the dividend by 8 makes 9 8 7 4 7 2 5 it eight times as great; therefore, to ob8 tain the true quotient, we must divide by 1000, a divisor eight times as great 7 8 9 9 718 0 0 Quotient. as 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. EXAMPLES FOR PRACTICE. 2. Divide 1728125 by 125. Ans. 13825. Ans. 3830106. Ans. 4729879. Ans. 3917 108 Ans. 66894703 66. The rule for dividing by 33}? The reason for the rule ? — 67. The rule for dividing by 125 ? The reason for the rule ? MISCELLANEOUS EXAMPLES, INVOLVING THE FOREGOING RULES. gain ? 1. A bought 73 hogsheads of molasses at 29 dollars per hogshead, and sold it at 37 dollars per hogshead ; what did he 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 Ans. 4257 cents. 7. A man traveled 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 traveled ? 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 one hundred-dollar bill ; how many dollars must Ames return to Peabody? Ans. 37 dollars. 13. Bought of P. Parker a cooking-stove for 31 dollars, 7 quintals of his best fish at 6 dollars per quintal, 14 bushels of rye at 1 dollar per bushel, and 5 mill-saws at 16 dollars each; in part payment for the above articles, I sold him eight thousand feet of rds at 15 dollars per thousand; how much must I pay him to balance the account? Ans. 47 dollars. 14. In 1 day there are 24 hours ; how many in 57 days? Ans. 1368 hours. 15. In one pound avoirdupois weight there are 16 ounces; how many ounces are there in 369 pounds ? Ans. 5904 ounces. 16. In a square mile there are 640 acres ; how many acres are there in a town which contains 89 square miles ? Ans. 56960 acres. 17. What cost 78 barrels of apples at 3 dollars per barrel ? Ans. 234 dollars. 18. Bought 500 barrels of flour at 5 dollars per barrel, 47 hundred weight of cheese at 9 dollars per hundred weight, and 15 barrels of salmon at 17 dollars per barrel ; what was the amount of my purchase ? Ans. 3178 dollars. 19. Bought 760 acres of land at 47 dollars per acre, and sold J. Emery 171 acres at 56 dollars per acre, J. Smith 275 acres at 37 dollars per acre, and the remainder I sold to J. Kimball at 75 dollars per acre ; how much did I gain by my sales ? Ans. 7581 dollars. 20. Bought a hogshead of oil containing 184 gallons, at 75 cents per gallon ; but 28 gallons having leaked out, I sold the remainder at 98 cents per gallon ; did I gain or lose by my bargain ? Ans. 1488 cents, gain. 21. Bought a quantity of flour, for which I gave 1728 dollars, there being 288 barrels ; I sold the same at 8 dollars per barrel ; how much did I gain ? Ans. 576 dollars. 22. Purchased a cargo of molasses for 9212 dollars, there being 196 hogsheads ; I sold the same at 67 dollars per hogshead; how much did I gain on each hogshead ? Ans. 20 dollars. 23. A farmer bought 5 yoke of oxen at 87 dollars a yoke; 37 cows at 37 dollars each; 89 sheep at 3 dollars apiece. He sold the oxen at 98 dollars a yoke ; for the cows he received 40 dollars each ; and for the sheep he had 4 dollars apiece. How much did he gain by his trade ? Ans. 255 dollars. |