97. Abbreviated method of long division. Ex. 1. Divide 34634 by 134. OPERATION. The operation we perform 134) 3 4 6 34 (258 Ans. thus: finding the first quotient 783 1134 134 figure to be 2, we say 4 × 2: 8; 16 - 8 8; 3 X 2 = = 6, 7; 14-7 2, and 1 car 3 3; 3 - now bring down 3, and find 0 - 3; 5, and 1 = the next quotient figure to be 5, then, 4 × 5 3 X 5= = 15, and 2 carried 17; 8 7 = carried = 6; 7—6—1. We next bring down 4, and find the next quotient figure to be 8; then, 4 X 8 2; 3 X 8 24, and 3 carried 27; 13 - 7 = 6; 1 X 8 8, and 2+1 carried 0; 1-1 = 0. Hence, this method is that of ordinary long division (Art. 73), abridged by subtracting each figure of the product of the divisor and a quotient figure, as it is obtained, and writing down only the remainders. 3. A certain orchard contains 1088 trees in 34 rows; how many trees in each row? Ans. 32 trees. 4. A speculator sold 191 mules at a gain of 5157 dollars what was the gain on each? ; 5. The population of Massachusetts, in 1855, was 1,133,123; how many would that be to each of its 7750 square miles of surface? Ans. 1464928. 98. When the divisor or dividend has a fraction annexed. Ex. 1. Divide 3692 by 83. OPERATION. 83692 3 3 Ans. 426. We bring the divisor and dividend to the same fractional parts as those denoted by the given 26) 11076 (426 Ans. fraction, which are thirds, by mul 104 67 52 156 156 tiplying them both by 3, the number of thirds in 1. We thus change the divisor to 26 thirds, and the dividend to 3692 thirds, without changing the quotient (Art. 83); and dividing obtain 426, the answer required. RULE.-Reduce the divisor and dividend to the same fractional parts as are denoted by the given fraction, and then divide as in whole numbers. NOTE. In case there should be a remainder after the division, its true value may be found as in Art. 76. EXAMPLES. 2. Divide 13120 by 91. 3. Divide 766729 by 36. Ans. 141834. Ans. 2129383. Ans. 101 4. Divide 2090 by 205. 5. If 16 feet make a rod, how many rods are there in 10626 feet? 6. If 69 degrees make a mile, how many degrees in 12450 miles? Ans. 180 degrees. 7. How many barrels of pork, at 17 dollars a barrel, may be bought for 5591 dollars? 8. A merchant has sold 2667 yards of silk in dress patterns of 10 yards each; required the number of patterns. Ans. 254 patterns. 9. How many plats, each of 2721 square feet, are equal in extent to 136125 square feet? Ans. 500 plats. 10. How many days will 119 pounds of bread last a man, if Ans. 49 days. 11. How many casks of 31 gallons each will be required to hold 12968 gallons of cider? Ans. 415 casks. he consume 23 pounds per day? PROBLEMS, FOUNDED UPON THE FUNDAMENTAL RULES. 99. THE following problems are founded upon the general principles of addition, subtraction, multiplication, and division, the fundamental operations of arithmetic, which have already been explained. 1. The parts of a number being given, to find the number. Add the parts together (Art. 47). 2. The sum of two numbers and one of the numbers being given, to find the other number. - From the sum subtract the given number (Art. 50). 3. The difference between two numbers and the larger number being given, to find the smaller. From the larger number subtract the difference (Art. 52). 4. The difference between two numbers and the smaller number being given, to find the larger. Add the smaller number and the difference together (Art. 51). 5. The sum and the difference of two numbers being given, to find the numbers. - From the sum subtract the difference and divide the remainder by 2, for the smaller number; add the difference to the smaller number, for the larger (Art. 52). NOTE.-In like manner, when the sum and differences are given, may be found any number of required numbers. After subtracting the differences from the sum, if there are 3 required numbers, divide by 3 for the smaller number; if 4, divide by 4, and so on. 6. The product of two numbers, and one of the numbers being given, to find the other numbers, -Divide the product by the given number (Art. 62). 7. The product of three numbers, and two of the numbers being given, to find the other number. Divide the given product by the product of the two given numbers (Art. 72). 8. The dividend and quotient being given, to find the divisor. Divide the dividend by the quotient (Art. 77). 9. The divisor and quotient being given, to find the dividend. — Multiply the divisor and quotient together (Art. 74). EXAMPLES. 1. A carpenter has contracted to build one house for 2763 dollars, another for 4650 dollars, and a third for 8950 dollars. How much is he to receive for them all? Ans. 16363 dollars. 2. N. Chandler has invested in railroad stock and a small farm 929 dollars. If the amount invested in the stock was 279 dollars, how much did the farm cost him? Ans. 650 dollars. 3. Mount Black, the highest peak of the Blue Ridge, is 6476 feet high, which is 242 feet higher than Mount Washington, the highest peak of the White Mountains. What is the height of Mount Washington? 62341 4. The city of Mexico, in 1519, was taken by Cortes, and, 328 years after, by General Scott. In what year did it yield to Scott? Ans. 1847. 5. Two travellers, A and B, meeting on a journey, found they had both travelled 1963 miles, and that A had travelled 199 miles more than B. What distance had each travelled? Ans. A, 1081 miles; B, 882 miles. 6. A father gave his three sons 4698 dollars, of which James received 250 dollars more than George, and Edwin 410 dollars more than George. What sum did each receive? Ans. George $1346; James $1596; Edwin $1756. 7. There was paid for 217 chests of tea 8463 dollars. How much was that a chest? 8. How many weeks will 684 bushels of oats last 19 horses, each horse consuming 3 bushels a week? 9. On dividing 3808 dollars among a certain number of men, it was found that the share of each was 224 dollars. Required the number of men. Ans. 17 men. 10. A certain missionary society divides its income among 99 missions, giving to each an average of 575 dollars. What is its income? Ans. 56925 dollars. Ans. 30. 11. The product of 96, 22, and one other number is 63360. What is the other number? 12. The divisor being 13 and the quotient 1101, what is the dividend? Ans. 14313. MISCELLANEOUS EXAMPLES. 1. What is the distance by railroad from Boston to Galena, it being from Boston to Albany 200 miles, from Albany to Niagara Falls 305, from Niagara Falls to Detroit 230, from Detroit to Chicago 282, from Chicago to Galena 171? 2. Sold J. Weimer my best horse for 175 dollars, my secondbest chaise for 87 dollars, and a good harness for 31 dollars. He has paid me in cash 38 dollars, and has given me an order on S. Lantz for 12 dollars. How many dollars remain due? 3. Bought 97 barrels of molasses at $5 a barrel. Gave 17 barrels to support the poor, and the remainder was sold at $8 a barrel. Did I gain or lose, and how much? Ans. $ 155 gain. 4. It requires 1728 cubic inches to make one cubic foot; required the number of cubic inches in 3787 cubic feet. 2 5. If a garrison of 987 men are supplied with 175686 pounds of beef, how much will there be for each man? Ans. 178 lbs. 6. Albert Peyton sold off from his farm 120 acres, gave his son 80 acres, and had remaining 160 acres; what number of acres did his farm contain before he disposed of any portion of it? Ans. 360 acres. 7. The annual revenue of a gentleman being $8395, how much per day is that equivalent to, there being 365 days in a year? Ans. $23. 8. What is the difference between half a dozen dozen, and six dozen dozen? Ans. 792. 9. Bought of F. Johnson 8 barrels of flour at $7 per barrel, and 3 hundred-weight of sugar at $8 per hundred. What was the amount of his bill? 10. George Adams bought an equal number of cows and oxen for 3952 dollars. For the cows he paid 31 dollars each, and for the oxen 45 dollars each. How many of each kind did he buy? Ans. 52. 11. If a certain quantity of provisions will sustain 13 men 4 days, how long would it sustain 1 man? Ans. 52 days. 12. The Globe Manufacturing Company has a capital of 250,000 dollars, divided into 500 shares. share? How much is each -31 |