of $105 in implements and machinery. What was the aggregate value of the farms and implements ? Ans. $3,424,164,225. Find the values of the following expressions : 11. 24 x 55 — 73 ? Ans. 49,657. 12. 153 - (32 X 25) + 2082 - 9 X 24? Ans. 46,207. 13. 22 + 33 + 44 + 55 + 66 ? 14. In 1852 Great Britain consumed 1200000 bales of American cotton; allowing each bale to contain 400 pounds, what was its total weight? 15. If a house is worth $2450, and the farm on which it stands 6 times as much, lacking $500, and the stock on the farm twice as much as the house, what is the value of the whole ? Ans. $21550. 16. A flour merchant bought 1500 barrels of flour at 7 dollars barrel; he sold 800 barrels at 10 dollars a barrel, and the remainder at 6 dollars a barrel. How much was his gain ? 17. A man invests in trade at one time $450, at another $780, at another $1250, and at another $2275; how much must he add to these sums, that the amount invested by him shall be increased fourfold ? Ans. $14,265. 18. At the commencement of the year 1858 there were in operation in the United States 35000 miles of telegraph; allowing the average cost to be $115 per mile, what was the total cost ? 19. The cost of the Atlantic Telegraph Cable, as originally made, was as follows; 2500 miles at $185 per mile, 10 miles deepsea cable at $1450 per mile, and 25 miles shore ends at $1250 per mile. What was its total cost ? Ans. $1,258,250. 20. For the year ending June 30, 1859, there were coined in the United States 1401944 double eagles valued at twenty dollars each, 62990 eagles, 154555 half eagles, and 22059 three dollar pieces; what was the total value of this gold coin? Ans. $29,507,732. 1 DIVISION 103. Division is the process of finding how many times one number is contained in another. 104. The Dividend is the number to be divided. 105. The Divisor is the number to divide by. 106. The Quotient is the result obtained by the process of division. 107. The Reciprocal of a number is 1 divided by the number. Thus, the reciprocal of 15 is 1 ; 15, or is. Notes.-1. When the dividend does not contain the divisor an exact number of times, the part of the dividend left is called the Remainder, which must be less than the divisor. 2. As the remainder is always a part of the dividend, it is always of the same name or kind. 3. When there is no remainder the division is said to be exact. 108. The method of dividing any number by another depends upon the following principles : I. Division is the reverse of multiplication, the dividend corresponding to the product, and the divisor and quotient to the factors. II. If all the parts of a number be divided, the entire number will be divided. Since the remainder in dividing any part of the dividend must be less than the divisor, it can be divided only by being expressed in units of a lower order. Hence, III. The operation must commence with the units of the highest order. 1. Divide 2742 by 6. ANALYSIS. We write the divisor at the left of the dividend, separated from it by a line. 6) 2742 As 6 is not contained in 2 thousands, we take 457 Ans. the 2 thousands and 7 hundreds together, and proceed thus; 6 is contained in 27 hundreds 4 hundred times, and the remainder is 3 hundreds; we write 4 in hundreds' place in the quotient, and unite the remainder, 3 hundrede, OPERATION. to the next figure of the dividend, making 34 tens; then, 6 is contained in 34 tens 5 tens times, and the remainder is 4 tens; writing 5 tens in its place in the quotient, we unite the remainder to the next figure in the dividend, making 42; 6 is contained in 42 units 7 times, and there is no remainder; writing 7 in its place in the quotient, we have the entire quotient, 457. Note 1.–The different numbers which we divide in obtaining the successive figures of the quotient, are called partial dividends. 2. Divide 18149 by 56. ANALYSIS. As neither 1 nor 18 OPERATION. will contain the divisor, we take 56) 18149 (324%& Ans. three figures, 181, for the first par168 tial dividend. 56 is contained in 134 181 3 times, and a remainder; we 112 write the 3 as the first figure in 229 the quotient, and then multiply 224 the divisor by this quotient figure; 3 times 56 is 168, which subtracted 5 from 181, leaves 13; to this remainder we annex or bring down 4, the next figure of the dividend, and thus form 134, the next partial dividend ; 56 is contained in 134 2 times, and a remainder; 2 times 56 is 112, which subtracted from 134, leaves 22; to this remainder we bring down 9, the last figure of the dividend, and we have 229, the last partial dividend ; 56 is contained in 229 4 times, and a remainder; 4 times 56 is 224, which subtracted from 229, gives 5, the final remainder, which we write in the quotient with the divisor below it, thus completing the division, (35). Note 2.-When the multiplication and subtraction are performed mentally, as in the first example, the operation is called Short Division; but when the work is written out in full, as in the second example, the operation is called Long Division. The principles governing the two methods are the same. 109. From these principles and illustrations we derive the following general RULE. I. Beginning at the left hand, take for the first partial dividend the fewest figures of the given dividend that will contain the divisor one or more times ; find how many times the divisor is contained in this partial dividend, and write the result in the quotient; multiply the divisor by this quotient figure, and subtract the product from the partial dividend used. II. To the remainder bring down the next figure of the dividend, with which proceed as before; and thus continue till all the figures of the dividend have been divided. III. If the division is not exact, place the final remainder in the quotient, and write the divisor underneath. 110. Proof. There are two principal methods of proving division. 1st. By multiplication. Multiply the divisor and quotient together, and to the product add the remainder, if any; if the result be equal to the dividend, the work is correct. (108, I.) Note.—In multiplication, the two factors are given to find the product; in division, the product and one of the factors are given to find the other factor. 2d. By excess of 9's. 111. Subtract the remainder, if any, from the dividend, and find the excess of 9's in the result. Multiply the excess of 9's in the divisor by the excess of 9's in the quotient, and find the excess of 9's in the product; if the latter excess is the same as the former, the work is supposed to be correct. (85.) (1.) 6) 473832 EXAMPLES FOR PRACTICE. (3.) (4.) 12) 73042164 Quotients. 4732. 8721. 5280. 291042. 5. Divide 170352 by 36. 1. Divide 409887 by 47. 7. Divide 443520 by 84. 8. Divide 36380250 by 125. 9. Divide 1554768 by 216. 10. Divide 3931476 by 556. 11. Divide 48288058 by 3094. 12. Divide 11214887 by 232. 13. Divide 27085946 by 216. 14. Divide 29137062 by 5317. 15. Divide 4917968967 by 2359. Rem. Rem. 7. 194. 5219. 1255 16. What is the value of 721198 :- 291 ? Rem. 100. 17. What is the value of 3844419 = 657? 342. 18. What is the value of 536819237 = 907 ? 403. 19. What is the value of 571913007145 : 37149 ? 12214. 20. What is the value of 48659910 : 51001? 5009. 21. The annual receipts of a manufacturing company aro $147675; how much is that per day, there being 365 days in the year? Ans. $404315 22. The New York Central Railroad Company, in 1859, owned 556 miles in length of railroad, which cost, for construction and equipment, $30732518; what was the average cost per mile? Ans. $55,271536. 23. The Memphis and Charleston Railroad is 287 miles in length, and cost $5572470; what was the average cost per mile? Ans. $19,416.1987. 24. The whole number of Post offices in the United States, in 1858, was 27977, and the revenue was $8186793; what was the average income to an office? ABBREVIATED LONG DIVISION. 112. We may avoid writing the products in long division, and obtain the successive remainders by the method of subtraction employed in the case of several subtrahends. (76.) 1. Divide 261249 by 487. ANALYSIS. - Dividing the first partial 487) 261219 (536 OPERATION. dividend, 2612, we obtain 5 for the first 177 figure of the quotient. We now multi313 ply 487 by 5'; but instead of writing the 217 Rem. product, and subtracting it from the partial dividend, we simply observe what figures must be added to the figures of the product, as we proceed, to give the figures of the partial dividend, and write them for the remainder sought. Thus, 5 times 7 are 35, and 7 (written in the remainder,) are 42, a number whose unit figure is the same as the right hand figure of the partial dividend ; 5 times 8 are 40, and 4, the tens of the 42, are 44, and 7 (written in the remainder,) are 51; 5 |