" 10. What cost 42 horses at 175 dollars each? OPERATION. SOLUTION.—42 equals 6 times 7: if one horse costs 175 175 dollars, 7 horses will cost 7 times 175 dollars, which are 1225 dollars; and 6 times 7 horses, or 42 horses, will cost 6 times 1225 dollars, which are 7350 1225 dollars. Therefore, etc. 7350 11. What will 72 yards of cloth cost at the rate of $6.35 a yard ? Ans. $457.20. 12. What will 84 yoke of oxen cost at the rate of $135 a yoke? Ans. $11,340. 13. How much must I pay for grading 132 rods of railroad at $345 a rod ? Ans. $45,540. 14. A farm containing 144 acres of land was sold at the rate of $296 an acre; for what did it sell? Ans. $42,624. 27 CASE II. 112. When ciphers are at the right of one or both factors. 1. Multiply 2700 by 120. OPERATION. SOLUTION.—27 multiplied by 12 equals 324, hence 2700 2700 multiplied by 12 equals 100 times 324, or 32400; 120 and multiplied by 120 will be 10 times as much, or 324000. (Prin. 1. Art. 110.) 324000 Rule.- Take the product of the numbers denoted by the significant figures, and annex as many ciphers to the result as are found at the right of both factors. What is the value 2. Of 3560 x 360 ? Ans. 1281600. 3. Of 48700 x 450 ?, Ans. 21915000. 4. Of 30900 x 670 ? Ans. 20703000. 5. Of 28500 X 8500 ? Ans. 242250000. 6. Of 67400 x 9600 ? Ans. 647040000. 7. Of 865000 x 7800 ? Ans. 6747000000. 8. Of 723000 x 9700 ? Ans. 7013100000. 9. Of 3876000 x 35100 ? Ans. 136047600000. 10. Of 4264000 x 20400 ? Ans. 86985600000. CASE III. 113. When one part of the multiplier is a factor of another part. 1. Multiply 576 by 246. SOLUTION.—In this example 6, one OPERATION. part of the multiplier, is a factor of 24, 576 the other part; hence we may proceed 246 thus: 6 times 576 equals 3456; and 24 times 576 equals 4 times 3456, or 3456 Prod. by 6 units. 13824, which we write as tens: taking 13824 Prod. by 24 tens. the sum of the partial products, we 141696 Ans. have 141696. 2. Multiply 43526 by 24832. SOLUTION.—We first multiply by the 8 hundreds, OPERATION. writing the first figure in hundreds place; we then 43526 multiply this product by 4, writing the first figure in 24832 units place, which gives 32 times the number; we then multiply the first product by 3 and write the 348208 first figure in thousands place, which gives 24 thou 1392832 sand times the number : taking the sum of these par 1044624 tial products, we have the entire product. 1080837632 Rule.-I. Multiply the multiplicand by some term of the multiplier which is a factor of one or more parts of the multiplier. II. Multiply this product by a factor which, taken with the terms used, will produce other parts of the multiplier, and place the right hand term of the product under the right hand term of the part of the multiplier thus used. III. Continue thus until the entire multiplier is used ; tho sum of all the products will be the entire product. What is the value 3. Of 4675 X 355 ? Ans. 1659625. 4. Of 7608 x 369? Ans. 2807352. 7. Of 13524 x 428 ? Ans. 5788272.. 6. Of 37643 X 2807 ? Ans. 105663901. 7. Of 57316 X 35728 ? Ans. 2047786048. 8. Of 618504 X 24642 ? Ans. 15241175568. 9. Of 730592 x 408848 ? Ans. 298701078016. 10. Of 395076 x 576426 ? Ans. 227732078376. CASE IV. 114. When the multiplier differs but little from 100, 1000, 10000, etc. 1. Multiply 5607 by 996. OPERATION. SOLUTION.Since 996 equals 1000 minus 4, 996 5607 times 5607 is the same as 1000 times the number minus 996 4 times the number; 1000 times 5607 is 5607000, and 4 times 5607 is 22428, and the difference is 5584572. 5607000 22428 Hence the 5584572 Rule.- Annex to the multiplicand as many ciphers as there are terms in the multiplier ; multiply the multiplicand by the difference between the multiplier and 100, 1000, etc., and add or subtract the two results as the multiplier is greater or less than 100, 1000, etc. NOTE.—This rule is of especial value when the multiplier is a little less than 100, 1000, etc. What is the value 2. Of 76573 x 93 ? Ans. 7121289. 3. Of 53781 x 998 ? Ans. 53673438. 4. Of 64336 x 105 ? Ans. 6755280. 5. Of 397842 x 9994 ? Ans. 3976032948. 6. Of 587543 x 9989 ? Ans. 5868961027. 7. Of 473721 x 9970 ? Ans. 4722998370. 8. Of 5654321 X 99980 ? Ans. 565319013580. 9. Of 7733447 x 998800 ? Ans. 7724166863600. PRACTICAL PROBLEMS. 1. A clerk's salary is $25 a week; he pays $7.75 for his board, and $5.25 for other expenses; how much will he save in a year? Ans. $624. 2. A man is north of Cincinnati 50 miles; if he should travel south 15 days at the rate of 35 miles a day, how far would he be from Cincinnati ? Ans. 475 miles. 3. The library of an academy consists of 4 cases, each containing 16 shelves, and each shelf averaging 77 books; how many books are in the library ? Ans. 4928 books. 4. Stewart & Co. bought 18 cases French chintzes, each case containing 45 pieces, and each piece 34 yards, at 17 cents a yard; wbat was their bill ? Ans. $4681.80. 5. A shipping firm received the following freight: 25 hogsheads tobacco at $10.50 a bhd.; 3000 quarters of wheat at $2 a quarter, and 4000 barrels of petroleum at $1.10 a barrel; what was the amount received ? Ans. $10,662.50. 6. G. Pidcock & Co. bought Obio sheep as follows: 209, averaging 79 lbs. each, at 6 cents per lb.; 108, averaging 100 lbs., at 7 cents per lb.; 68, averaging 56 lbs., at 5 cents per lb.; and 90 Canadian lambs, averaging 77 lbs., at 8 cents per lb.; they sold the whole at an average price per head of $6; what was their profit, deducting $67.20 for expenses? Ans. $291.34. 7. A bankrupt failed for $100,000; his assets were as follows: a farm of 450 acres, worth $97 an acre; a house worth $25,000 ; 55 shares New York Central Railroad, at $101 a share; 99 shares Pacific Mail at $39 a share; and 155 shares Western Union Telegraph at $82 ; wbat remains unpaid ? Ans. $9224. 8. A commission merchant in Philadelpbia sold the following consignment from Cincinnati: 200 barrels prime mess pork at $19 per barrel; 1990 lbs. pickled hams at 11 cents per lb. ; 996 lbs. smoked hams at 13 cents a lb., and 1080 lbs. of lard at 14 cents per lb.; in return he forwarded 150 barrels crushed sugar, 200 lbs. each, at 11 cents per lb., and the remainder he paid by draft; what was the amount of the draft ? Ans. $999.58. 9. A wool dealer in New York, on making up his accounts for the week ending December 18, 1875, found his purchase to be as follows: Smyrna unwashed fleeces, 450 lbs., at 20 cents; Syrian washed, 230 lbs., at 33 cents; Donskoi washed, 140 lbs., 31 cents; Cape of Good Hope, 75 lbs., 36 cents. During the same time his sales were as follows: Texas fine, Eastern, 250 lbs., 30 cents; medium, 190 lbs., 29 cents; Texas, Western, 81 lbs., 20 cents; California Spring Clip, superior, unwashed, 150 lbs., 33 cents; coarse, 120 lbs., 22 cents; burry, 75 lbs., 15 cents; what are the amounts of the purchases and sales for the week, and what is the balance of the account? Ans. $2.85. DIVISION. 115. Division is the process of finding the quotient of two numbers. 116. The Quotient of two numbers is a number which expresses how often one number is contained in another. 117. The Dividend is the number to be divided. 118. The Divisor is the number by which we divide. 119. The Remainder is the number which is sometimes left after dividing: 120. The Terms in Division are the Dividend, the Divisor, and the Quotient. 121. The Sign of Division is • , and is read divided by. It denotes that the number preceding it is to be divided by the number following it. Division is also indicated by writing the divisor beneath the dividend with a line between them; or by writing the divisor at the left of the dividend with a curved line between them; thus, 24, also 8)24. NOTES.—The Sign of Division is a short line, in the line of writing, with lots above and below the middle of it. 2. The symbol ; was introduced by Dr. John Pell, an English mathematician, born in 1610. PRINCIPLES. 1. The divisor and dividend are always similar numbers. For, it is evident that any number can be contained only in a similar number; and also that no number of times one concrete number can equal a concrete number of another kind. 2. The quotient is always an abstract number. For, since the quotient shows how many times the divisor is contained in the dividend, it cannot be apples times or peaches times, but simply an abstract number of times. 3. The remainder is a number similar to the dividend. For, since it is an undivided part of the dividend, it must be of the jame unit as the dividend. 4. If all the parts of the dividend be divided by the divisor, the whole dividend will be divided by it. For, all the partial quotients taken together will evidently be equal to the entire quotient. Note.—These principles are theoretically true, though in practice we lo sometimes divide by an abstract number and obtain a concrete quotient. |