22. If two persons should start from the same place and travel in opposite directions, one at the rate of 45 miles a day and the other at the rate of 67 miles a day, how far will they be apart in 67 days? Ans. 7504 miles.

23. A planter put up his cotton into 275 bales, averaging 576 pounds each, and sold it at 15 cents per pound; what did he receive for the whole? Ans. $23,760.

24. Mr. Angell's annual income is $9750, and his average daily expenditure is $9.75; what can he save in a year of 865 days? Ans. $6191.25.

25. A bank teller receives $2500 salary, and spends $520 for board, $475 for clothing, $112.65 for books, and $117.25 for other expenses annually; how much will he save in 5 years? Ans. $6375.50.

26. How many miles will a railroad conductor travel in a year if he makes a trip from Harrisburg to Philadelphia (106 miles) and back, every day except Sundays, reckoning 52 Sundays to a year? Ans. 66,356 miles.

27. A shoe-dealer bought 35 cases of French calf boots, each case containing 12 pairs, at $6.50 a pair; what did the bill amount to? Ans. $2730. 28. A division consisted of 4 regiments, each regiment of 9 companies, and each company of 98 men; how many men in the division ? Ans. 3528 men.

29. In a block of houses there are 45 buildings, each building containing 32 windows, and each window 12 panes of glass; how many panes of glass in the whole block of buildings? Ans. 17,280 panes. 30. If it requires 124 tons of iron worth $72 a ton to build one mile of a railroad, what will be the cost of iron enough to build a railroad 236 miles long? Ans. $2,107,008.

81. A field contains 5076 rows of corn, each row contain ing 4005 hills, and each hill 4 stalks; how many stalks of corn in the field? Ans. 81,317,520 stalks

4

CONTRACTIONS IN MULTIPLICATION.

78. Contractions in Multiplication are abbreviated methods of multiplying.

79. A Composite Number is the product of two or more numbers, each greater than a unit, called factors. Thus, 24 is a composite number, whose factors are 4 and 6, or 3 and 8, or 2, 3, and 4.

CASE I.

80. When the multiplier is a composite number. 1. Multiply 53 by 24.

SOLUTION.-24 equals 4 times 6, hence 24 times 53 equals 4 times 6 times 53. 6 times 53 equals 318, and 4 times 6 times 53 equals 4 times 318, which equals 1272; therefore, 53 multiplied by 24 equals 1272. Hence the

OPERATION

53

6

318

4

1272

Rule.-Multiply the multiplicand by one factor, this product by another factor, and thus continue until all the factors have been used; the last product will be the result required. Multiply

Ans. 375354.

5. 87 by 56. 6. 123 by 63.

Ans. 7749. 12. 8937 by 42. Ans. 13392. 13. 40729 by 72. 14. What cost 35 cows at 28 dollars apiece?

7. 248 by 54.

SOLUTION.-35 equals 5 times 7. If one cow costs 28 dollars, 7 cows will cost 7 times 28 dollars, which are 196 dollars; and 35 cows, which are 5 times 7 cows, will cost 5 times 196 dollars, which are 980 dollars. Therefore, etc.

Ans. 2932488

OPERATION.

28

7

196

5

980

15. What cost 21 yards of muslin, at 14 cents a yard?

Ans. $2.94.

Ans. $195.

16. What cost 15 barrels of fish, at $13 a barrel?

17. What cost 24 horses, at the rate of $165 each?

Áns. $3960

18. What will 42 cows cost at the rate of $27 apiece?

Ans. $1134.

19. What must I pay for 84 books, at the rate of $3.25 each? Ans. $273.

20. If one bushel of rye is worth $0.84, how much ar 64 bushels worth? Ans. $53.76. 21. If a yoke of oxen cost $137, what will 56 yoke cost at the same rate? dns. $7672. 22. What cost 64 acres of land, at $256 an acre; and what cost 81 acres at the same rate? Ans. 2d, $20,736

CASE II.

81. When there are ciphers at the right of one or both factors.

Principle.-Annexing one cipher to a number multiplie it by 10; annexing two ciphers multiplies it by 100; annexing three, multiplies it by 1000, etc.

For, adding one cipher removes each term one place the left, and thus makes it denote 10 times as many units as before, hence the entire number is ten times as great as before.

1. Multiply 26 by 140, also 2600 by 140. SOLUTION 1ST.-26 multiplied by 14 equals 364; hence 26 multiplied by 140, which is 10 times 14, equals 10 times 364, which, by annexing one cipher, equals 3640.

SOLUTION 2D.-14 times 26 equals 364, hence 14 times 26 hundred equals 100 times as much, which by annexing

OPERATION.

OPERATION.

26

2600

140

140

104

104

26

364000

26

3640

two ciphers, is 36400; and 140 times 2600 equals 10 times 36400 which, by annexing one cipher, is 364000. Hence the following rule:

Rule.-Take the product of the numbers denoted by the significant figures, and annex as many ciphers to the result 18 are found at the right of both factors.

IN ADDITION, SUBTRACTION, AND MULTIPLICATION.

1. I bought 3 spades at $1.50 apiece, 5 shovels at $1.25 apiece, and 4 rakes at $0.75 apiece; what was the entire bill? Ans. $13.75.

2. A's income is $5875 a year; he pays $975 for houserent, and 3 times as much for other expenses; how much can he save annually? Ans. $1975.

3. An army of 24,500 men lost in a battle 246 killed, 4 times as many, plus 145, wounded, and 1687 captured; how many remained in the army? Ans. 21,438 men.

4. Mary and Martha tried which could count the greater oumber in 25 minutes; Mary averaged 76 a minute, and Martha 81 a minute; how many did Martha count more than Mary? Ans. 125.

5. A poor boy attempted to walk from Philadelphia to Pittsburgh, a distance of 353 miles; after walking 18 miles a day for 15 days, how far was he from Pittsburgh?

Ans. 83 miles.

6. The light from a certain star is 7 years in coming to the earth; what is its distance, supposing there are 86,400 sec

onds in one day, and 365 days in a year, light moving at the ate of 192,500 miles a second? Ans. 42494760 millions.

7. A wholesale druggist bought as follows: 500 lb. opium at $7.20 per lb.; 375 lb. morphine at $4.75 per lb.; 240 lb. quinine, at $2.70 per lb.; 200 lb. camphor, at $0.29 per lb.; what was his whole bill? Ans. $6087.25.

8. A clerk's salary is $1500 a year; he pays $5 a week for his board and washing, $4 a month for car fare, and his other expenses amount to about $2 a day; how much can he save in a year? Ans. $462.

9. Mrs. Wilson bought at a furniture store an oiled walnut chamber set for $150, two cottage sets for $55 each, a dozen dining-room chairs at $3.50 each, and 3 camp chairs at $4.25 each; what was her bill? Ans. $314.75.

10. A shipping firm bought 150 lb. Manila bolt rope yarns at 16 cents per lb.; 450 lb. Manila whale lines at 17 cents a pound; 260 lb. Russia hemp tarred cordage, at 13 cents per lb.; and 850 lb. Sisal rope at 11 cents per lb.; what was the whole cost? Ans. $227.80.

11. A dry goods merchant sold to a Western customer as follows; 125 yards Cocheco fancy prints, at 7 cents a yard; 147 yards Cocheco robes, 8 cents; 156 yards Merrimac robes, 8 cents; 122 yards Merrimac checks and stripes, 7 cents; 125 yards French cashmeres, $1.15; what was the amount of the bill? Ans. $185.28.

12. James Chauncey & Co. sold to Barton Brothers, Dec. 23, 1875, the following: 450 bushels White Western oats, at 38 cents a bushel; 375 bushels New York No. 2 White oats, 43 cents; 240 bushels choice new Southern Black-eye peas, $1.45. Jan. 10, 1876, J. C. & Co. bought of Barton Brothers, 460 bushels Pennsylvania rye, 91 cents; 175 bushels No. 2 Canada barley, $1.10; 720 bushels common Winter Red Western wheat, $1.25; required the balance of the account. Ans. $830.85 favor of Barton Brothers.