MISCELLANEOUS EXAMPLES.

1. If the population of the world be as follows: America, 57,650,000; Europe, 263,517,496; Asia, 626,400,000; Africa, 100,000,000; Australia, 1,445,000; Polynesia, 1,500,000; and the average length of life be 33 years, what must be the average number of deaths annually? Ans. 31,833,712.

2. A farmer has in 3 bins 755 bushels of grain; there being in the first 125 bushels, and in the second 96 bushels, more than in the third; how many bushels in the second and third? Ans. 363 in the second, 267 in the third.

3. There is a certain island 30 miles in circumference. If A and B commence travelling round it, A at the rate of 3 miles an hour, and B at the rate of 5 miles an hour, how far apart will they be at the end of 30 hours?

4. Having money to invest, I purchased two farms at $ 1,750 each, and 19 shares of bank stock at $103 per share, and have left $113; how much money had I? Ans. $5,570.

5. It has been agreed by 12 men to gather 960 bushels of cranberries, and receive for their labor one half of the quantity gathered; after one half was gathered, one third of the men withdrew, leaving the others to complete the job. How many bushels should each man receive? Ans. Those who left, 20 bushels each; those who remained, 50 bushels each.

6. A drover made $ 652.00 by selling a lot of sheep, at a profit of 50 cents each; how many did he sell?

7. What will it cost to carpet a floor that is 18 feet wide and 27 feet long, provided the carpeting cost $2.25 per sq. yd.?

8. If a young man, by early rising and economy of time, can save for study and improvement of mind two and a half hours a day, how many years' study, of 12 hours per day, can thus be gained in 20 years? Ans. 4y. 60d. 10h.

9. From 4 piles of wood, the first containing 7c. 76ft. 1671in., the second 16c. 28ft. 56in, the third 29c. 127ft. 1000in., the fourth 29c. 10ft. 1216in., I have sold 45 cords and 6 cord feet; how much remains? Ans. 37c. 19ft. 487in.

10. Boston is in north latitude 42° 21'; Portland is in latitude 1° 15′ north of Boston; and Charleston is in latitude

10° 40' south of Portland. What is the latitude of Charleston? Ans. 32° 56' north. 11. If 1 cubic foot of anthracite coal weighs 54 pounds, how many cubic feet of space are required to stow 2 tons of 2000 pounds each? Ans. 742 cubic feet. 12. Two engineers, A and B, surveyed a certain house-lot. A made its contents 3R. 18p. Oyd. 6ft. 64sq. in., but B made its contents 3R. 17p. 30yd. 8ft. 100sq. in. How much did the one differ from the other?

13. The products of the industry of 250,000 persons in Massachusetts, during the year 1855, amounted to $295,300,000. What was the average amount to each individual, and how much was added to the capital of the State, if one fourth of the whole amount was saved? Ans. $1,181.20 to each individual; $73,825,000 added to the capital.

14. The capacity of a certain cistern is 216 cubic feet; how many hogsheads of water will it contain?

15. What day of the month and what day of the year is the second Monday of May, in a common year commencing on Thursday? Ans. 11th day of May; 131st day of the year.

16. Purchased 18T. 17cwt. 3qr. 20lb. of copperas, at 4 cents per pound. I sold 4T. 6cwt. 1qr. 14lb. at 5 cents per pound, and 7T. 1cwt. 3qr. 10lb. at 6 cents per pound. Moses Atwood purchased one fourth of the remainder at 6 cents per pound. One half of what then remained I sold to J. Gale at 10 cents per pound. The remainder I sold to J. Smith at 12 cents per pound; but he has become a bankrupt, and I lose half my debt. What have I gained by my purchase? Ans. $894.07.

17. The distance between Boston and San Francisco is 2691 miles. If Nathan Swift of San Francisco and Oliver Fleet of Boston, on Thursday, the first day of January, 1857, set out to meet each other, Swift travelling 3 miles 7 furlongs 29 rods 15 feet per hour, and Fleet 5 miles 10 rods and 12 feet per hour, both travelling 6 hours per day, commencing at 8 o'clock, A. M., provided they rest on the Sabbath, in what year and month, and on what day of the month, and at what time of the day, will they meet, and how far will each have travelled?

Ans. On Monday, February 23, 1857, 2h. 30m. P. M. Swift, 1186m. 4fur. 22rd. 13ft. 6in.; Fleet, 1504m. 3fur. 17rd. 3ft.

EXAMPLES BY ANALYSIS.

1. If 7 pairs of shoes cost $8.75, what will one pair cost? what will 20 pairs cost? Ans. $25.00.

2. If 5 tons of hay cost $ 85, what will 1 ton cost? what will 17 tons cost? Ans. $289.00. 3. When $0.75 are paid for 3gal. of molasses, what is the value of 1gal.? What cost 37 gal.?

4. Gave $1.92 for 4lb. of tea; what cost 1lb.? what cost 37lb.? Ans. $17.76. 5. For 12lb. of rice I paid $ 1.08; what was paid for 1lb.; and what must I give for 25lb.? Ans. $2.25.

6. Gave S. Smith $ 63.00 for 9 tubs of butter; what was the cost of 1 tub? What cost 27 tubs ? Ans. $189.00.

7. T. Swan can walk 20 miles in 5 hours; how far can he walk in 1 hour? How long would it take him to walk from Bradford to Boston, the distance being in a straight line 28 miles?

8. If a hungry boy would eat 49 crackers in 1 week, how many would he eat in 1 day? how many would be sufficient to last him 19 days? Ans. 133 crackers.

9. Gave $20 for 5 barrels of flour; what cost 1 barrel? what cost 40 barrels ? Ans. $160.00. 10. For 3lb. of lard there were paid 36 cents; what was the cost of 371b.?

11. Paid F. Johnson 72 cents for 9 nutmegs; how many cents were paid for 1 nutmeg; and what should be charged for 37 nutmegs ? Ans. $2.96.

12. Paid 2£. 17s. 5d. for 52lb. of sugar; what cost 1lb.? what cost 76lb.?

13. Paid 4£. 3s. 11d. for 761b. of sugar; what cost 52lb.? 14. If a man walk 17m. 4fur. 28rd. in 6 days, how far will he walk in 100 days? Ans. 293m. 1fur.

15. If a farmer feed to his stock in 7 months 41bu. 3pk. 4qt. 1pt. of grain, how much is required for 1 month? how much for 7 years?

Ans. 502bu. 2pk. 6qt. 5p. 8yd. 6ft. 108in. will How large a field will pas

16. A field containing 39A. 2R. pasture 8 cows during the season. ture 1 cow? How large a field 72 cows?

17. If 4 casks of vinegar contain 63gal. 3qt., what are the contents of one cask? What are the contents of 37 casks? *Ans. 589gal. 2qt. 1pt. 2gi. 18. When 5yd. 3qr. 1na. of cloth cost $4, how much cloth can be bought for $1? How much for $36?

Ans. 52yd. 1qr. 1na. 19. If 11T. 3cwt. 2qr. of hay be sufficient to keep 4 horses 7 months, how much will keep 1 horse the same time? How much 23 horses? Ans. 64T. 5cwt. 12lb. 8oz.

20. If 12 men can dig a certain ditch in 286 days 4h. 33m., how long will it require 1 man to do the same labor? How long 72 men? Ans. 47 days 16h. 45m. 30sec.

21. If 27yd. 1qr. of cloth be required to make 21 coats, how many yards will be required to make 11 coats?

22. If a train of cars move at the rate of 174m. 26rd in 7 hours, how far will it move in 1 hour? How far in 10 hours? Ans. 248m. 5fur. 20rd.

23. If 4 cases of shoes, containing 60 pairs each, cost $192, what will 1 pair cost? What will 25 cases cost?

24. When 3A. 2R. 20rd. of land will buy 4 hogsheads of molasses, how much land will buy 1 hogshead? How much 30 hogsheads? Ans. 27A. OR. 30rd. 25. If a man can travel 20deg. 49m. 5fur. 35rd. 5yd. 3in. in 9 weeks, how far would he travel in 1 week? How far in 90 weeks? Ans. 207deg. 13m. 1fur. 25rd. 5yd.

PROPERTIES OF NUMBERS.

DEFINITIONS.

162. An integer is a whole number; as 1, 7, 16. All whole numbers are either prime or composite.

163. A prime number is a number which can be exactly divided only by itself and 1; as 1, 3, 5, 7, 11.

A composite number is a number which can be exactly divided by some number besides itself and 1; as 6, 9, 14, 18.

164. A factor of a number is such a number as will, by being taken an entire number of times, produce it; as, 3 is a factor of 9, and 4 a factor of 16.

165. A prime factor of a number is a prime number that will exactly divide it; thus the prime factors of 10 are the prime numbers 1, 2, and 5.

NOTE. Unity or 1 is not generally regarded as a prime factor, since multiplying or dividing any number by 1 does not alter its value. It therefore will be omitted when speaking of the prime factors of numbers.

A composite factor of a number is a composite number that will exactly divide it; thus, 6 and 8 are composite factors of 48.

166. Numbers are prime to each other when they have no factor in common; thus, 4, 9, and 23 are prime to each other.

167. An aliquot part of a number is such a part as will exactly divide it; as, 1, 3, and 5 are aliquot parts of 15.

The aliquot parts of a number include all its factors, prime and

An aliquant part of a number is such a part as will not exactly divide it; as 2, 4, 5, 7, and 8 are aliquant parts of 9.

168. The reciprocal of a number is the quotient arising from dividing 1 by the number; thus, the reciprocal of 2 is .

169. The power of a number is the product obtained by taking the number a certain number of times as a factor; thus 25 is a power of 5.

NOTE. When the number is taken once. it is called its first power; when taken twice, as a factor, the product is called its second power; and so on. The second power of a number is sometimes termed its square, and the third power, its cube.

170. The exponent of a power is a figure written at the right of a number, and a little above it, to show how many times it is taken as a factor; thus, in the expression 42, the exponent is the 2, and the whole is read 4 second power; and in 7, it is the 3, and the whole read 7 third power.

NOTE. The first power of a number being always the number itself, its exponent is not expressed.