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EXAMPLES BY ANALYSIS.
1. If 7 pairs of shoes cost $ 8.75, what will one pair co£t? 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 I gal.? What cost 37 gal.?
4. Gave $ 1.92 for 41b. of tea; what cost l1b.? what cost 371b.? Ans. $ 17.76.
5. For 121b. of rice I paid $ 1.08; what was paid for lib.; and what must I give for 251b.? 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 31b. 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 521b. of sugar; what cost l1b.? what cost 761b.?
13. Paid 4£. 3s. lid. for 761b. of sugar; what cost 521b.?
14. If a man walk 17m. 4fur. 28rd. in 6 days, how far will he walk in 100 days? Ans. 293m. lfur.
15. If a farmer feed to his stock in 7 months 41bu. 3pk. 4qt. lpt. of grain, how much is required for 1 month? how much for 7 years? Ans. 502bu. 2pk. 6qt.
16. A field containing 39A. 2R. 5p. 8yd. 6ft. 108in. will pasture 8 cows during the season. How large a field will pasture 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. lpt. 2gi.
18. When 5yd. 3qr. lna. of cloth cost $ 4, how much cloth can be bought for $ 1? How much for $ 36?
Ans. 52yd. lqr. lna.
19. If 11T. 3cwt. 2qr. of hay be sufficient to keep 4 horses 7f\ months, how much will keep 1 horse the same time? How much 23 horses? Ans. 64T. ocwt. 121b. 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. lqr. 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. SOrd.
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. lfur. 25rd. 5yd.
PROPERTIES OF NUMBERS.
162. An integer is a whole number; as 1, 7, 16.
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.
Note. — The aliquot parts of a number include all its factors, prime and composite.
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 73, 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.
PROPERTIES OF PRIME NUMBERS.
171. No direct process of detecting prime numbers has been discovered.
Note.— A few fncts, such as are given below, if kept in mind, will aid «omewhat in ascertaining whether a number is prime or not .
172. TJte only even prime number is 2; since all other even numbers, as 4, 6, 8, and 10, it is evident, can be exactly divided by 2, and therefore must be composite.
173. The only prime number having 5 for a unit or righthand figure is 5; since every other whole number thus terminating, as 15, 25, 35, and 45, can be exactly divided by 5, and therefore must be composite.
174. Every prime number, except 2 and 5, must have 1, 3, 7, or 9 for the right-hand figure; since all other numbers are composite.
175. Every prime number above 3, when divided by 6, must leave 1 or 5 for a remainder; since every prime number above 3 is either 1 greater or 1 less than 6, or some exact number of times 6.
176. In a series of odd numbers written in their proper or natural order, if beginning with 3 every Third number, with 5 every Fifth, with 7 every Seventh, be cancelled, as composite, the remaining numbers, with 2, will be the prime numbers of the natural series. Thus, in the series 1, 3, 5, 7, 0, 11, 13, 1$, 17, 19, fti, 23, ftfi, %t, 29, 31, #3, &$, 37, 30, 41, 43, i$, 47, ^0, every third number from the 3, every fifth from the 5, every seventh from the 7, every ninth from the 9, and so on, being cancelled, the remaining numbers, with 2, are all the prime numbers under 50.
Note 1. —In the series, every third number from the 3 contains that number as a factor; every fifth number from the 5, that number as a factor; and so on.
Note 2. —The whole number of prime numbers from 1 to 100,000 is 9,593. Although all of these, except 2 and 5, end in 1, 3, 7, or 9, there are, within the same range, no less than 30,409 composite numbers terminating with some one of the same figures.