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17. If a family consume in 1 year 598gal. 2qt. of molasses, how much will be necessary for 1 month ?
18. John Smith has 12 silver spoons, weighing 31b. 10oz. llpwt. ; what is the weight of each spoon ?
19. Samuel Johnson bought 7 loads of timber, measuring 56T. 19ft. ; what was the quantity in each load ?
20. If the moon, in 10 days, move in her orbit 4S. 11° 55' 50", how far does she move in 1 day?
21. If $ 9 will buy 241b 83 33 13 10gr. of ipecacuanha, how large a quantity will $ 1 purchase ?
22. When $ 12 will buy 34A. OR. 32p. 8yd. 5ft. 48in. of wild land, how much will $ 1 buy?
23. Joseph Doe will cut 24 cords 105 feet of wood in 9 days; how much will he cut in 1 day?
24. When 8 acres of land produce 25ch. 17bu. 3pk. 4qt. of grain, what will 1 acre produce?
110. When the divisor is a composite number, and none of its factors exceed 12.
Ex. 1. When 24 yards of broadcloth are sold for 57£ 10s. Od., what is the price of 1 yard?
Ans. 2£ 7s. 11d.
24 is equal to 6 X 4. 6) 57 10 ) 0 price of 24 yards.
We therefore divide the
price by one of these fac4)9 11
8 = price of 4 yards. tors, and the quotient aris2 7 11=price of 1 yard.
ing by the other. RULE. - Divide by the factors of the composite number in succession. !
EXAMPLES FOR PRACTICE.
2. If 360 tons of iron cost 6409£ 10s. Od., what is the cost of 1 ton?
3. If a man travel 117m. 7fur. 20rd. in 30 days, how far will he travel in 1 day?
4. If 84 loads of hay weigh 201 tons 6cwt. Oqr. 121b., what will 1 load weigh ?
5. When 72 ladies require 567yd. Oqr. Ona. for their dresses, how many yards will be necessary for one lady?
110. How does it appear that dividing by 6 in Ex. 1 gives the price of 4 yards? How do you divide by a composite number?
6. When 132 sailors require 470yd. Iqr. of cloth to make their garments, how many yards will be necessary for 1 sailor ?
111. When the divisor is not a composite number, and exceeds 12, or, if a composite number, and any of its factors exceed 12.
Ex. 1. If 23cwt. of iron cost 171£ 1s. 3d., what cost lcwt. ?
Ans. 7£ 8s. 9d.
We divide the pounds by 23, and obtain 7£ for the quotient, and 10£ remaining, which we reduce to shillings, and add the 1s., and again divide by 23, and obtain 8s. for the quotient. The remainder, 17s., we reduce to pence, and add the 3d., and again divide by 23, and obtain 9d. for the quotient; and, by uniting the several quotients, we obtain 7£ 8s. 9d. for the answer. Therefore,
The method of operation is like that by the general rule (Art. 109), excepting that more of the work is written down.
2. If $ 62 will buy 1095lb. 14oz. 6dr. of beef, how much may be obtained for $1?
3. Paid 280£ 5s. 9 d. for 97 tons of lead ; what did it cost
per ton ?
4. If a man travel 662m. 4fur. 28rd. 3yd. 2ft. 2in. in 38 days, how far will he travel in 1 day ?
5. When 98 acres produce 2739bu. 1pk. 5qt. of grain, what will 1 acre produce ?
6. A tailor made 347 garments from 2732yd. 2qr. 2na. of cloth; what quantity did it take to make 1 garment?
7. When 19 tons of iron will purchase 262A. 3R. 37p. 25yd. ist. 40in. of land, low much may be obtained for 1 ton ?
8. 451 tons of copper ore will purchase 8003T. 17cwt. lqr. 121b. 10oz. of iron ore, how much will 1 ton purchase ?
111. When the divisor is large, and not a composite number, how is the division performed ?
1. Bought 30 boxes of sugar, each containing 8cwt. 3qr. 2016., but having lost 68cwt. 2qr. Olb., I sold the remainder for 1£ 17s. 6d. per cwt. ; what sum did I receive ?
Ans. 375£. 2. A company of 144 persons purchased a tract of land containing 11067A. 1R. 8p. John Smith, who was one of the company, and owned an equal share with the others, sold his part of the land for 1s. 91d. per square rod; what sum did he receive?
Ans. 1101£ 12s. 1 d. 3. The exact distance from Boston to the mouth of the Columbia River is 2644m. 3fur. 12rd. A man, starting from Boston, traveled 100 days, going 18m. 7fur. 32rd. each day ; required his distance from the mouth of the Columbia at the end of that time.
Ans. 746m. 7fur. 12rd. 4. James Bent was born July 4, 1798, at 3h. 17m. A. M. ; how long had he lived Sept. 9, 1807, at 11h. 19m. P. M., reckoning 365 days for each year, excepting the leap year 1804, which has 366 days ?
Ans. 3353da. 20h. 2m. 5. The distance from Vera Cruz, in a straight line, to the city of Mexico, is 121m. 5fur. If a man set out from Vera Cruz to travel this distance, on the first day of January, 1848, which was Saturday, and traveled 3124rd. per day until the eleventh day of January, omitting, however, as in duty bound, to travel on the Lord's day, how far would he be from the city of Mexico on the morning of that day?
Ans. 43m. 4fur. 8rd. 6. Bought 16 casks of potash, each containing 7cwt. 3qr. 181b., at 5 cents per pound. I disposed of 9 casks at 6 cents per pound, and sold the remainder at 7 cents per pound; what did I gain?
Ans. $ 182.39. 7. A merchant purchased in London 17 bales of cloth for 17£ 18s. 10d. per bale. He disposed of the cloth at Havana for sugar at 1£ 17s. 6d. per cwt. Now, if he purchased 144cwt. of sugar, wbat balance did he receive ?
Ans. 35£ Os. 2d. 8. A and B commenced traveling, the same way, round an island 50 miles in circumference. A travels 17m. 4fur. 30rd. a day, and B travels 12m. 3fur. 20rd. a day ; required how far they are apart at the end of 10 days.
Ans, lm. 4fur. 20rd,
9. Bought 760 barrels of flour at $ 5.75 per barrel, which I paid for in iron at 2 cents per pound. The purchaser afterwards sold one half of the iron to an ax manufacturer ; what quantity did he sell ?
Ans. 54T. 12cwt. 2qr. 10. Bought 17 house-lots, each containing 44 perches, 200
From this purchase I sold 2A. 2R. 240ft., and the remaining quantity I disposed of at 1s. 2 d. per square foot ; what amount did I receive for the last sale ?
Ans. 5914£ 193. 5 d. 11: J. Spofford's farm is 100 rods square.
From this he sold H. Spaulding a fine house lot and garden, containing 5A. 3R. 17p., and to D. Fitts a farm 50rd. square, and to R. Thornton a farm containing 3000 square rods ; what is the value of the remainder, at $ 1.75 per square rod ?
Ans. $ 6235.25. 12. Bought 78 A. 3R. 30p. of land for $ 7000, and, having sold 10 house-lots, each 30rd. square, for $8.50 per square rod, I dis-pose of the remainder for 2 cents per square foot. How much do I gain by my bargain ?
Ans. $ 89265.35.
PROPERTIES OF NUMBERS.
112. An Integer is a whole number ; as 1, 6, 13. All numbers are either odd or even.
An Odd Number is a number that cannot be divided by 2 without a remainder ; thus, 3, 7, 11.
An Even Number is a number that can be divided by 2 without a remainder; thus, 4 8, 12.
Integers are also either prime or composite numbers.
A Prime Number is a number which can be exactly divided by no integer except itself or 1; as, 1, 3, 5, 7.
A Composite Number is a number which can be exactly divided by an integer other than itself or 1; as, 6, 9, 14.
Numbers are prime to each other when they have no factor (Art. 41) in common; thus, 7 and 11 are prime to each other, as are also 4, 15, and 19.
112. What is an integer ? What are all integers ? What is an odd number? An even number? What other distinctions of nunibers are mentioned? What is a prime number? When are numbers prime to each other? What is a composite number?
All the prime numbers not larger than 1109 are included in the following
113. A Prime Factor of a number is a prime number that will exactly divide it ; thus, the prime factors of 21 are the prime numbers 1, 3, and 7.
A Composite Factor of a number is a composite number (Art. 41) that will exactly divide it ; thus, the composite factors of 24 are the composite numbers 4 and 6.
Note 1.- Unity or 1 is not commonly regarded as a prime factor, since multiplying or dividing any number by i does not alter its value ; it will be omitted when speaking of the prime factors of numbers. Note 2.
- No direct process of finding prime numbers has been discovered. The following facts, however, will aid in ascertaining whether a number is prime or not; and, if not prime, will indicate one or more of its factors :
1. 2 is the only even prime number.
3. 3 is a factor of every number the sum of whose digits 3 will exactly divide ; thus, 15, 81, and 546 have each 3 as a factor.
4. 4 is a factor of every number whose two right-hand figures 4 will exactly divide ; thus, 316, 532, and 1724, have each 4 as a factor.
5. 5 is the only prime number having 5 for a unit or right-hand figure.
113. What is a prime factor? What is a composite factor? How is unity or 1 regarded ? Is there any direct process for determining prime numbers? Which is the only even prime number? Of what numbers is 2 a factor? Of what numbers is 3 a factor? Of what numbers is 4 a factor ?