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What indicates the degree

What is the sign for a power? for a root?

of root? What is an even number? an odd? DIVISIBILITY OF NUMBERS.

- When are numbers divisible by 2? by 3? by 4? by 5? by 6? by 8? by 9? by 10? by 11? by any composite number? How shall we ascertain whether any given number is prime? Describe Eratosthenes' sieve?

FACTORING OF NUMBERS. What is the simplest way of resolving numbers into their prime factors? scribe, and when would you use it? method, and explain the process. ond method, and explain the process.

What other method can you de-
Find the factors of 180 by first
Find the factors of 10296 by sec-

GREATEST COMMON DIVISOR.-What is a divisor of a number? a common divisor of two or more numbers? the greatest common divisor? Find the G. C. D. of three numbers by the first method given. Explain and give the rule. Find the G. C. D. of three numbers by second method. Explain and give the rule. In what cases is the second method the better? When is it necessary to find the G. C. D. of numbers?

FRACTIONS. - What is a fraction? Name and describe its terms. Name the different kinds of fractions of which you have learned. Define a common fraction; a decimal fraction; a proper fraction; an improper fraction; a mixed number; a compound fraction; a complex fraction. Give an example of each. Explain the expression. Upon what does the value of a fraction depend? Which of the fundamental rules is indicated by a fraction? What effect does multiplying the numerator of a fraction have upon that fraction? Why? In what other way could you produce the same effect, and why? What effect does dividing the numerator have upon a fraction? Why? In what other way could you produce the same effect, and why? What effect does multiplying both terms of a fraction by the same number have it? Why? What effect does dividing both terms of a fraction have upon it? Why?

upon

REDUCTION OF FRACTIONS. - How do you reduce fractions to lower terms? What is cancellation? How do you reduce whole or mixed numbers to improper fractions? How do you reduce improper fractions to whole or mixed numbers

?

MULTIPLICATION OF FRACTIONS. - How do you multiply a fraction by a whole number? a mixed number by a whole number? Explain, by an example, the method of multiplying a whole number by a fraction. Multiply a fraction by a fraction; explain and give the rule. How do you multiply a mixed number by a mixed number or a fraction? How

do you reduce compound fractions to simple ones? Can you give one general rule for multiplying fractions, whole or mixed numbers, by fractions?

DIVISION OF FRACTIONS.· How do you divide a fraction by a whole number? a mixed number by a whole number? a whole number by a fraction? Explain, by an example, the method of dividing a fraction by a fraction, and give the rule. Give one general rule for dividing a fraction, a whole or mixed number by a fraction. How do you reduce complex fractions to simple ones? How do you find what part of one number another is?

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LEAST COMMON MULTIPLE. -Define a multiple; a common multiple of two or more numbers; the least common multiple. When do you make use of the L. C. M.? Give and explain the first method of finding it; the second. What does the L. C. M. of prime numbers

equal?

COMMON DENOMINATOR. When are fractions said to have a common denominator? In what operations upon fractions do we first reduce them to those having the same denominator? Can we change fractions to those of any denominator? How? (Ans. By dividing or multiplying the numerator by the same number by which we divide or multiply the denominator to produce the denominator required.) What denominator is generally chosen? Reduce a simple, a compound, and a complex fraction to those of the same denominator, explain the process, and give the rule.

- How do you add

ADDITION AND SUBTRACTION OF FRACTIONS. fractions of different denominators ? How do you subtract one fraction from another? How do you add mixed numbers? In subtraction of one mixed number from another, how do you proceed when the fraction in the subtrahend exceeds that in the minuend?

G. C. D. AND L. C. M. OF FRACTIONS*- How do you find the G. C. D. of fractions? How do you find the L. C. M. of fractions? Find the G. C. D. of, and 13, and explain. Find the L. C. M. of †, §, and, and explain.

148. MISCELLANEOUS EXAMPLES.

1. Into strips of what widths may I cut cloth which is 36 inches wide, that none may be wasted, the width of the strips to be expressed in inches?

2. How many gallons in the largest vessel which will exactly measure 3 hogsheads, containing severally 128, 94, and 158 gallons?

* Optional.

3. What will 16

4. What cost 9

5. What cost 271

6

yards of cloth cost at $.53 a yard? bushels of corn, at $.87 a bushel?

acres of land, at $313 per acre?

of 2 of 56 times what number equals

?

7. I paid $.65 for 2 boxes of strawberries; what will be the cost of 45 boxes at the same rate?

8. What is my bill for 7 pear trees, $3 apiece for the trees, and $2 a dozen for setting?

9. What do I receive per pound by selling 15 pounds of coffee for $31?

10.

of a man's property is in land, and is valued at $2324§; what is the value of his whole property?

11. Bought of an acre of land for $40.75; what would 1 acre cost at the same rate?

12. What costs 3 pieces of calico, 37 yards in a piece, at 191 cents per yard?

13. If 32 acres of land cost $1100, what costs 1 acre?

14. Sold my house and farm of 47 acres for $6150; allowing $3500 for the house, what did I receive per acre for the land?

15. How long will a barrel of flour last a family of 8 persons, if it lasts 3 persons 4 months?

16* What number is that from which if

mainder will be ¿?

you take, the re

17. What number is that to which if you add 97, the sum will be 124g?

18. What is that number to which if you add of 261, the sum will be 147?

19. Bought 7 yards broadcloth at $5 per yard, 14 kerseymere at $1 per yard, 4 yards of silk at $ and yards of doeskin at $41 per yard, for which I ment a $100 bill. What balance is due me?

yards of

per yard,

gave

in pay

20. I have paving stones 12 inches long and 10 inches wide; what must be the width of a walk which will just receive these stones, laid either lengthwise or widthwise?

21 What is the smallest sum of money which can be exactly paid in pieces of money worth either $.163 or $.121?

22. How long will 200 pounds of meat last 9 persons at the rate of 2 pounds a day for each person?

23. What length of time would a man require to travel around the earth, the distance being 25000 miles, if he travel at the rate of 31 miles per day?

24. If a man can build 2 rods of wall in a day, how much can he build in 63 days?

25. What is that number
NOTE.; if be 2, 20 will equal 20 X

by 2?

of which exceeds

2 = 40, Ans.

by 112?

26. What number is that of which exceeds

27. How many bushels of wheat can a man purchase for $2724, at 31 cents per bushel?

28. What is (71⁄2 of § of 81). 29. What is (of 4) ÷ (1 of 3)?

30. If I buy 125 bushels of corn at 413 cents per bushel, and sell it at 52 cents per bushel, what do I gain?

31. What number divided by equals 1254?

32. What are the contents of 3 floors measuring as follows: 13 square yards, 32 square yards, and 4921 square yards?

33. The product of three numbers is 744; two of them are 84 and 6; what is the third?

34. Exchanged 42 tubs of butter, averaging 485 pounds, at 21 cents per pound, for 42 barrels of flour, at $93 per barrel, and received the balance in cash; required the balance.

35. I have three boxes of cloth, each containing 12 pieces, each piece containing 4 yards, weighing 34 pounds to the yard; what is the weight of the whole?

36. What will 423 quires of paper weigh at 37. A grocer has five casks of raisins of the 115 pounds, 117 pounds, 99 pounds, 100 pounds; what is the average weight per cask?

pound per quire? following weights: pounds, and 121

38. What is the cost of the above at 83 cents per pound? 39. Owning of a paper-mill, I sold of my share for $1750; what is the value of the whole mill at the same rate?

40. A man sold 50 yards of cloth at the rate of 1 yards for 2 dollars; what did he receive for it?

41 Mr. Gray raised 212 bushels of potatoes, of which he stored in 4 equal-sized bins; what did each bin hold? He sold the other at the rate of 3 bushels for 2 dollars; what did he

receive for the potatoes which he sold?

42. If of a ton of lead cost $1342, how many ounces of gold at $19 per oz. will pay for one ton of lead?

43. When hay was $15 per ton, I gave 2 of a ton for 1 tons of coal; what was the coal worth per ton?

44. If 2 of a yard of cloth will pay for six hats worth $41 per dozen, what is the price of the cloth per yard?

45. If of an acre of land cost $2803, what will 5 acres cost?

46. If a man walks 9 miles in 24 hours, how far will he walk in 42 hours?

47. At the rate of 4 miles an hour, what time will be required to walk 122 miles?

48. In 1860, I purchased

sold in 1862 at 907 cents.

cotton at 83 cents a pound, which I What did I gain on 1000 lbs.?

49. If a man can earn $2 per day, how many days' work will he have to give for a suit of clothes, of which the coat cost $25, the pants $81, and the vest $51?

50* The longest canal in the world, is the Grand Canal in China; + 1 + 1 + of its length is 331388 miles; what is its entire length? Ans. 650 miles.

51. If of of a ship cost $42,000, what is of her worth? 52. In a certain manufactory, of the operatives are Germans, Dutch, Scotch, English, Canadians, and the remainder, 140, native Americans; what is the whole number, and the number of each nationality?

53. of my money is in gold, of the remainder in silver, and the balance, $360, is in bank notes; how much money have I in all?

54* If 17 boxes of raspberries cost $2.83, what part of a box can I buy for 12 cents?

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