13. § of 40 is 2 of what number?

ANALYSIS.— of 40 is 5; ğ is 5 times 5, or 25. If 25 is of some number, of 25 is of the same number. of 25 is 12. If 12 is 4,7 times 12 is ; 7 times 12 is 84, and 7 times is

added to 84 = 87.

2

7

31⁄2, which Therefore, & of 40 is 2 of 871⁄2.

14. of 30 is of what number?

15. of 81 is of what number?

9

15

16. If 40 pounds of sugar cost $3.20, how much will 9 barrels cost?

17. If 7 tons of hay cost $31, what will 8 tons cost? 18. If 6 men can do a job of work in 13 days, in what time can 11 men do it?

19.12 pounds of coffee will last a family of 5 persons 8 weeks; how long will it last them if 3 persons are added to the family?

20. If 7 men can dig a ditch in 13 days, in what time can 11 men dig one 3 times as long?

21. When potatoes are 50 cents a bushel, how many bushels must I give in exchange for 18 bushels of wheat, at 75 cents a bushel?

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22. How many yards of carpeting that is of a yard wide, are equal to 17 yards that is of a yard wide?

23. What number is that to which, if its third and 64 more be added, the amount will be double the number?

24. If of a pole stand in the water, in the mud, and 10 feet above the water, what is the length of the pole?

9 20

and are equal to ; hence part above the water.

11

10 ft., the

25. A messenger traveling at the rate of 6 miles an hour, was sent to Mexico with dispatches for the American army; after he had been gone 7 hours, an

other was sent with countermanding orders, who could travel 15 miles in the same time that the first could go 12; how long would it require for the latter to overtake the former; and how far must he travel?

SOLUTION. From the conditions of the question, it appears that the second gained on the first 11⁄2 miles per hour; and will, therefore, be as many hours in overtaking him, as the number of times that 1 is contained in 7 times 6 or 42, namely, 28 hours: and 28 times 74m. 210m., the distance traveled.

26. The head of a certain fish was found to be 9 inches long; its tail was as long as its head and 1⁄2 of its body, and the body was as long as the head and tail both; what was the length of the body?

SOLUTION. Since the body was as long as the head and tail, it follows that it was the length of the fish, and since the tail equals the head and the body, it must have been 9 in. plus of the fish.

4

If now we draw a line representing the fish, we shall readily discover his length.

27. A hare starts 50 leaps before a greyhound, and takes 4 leaps to the hound's 3; but 2 of the hound's leaps are equal to 3 of the hare's; how many leaps must the hound take to overtake the hare?

SOLUTION.--Since 2 of the hound's leaps

hare's, 1 of the hound's

14 of the hare's.

3 of the

Hence,

while the hare is taking 4 leaps, the hound advances

3 times 1 of the hare's leaps. 11 × 3

41; there

fore, in taking 3 leaps, the hound gains of the hare's leaps, and must take 6 leaps to gain 1 of the hare's leaps; and to gain the 50 leaps, he must take 50 times 6 300 leaps.

28. A workman was engaged for 40 days upon this condition, that he should receive 20 cents for every day he wrought, and should forfeit 10 cents for every day he was idle; at settlement he received 5 dollars; how many days did he work, and how many days was he idle?

29. A, B, and C, bought a quantity of wine for $340, of which sum A paid three times more than B, and B four times more than C; how much did each pay?

30. A man and his wife can eat a barrel of beef in 15 weeks, but after eating 9 weeks, it was found that his wife could eat what remained in 30 weeks; in what time could either have eaten the whole?

31. A young man having a patrimony of $36,000, spent & of it in gambling and dissipation; how many acres of land, at $5 an acre, could he buy with the remainder, after paying $500 for farming implements? 32. If a yard of silk cost 50 cents, what will 360 yards cost?

Note.-The pupil will note that 50 cents is equal to $1.

1

33. A pillar of granite stands 40 feet above the surface of the snow, in the snow, and in the ground; what is the whole length of the pillar?

16

34. What number is that to which, if

self be added, the sum will be 164?

and of it

35. If 12 horses consume 720 bushels of oats in 3 months, in what time will 18 horses consume 1200 bushels ?

36. If 10 horses consume 20 tons of hay in 30 weeks, what number of horses will it require to consume 40 tons in 15 weeks?

37. 36 barrels of flour will last 840 men 5 days; how many barrels will last 1200 men 11 days?

38. A returned Californian being asked how much money he had, answered that , 2, and § of it made $147,000; what amount had he?

12

39. What number is that, 3, 4 and of which is 390 ?

40. What number is that, of which exceeds of it by 18?

41. What number is that to which, if its and be added, the sum will be equal to 5 times 11?

2

42. A farmer being asked how many sheep he had, replied that, if he had as many more, as many more, and 2 sheep, he should have 100; what number of sheep had he?

43. A clothier bought 8 pounds of indigo for $205; what would have been the cost of 5 pounds, at the same rate?

44. The distance from A to B, which is 40 miles, is of the distance from C to D; what is the distance from C to D?

45. A has

as C, who has

5

as much money as B, and as much as much as D, and he has $1800. What are the respective sums owned by A, B, and C ?

6

CHAPTER XIII.

§ 86.-Note. In this chapter the same principles are used as in previous sections, the chief difference being, that larger numbers are used. No mental exercise is of more value than that of adding large numbers mentally. The books of the class being closed, let the following and similar examples be much

dwelt upon. The teacher should solve the questions, as well as the pupils. The practice will be of value, and will also furnish a guide, as to the rapidity with which the numbers should be read. Let all remain silent until the final result is secured.

1. What is the sum of 18, 13, 19, 15, 17, and 14? 2. What is the sum of 16, 20, 29, 25, 24, and 27 ? 3. What is the sum of 23, 28, 32, 37, 38, 41 and 49? 4. What is the sum of 45, 53, 44, 56, 48, 47 and 58 ? 5. What is the sum of 38, 48, 58, 83, 99, 75 and 98 ?

ARITHMETICAL SIGNS.

§ 87. The sign of ADDITION consists of two lines, one horizontal and the other perpendicular, thus, +, and shows that the numbers between which it is placed, are to be added together. It is usually read plus. Thus the expression 4 +7, is read 4 plus 7.

§ 88. The sign of EQUALITY consists of two parallel horizontal lines, thus, and shows the numbers or quantities between which it is placed, are equal to each other.

Thus 4 + 7 = 11, is read 4 plus 7 equal 11.

§ 89. The sign of MULTIPLICATION is commonly denoted by two oblique lines crossing each other, thus X, and shows that the numbers between which it is placed, are to be multiplied together. Thus 4 × 7 X 28, is read 4 multiplied by 7 = 28. A dot or point placed between the numbers, also signifies the same thing. Thus 4.7 signifies the same as 4 X 7, and is read in the same manner.

§ 90. The sign of SUBTRACTION is represented by a short horizontal line, thus

,

and is read minus.