76. The interest of 1 dollar for 2 months, is the same as the interest of how many dollars for 1 month?

77. The interest of 2 dollars for 2 months, is the same as the interest of how many dollars for 1 month?

78. The interest of a dollars for t years, is the same as the interest of how many dollars for 1 year? Ans. at.

79. A and B engage in trade, in partnership. A has 100 in trade for 3 years. B has 200 in trade. How long should B's be in trade, that they may have an equal share of the profits?

A's money, 100, for 3 years, is the same as 300 for 1 year. B's 200 dollars must remain in trade until it is equivalent to 300 for 1 year.

Let x = the number of years B's must be in trade.

Then, since 200 dollars for x years, is the same as x times 200, or 200 x for 1 year.

200 x= 300
x = 11/2/

80. A and B engage in trade in partnership; A puts in 400 dollars and B 900. A's money was in trade 2 years longer than B's; and, he received only of the profits. How long was the money of each in trade?

81. A and B trade upon a joint stock of 1000 dollars. After they had been in partnership for 3 years, they took in a third partner, C, who contributed 800 dollars to the stock. After several years they dissolved partnership; they had gained 1500 dollars, of which C took 600 dollars. How long did the partnership last?

82. A, B and C had, each, 100 barrels of flour, which they agreed to ship, together, to New Orleans. A purchaser upon inspecting the flour at the place of its destination, said that the flour of A and B was of the same quality; but, that C's flour was worth 20 per cent. more than theirs; and, that he would give 3000 dollars for the whole. What would each of the owners of the flour receive per barrel, at that rate?

83. Four hundred dollars are to be paid in 2 months, and 600 in 6 months. Suppose the whole to be paid at one time; when should the time of payment be?

84. A owes 100 to be paid in 6 months, 200 to be paid in 8 months, and 300 to be paid in 12 months; but, he wishes to pay it all at once. When should the time of payment be?

85. There is a number expressed by two digits; the second of which is double the first; and, if 9 be added to the number, it will make the number expressed by the digits inverted. What is the number?

86. There is a number expressed by two digits; the second is of the first; and, if the number be divided by 8, the quotient will be 2 less than the sum of its digits. What is the number?

87. There is a number expressed by two digits; the second 5 less than the first; and, if of the number be subtracted from the number, the digits will be inverted. What is the number?

SECTION LXXXIII.

On the Lever.

A lever is an inflexible rod or beam, [SEE FIG. BELOW,] as A B resting on a support at C, called a fulcrum. A weight may be attached to each end, A and B. When the lever remains stationary, or does not turn about the point C, the lever is said to be in equilibrio. The parts A C and B C are called arms of the lever.

The following proposition respecting the lever is proved in Mechanical Philosophy, both by theory and experiment: viz. PROPOSITION. When the lever is in equilibrio, the weight at B, multiplied by the length of the arm C B, is equal to the weight at A, multiplied by the length of the arm A C.

1. A lever is in equilibrio; the weight at A is 2 pounds, the length of the arm A C is 2 feet, the length of C B 8 feet. What is the weight at B?

Let x= the weight at B,

Then, by the proposition

4=8x
8x=4

x=÷
x = = =
124

2. There is a lever, A B, 10 feet long. The weight at A is 6 pounds, and the weight at B is 10 pounds. What is the length of the arms of the lever?

Let

x= the length of one arm,

Then 10-x= the length of the other,

.*

3. There is a lever 20 feet long; the weights at the two ends are 12 and 16 pounds. What are the lengths of the arms, when the lever is in equilibrio?

4. A lever, A B, is 30 inches long. The weights at the two ends are 12 and 20, respectively. What are the lengths of the arms, when the lever is in equilibrio?

5. The lengths of the two arms of a lever are 12 and 16 inches, respectively. The sum of the weights at the two ends is 120 pounds; the lever is in equilibrio. What are the weights at each end?

6. The difference in the lengths of the arms of a lever, is 12 inches. The weight on one arm is 20 pounds, and the weight on the other is 30; the lever is in equilibrio.

its length?

What is

7. There is a lever, A B, one arm of which is 10 feet long. Upon the extremity of the other arm, at B, is placed a weight of 14 pounds, at a distance of 3 feet from the fulcrum. The weight at A, the other extremity of the lever, is found just 4 pounds too heavy to balance the weight at B. What is the weight at A?

8. There is a lever, A B; at B, one extremity, 4 inches from the fulcrum, a weight is suspended. The weight at A, the other extremity, 14 inches from the fulcrum, is found too heavy to balance the weight at B, by 4 pounds; but, when the weight at A is moved 6 inches nearer the fulcrum, it is, then, too light to balance the weight at B, by 3 pounds. What is the weight at A? What is the weight at B?

PROPOSITION.-If several weights are suspended on one or both arms of the lever, as shown in the figure below,

multiply each weight, on one side of the fulcrum, by its distance from the fulcrum, and add the products together. Then multiply each weight on the other side of the fulcrum,

by its distance from the fulcrum, and add the products together, and the two sums will be equal, when the lever is in equilibrio.

9. There is a lever 36 inches long, having a weight of 4 pounds suspended at the extremity A, and a weight of 6 pounds at the extremity B. These will not balance each other; but, if two additional weights are suspended on different sides of the fulcrum, one of 9 pounds, at a distance of 3 inches from A, and the other, of 4 pounds, at a distance of 6 inches from B; the lever will then be in equilibrio. What are the lengths of the arms of the lever?

10. There is a lever, one arm of which is 6 feet longer than the other, which is not balanced by weights of 20 and 15 pounds, suspended at the extremities of the shorter and longer arms, respectively; but, if a weight of 2 pounds be placed upon the longer arm, at a distance of 4 feet from its extremity the lever will then be in equilibrio. What are the lengths of the arms of the lever?

11. There are two weights, one of 100 pounds, and the other of 30, attached to the extremities of a lever of 60 inches in length. Where must the fulcrum be placed, that the lever may be in equilibrio ?

A B C represents an inclined plane; A B is the height of the plane, B C its length, w is a weight placed on the plane, to which a cord is attached, as represented in the figure, which passes over a pulley at B. p is a weight attached to the other end of the cord. If the two weights remain stationary, they are said to be in equilibrio.

In mechanical philosophy, the following proposition is proved, both by theory and experiment: viz.

When the two weights, p and w, are in equilibrio, the weight p, multiplied by the length B C, of the plane, is equal to the weight w, multiplied by the height of the plane A B.

1. If the length of the plane be 10 feet, the height 2 feet, and the weight w, is 30 pounds; what is the weight p?

Let
Then

2. If p 4 pounds, w= 10 pounds, and the length of the plane exceeds its height by 3 feet; what is the height, and what is the length of the plane?

3. If the sum of the height and length of the plane is 20 feet, p= 12 pounds and w = 60 pounds; what are the height and length of the plane?

4. If the difference of the weights, p and w, is 15 pounds, and the plane be 20 feet high and 30 feet long; what are the weights p and w?

5. If the sum of the weights be 500 pounds, the height of the plane 6 feet, and the length 30 feet; what are the weights w and p?

SECTION LXXXV.

On the Weights of Solids in Fluids.

If any substance, of any bulk, be immersed in water or any other fluid, it will remain in any position in which it is placed, that is, it will neither rise nor sink, provided it weighs just as much as an equal bulk of the fluid. If the substance immersed weigh more than an equal bulk of the fluid, its tendency to sink is just equal to the difference between the weight of the body and the weight of an equal bulk of the fluid. Hence, if the immersed substance be attached by a cord to one end of a pair of scales, it will require to support it, just as much weight in the opposite scale as is equal to the difference between the weight of the substance, and the weight of an equal bulk of the fluid. If, then, a cubic foot of iron be weighed in water, it will lose just as much of its weight as a cubic foot

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