63. A owed B $900, to be paid in 3 years; but at the expiration of 9 months A agreed to pay $300 if B would wait long enough for the balance to compensate for the advance; how long should B wait after the expiration of the 3 years? Ans. 13 mo. 64. A certain clerk receives $800 a year; his expenses equal of what he saves; how much of his salary does he save yearly? 65. A merchant sold cloth at $1 per yard, and made 10 per cent. profit; what would have been his gain or loss had he sold it at $.871 per yard ? Ans. Loss, 3 per cent.

66. What is the cube of

21

29

63

67. What is the cube root of

1491

68. A miller is required to grind 100 bushels of provender worth 50 cents a bushel, from oats worth 20 cents, corn worth 35 cents, rye worth 60 cents, and wheat worth 70 cents per bushel; how many bushels of each may he take?

69. A man owes $6480 to his creditors; his debts are in arithmetical progression, the least being $40, and the greatest $500; required the number of creditors and the common difference between the debts. S 24 creditors. Ans. 320 difference.

70. Two ships sail from the same port; one goes due north 128 miles, and the other due east 72 miles; how far are the ships from each other? Ans. 146.86+ miles.

71. If 10 pounds of cheese be equal in value to 7 pounds of butter, and 11 pounds of butter to 2 bushels of corn, and 14 bushels of corn to 8 bushels of rye, and 4 bushels of rye to 1 cord of wood; how many pounds of cheese are equal in value to 10 cords of wood? Ans. 550.

72. A and B traded until they gained 6 per cent. on their stock; then of A's gain was $18; if A's stock was to B's as to, how much did each gain, and what was the original stock of each? A's gain, $45; stock, $750. B's 66 $37.50; "

Ans.

$625.

73. If 20 men, in 21 days, by working 10 hours a day, can dig a trench 30 ft. long, 15 ft. wide, and 12 ft. deep, when the ground is called 3 degrees of hardness, how many men, in 25 days, by working 8 hours a day, can dig another trench 45 ft. long, 16 ft. wide, and 18 ft. deep, when the ground is estimated at 5 degrees of hardness? Ans. 84.

74. Wishing to know the height of a certain steeple, I measured the shadow of the same on a horizontal plane, 27 feet; I then erected a 10 feet pole on the same plane, and it cast a shadow of 2} feet; what was the height of the steeple? Ans. 1031 ft.

75. A can do a piece of work in 3 days, B can do 3 times as much in 8 days, and C 5 times as much in 12 days; in what time can they all do the first piece of work?

Ans. & da.

76. A person sold two farms for $1890 each; for one he received 25 per cent. more than its true value, and for the other 25 per cent. less than its true value; did he gain or lose by the sale, and how much? Ans. Lost $252.

77. Three men paid $100 for a pasture; A put in 9 horses, B 12 cows for twice the time, and C some sheep for 2 times as long as B's cows; C paid one half the cost; how many sheep had he, and how much did A and B each pay, provided 6 cows eat as much as 4 horses, and 10 sheep as much as 3 cows? Chad 25 sheep. Ans. A paid $18.

B 66

$32.

78. A man purchased goods for $10500, to be paid in three equal installments, without interest; the first in 3 months, the second in 4 months, the third in 8 months; how much ready money will pay the debt, money being worth 7 per cent. ? Ans. $10203.94+

79. A farmer sold 50 fowls, consisting of geese and turkeys; for the geese he received $.75 apiece, and for the turkeys $1.25 apiece, and for the whole he received $52.50; how many were there of each ? Ans. 20 geese, 30 turkeys.

80. There is an island 73 miles in circumference, and 3 footmen start together and travel around it in the same direction; A goes 5 miles an hour, B 8, and C 10; in what time will they all come together again if they travel 12 hours a day? Ans. 6 da. 1 h.

81. A, B and C are to share $100000 in the proportion of,, and, respectively; but C dying, it is required to divide the whole sum proportionally between the other two; how much is each one's A's, $57142.855. Ans. B's, $42857.14%.

share?

82. A, B, and C have 135 sheep; A's plus B's are to B's plus C's as 5 to 7, and C's minus B's to C's plus B's as 1 to 7; how many has each ? Ans. A, 30; B, 45; C, 60.

83. A man sold one hog, weighing 250 pounds, at 4 cents per pound; a second, weighing 300 pounds, at 4 cents; and a third, weighing 369 pounds, at 5 cents; what was the average price per pound for the whole ? Ans. 45 cents.

84. In a certain factory are employed men, women and boys; the boys receive 3 cents an hour, the women 4, and the men 6; the boys work 8 hours a day, the women 9, and the men 12; the boys receive $5 as often as the women $10, and for every $10 paid to the women, $24 are paid to the men; how many men, women, and boys are there, the whole number being 59?

Ans. 24 men, 20 women, 15 boys. 85. A fountain has 4 receiving pipes, A, B, C, and D; A, B, and C will fill it in 6 hours, B, C, and D in 8 hours, C, D, and A in 10 hours, and D, A, and B in 12 hours; it has also 4 discharging pipes, W, X, Y, and Z; W, X, and Y will empty it in 6 hours, X, Y, and Z in 5 hours, Y, Z, and W in 4 hours, and Z, W, and X in 3 hours; suppose the pipes all open, and the fountain full, in what time would it be emptied? Ans. 6 h.

86. How many building lots, each 75 feet by 125 feet, can be laid out or 1 A. 1 R. 6 P. 18 sq. yd. ?

Ans. 6.

87. A man bought a house, and agreed to pay for it $1 on the first day of January, $2 on the first day of February, $4 on the first day March, and so on, in geometrical progression, through the year; what was the cost of the house, and what the average time of payment ? Ans. Average time, Nov. 1.

88. A man sold a rectangular piece of ground, measuring 44 chains 32 links long by 36 chains wide; how many acres did it contain ?

Ans. 159 A. 2 R. 8.32 P. 89. What number is that which being increased by its half, its third, and 18 more, will be doubled? Ans. 108.

90. A merchant has 200 lb. of tea, worth $.62 per pound, which he will sell at $.56 per pound, provided the purchaser will pay in coffee at 22 cents, which is worth 25 cents per pound; does the merchant gain or lose by the sale of the tea, and how much per cent. ? Ans. gained 11%.

91. A man owes a debt to be paid in 4 equal installments at 4, 9, 12, and 20 months, respectively; discount being allowed at 5 per cent., he finds that $750 ready money will pay the debt; how much did he owe? Ans. $784.74+.

92. A and B traded upon equal capitals; A gained a sum equal to of his capital, and B a sum equal to 1 of his; B's gain was $500 less than A's; what was the capital of each? Ans. $4000.

93. I purchase goods in bills as follows: June 4, 1859, $240.75; Aug. 9, 1859, $137.25; Aug. 29, 1859, $65.64; Sept. 4, 1859, $230.36; Nov. 12, 1859, $36. If the merchant agree to allow credit of 6 mo. on each bill, when may I settle by paying the whole amount ?

Ans. Feb. 1, 1860. 94. A young man inherited a fortune, of which he spent in 3 months, and of the remainder in 10 months, when he had only $2524 left; how much had he at first? Ans. $5889.33 +. 95. A man bought a piece of land for $3000, agreeing to pay 7 per cent. interest, and to pay principal and interest in 5 equal annual installments; how much was the annual payment?

Ans. $731.67+. 96. I have three notes payable as follows: one for $200, due Jan. 1. 1859, another for $350, due Sept. 1, and another for $500, due April 1, 1860; what is the average of maturity? Ans. Oct. 24, 1859.

97. A man held three notes, the first for $600, due July 7, 1859; the second for $530, due Oct. 4, 1859; and the third for $400, due Feb. 20, 1860; he made an equitable exchange of these with a speculator for two other notes, one of which was for $730, due Nov. 15, 1859; what was the face of the other, and when due?

MENSURATION OF LINES AND SUPERFICIES.

460. In taking the measure of any line, surface, or solid, we are always governed by some denomination, a unit of which is called the Unit of Measure. Thus, if any lineal measure be estimated in feet, the unit of measure is 1 foot; if in inches, the unit is 1 inch. If any superficial measure be estimated in feet, the unit of measure is 1 square foot; if in yards, the unit is 1 square yard.

461. If any solid or cubic measure be estimated in feet, the unit of measure is 1 cubic foot; if in yards, the unit is 1 cubic yard. · 462. The area of a figure is its superficial contents, or the surface included within any given lines,

without regard to thickness.

463. An Oblique Angle is an angle greater or less than a right angle; thus, A B C and C B D are oblique angles.

CASE I.

D

B

464. To find the area of a square or a rectangle.

465. A Square is a figure having four equal sides and four right angles.

466. A Rectangle is a figure having four right angles, and its opposite sides equal.

RULE. Multiply the length by the breadth, and the product will be the square contents.

EXAMPLES FOR PRACTICE.

1. How many square inches in a board 3 feet long and 20 inches wide? Ans. 720. 2. A man bought a farm 198 rods long and 150 rods wide, and agreed to give $32 an acre; how much did the farm cost him?

Ans. $5940. 3. A certain rectangular piece of land measures 1000 links by 100; how many acres does it contain ?

CASE II.

Ans. 1 A.

467. To find the area of a rhombus or a rhomboid.

468. A Rhombus is a figure having four equal sides and four oblique angles.

469. A Rhomboid is a figure having its opposite sides equal and parallel, and its angles oblique.

NOTE. The square, rectangle, rhombus, and rhomboid, having their opposite sides parallel, are called by the general name, parallelogram.

It is proved in geometry that any parallelogram is equal to a rectangle of the same length and width; hence the

RULE. Multiply the length by the shortest or perpendicular distance between two opposite sides.

EXAMPLES FOR PRACTICE.

1. A meadow in the form of a rhomboid is 20 chains long, and the shortest distance between its longest sides is 12 chains; how many days of 10 hours each will it take a man to mow the grass on this meadow, at the rate of 1 square rod a minute? Ans. 6 da. 4 h, 2. The side of a plat in the form of a rhombus is 15 feet, and a perpendicular drawn from one oblique angle to the side opposite, will meet this side 9 feet from the adjacent angle; what is the area of the plat ? Ans. 180 sq. ft.

CASE III.

470. To find the area of a trapezoid.

471. A Trapezoid is a figure having four sides, of which two are parallel.

The mean length of a trapezoid is one

half the sum of the parallel sides; hence the

RULE. Multiply one half the sum of the parallel sides by the perpendicular distance between them.

EXAMPLES FOR PRACTICE.

1. What are the square contents of a board 12 feet long, 16 inches wide at one end, and 9 at the other?

Ans. 121 sq. ft. 16 inches wide at Ans. 8 sq. ft. 3. One side of a field is 40 chains long, the side parallel to it is 22 chains, and the perpendicular distance between these two sides is 25 chains; how many acres in the field? Ans. 77 A. 5 sq. ch.

2. What is the area of a board 8 feet long, each end, and 8 in the middle?

CASE IV.

472. To find the area of a triangle.

473. The Base of a triangle is the side on which it is supposed to stand.

474. The Altitude of a triangle is the perpendicular distance from the angle opposite the base to the base, or to the base produced or extended.

475. A Triangle is one half of a parallelogram of the same base and altitude; hence the

RULE. Multiply one half the base by the altitude, or one half the altitude by the base. Or, Multiply the base by the altitude, and divide the product by 2.

EXAMPLES. FOR PRACTICE.

1. How many square yards in a triangle whose base is 148 feet, and perpendicular 45 feet?

Ans. 370 yds.