71. If the freight for 15 boxes of sugar, each weighing 300 pounds, 40 miles, cost $6, what must be paid for carrying 50 boxes, weighing 360 pounds each, 95 miles?

72. If a family of 8 persons spend $900 in 6 months, how many dollars will be required for a family of 12 persons for 16

months?

73. If a family of 8 persons spend $900 in 6 months, how many months will $ 3600 sustain a family of 12 persons?

74. If a family of 8 persons spend $900 in 6 months, how large, a family may be sustained 16 months for $3600 ?

75. If $100 gain $6 in 1 year, what will $600 gain in 9 months?

76. If $600 gain $ 27 in 9 months, what is the rate?

77. If $100 gain $ 6 in 1 year, in what time will $600 gain $27?

78. At 6 % what principal will gain $ 27 in 9 months?

79. If 12 horses eat 10 hektoliters of oats in 8 days, how many hektoliters will 30 horses eat in 40 days?

80. Of a chimney, which is to be 45 feet high, 9 feet was completed by 3 men in 2 days; how many men must be employed to build the remainder of it in 8 days?

81. If a 5-cent loaf weighs 9 oz. when wheat is $1 a bushel, how much bread ought to be bought for $0.50 when wheat is worth $0.75 a bushel?

82. If 4 barrels of flour serve a family of 6 persons 12 months, how many barrels will serve a family of 15 persons 18 months?

83. If 9 men dig a trench 84 ft. long, 5 ft. wide, and 3 ft. deep, in 4 days, how many men can dig a trench 420 ft. long, 3 ft. wide, and 2 ft. deep, in 6 days?

NOTE 2. All the dimensions of a surface, or a solid, always belong in the same term of a proportion; so that, having decided where one dimension belongs, the others can be written under this one at once. Thus, in Ex. 83, having decided that the 84 ft. (length) belongs in the first term, write the 5 ft. (width) and the 3 ft. (depth) under the 84; and under the 420 write the 3 and the 2.

84. If 4 men dig a trench 60 ft. long, 4 ft. wide, and 2 ft. deep in 6 days, how many days will it take 7 men to dig a trench 480 ft. long, 3 ft. wide, and 4 ft. deep?

85. If 8 men, working 9 hours a day, dig a trench 45 ft. long, 4 ft. wide, and 3 ft. deep in 6 days, how many men, working 10 hours a day, can dig a trench 225 ft. long, 5 ft. wide, and 2 ft. deep in 3 days?

86. If 6 men, in 8 days of 10 hours each, build a wall 60 ft. long, 4 ft. wide, and 54 ft. deep, how many days of 8 hours each will it take 15 men to build a similar wall 140 ft. long, 4 ft. wide, and 44 ft. deep?

87. A garrison of 600 men have bread enough to allow 16 ounces a day to each man for 15 days; but, the garrison being reinforced by 200 men, how many ounces a day may each man have in order that the bread may last 20 days?

88. If a man travels 180 miles in 6 days, travelling 10 hours each day, how many miles will he go in 18 days, travelling, at the same rate, only 9 hours each day?

89. If 2 compositors, in 3 days of 10 hours each, set type for 40 pages, each page consisting of 34 lines of 48 letters each, how many compositors will set 576 pages of 40 lines of 51 letters each, in 12 days of 9 hours each?

90. If it takes 22 reams of paper to make 1000 copies of a book of 11 sheets, how many reams will be required to make 4500 copies of a book of 7 sheets?

91. If 18 horses or 15 oxen eat 3 tons of hay in 8 weeks, how much hay will 28 horses and 35 oxen eat in 9 weeks?

92. If a man, walking 11 hours a day for 9 days, travels 297 miles, in how many days of 10 hours each would he walk 540 miles, travelling at the same rate?

93. If 900 tiles, each 8 inches square, will pave a court, how many tiles that are 12 inches long and 6 inches wide will pave another court which is 3 times as long and half as wide?

94. If the freight for 8 hhd. of sugar, each weighing 700 lbs., 80 miles, cost $ 20, what must be paid for carrying 70 hhd., each weighing 350 lbs., 400 miles?

95. If 4 men, in 18 days of 10 hours each, build a wall 40 ft. long, 7 ft. high, and 3 ft. thick, how many men will be required to build a wall 49 ft. long, 6 ft. high, and 4 ft. thick, in 14 days of 9 hours each?

96. If $ 400 gain $ 15 in 9 months, what is the rate per cent? 97. If 8 shares in a bank yield their owner $27.50 in 3 months, how much will 15 shares yield in 3 years?

407. What is a ratio? 408. How is ratio indicated? What are the names of the terms of a ratio? 409. What is the value of a ratio? 410. What is an inverse, or reciprocal, ratio? 411. Explain the effect of multiplying or dividing either or both terms of a ratio. 412. How are antecedent, consequent, and ratio related to each other? 413. What is a simple ratio? 414. A compound ratio? 416. What is a proportion? How is a proportion written? 417. What are the extremes? The means? 418. What is a mean proportional? 419. Prove that the product of the extremes is equal to the product of the means. 420. Whence the name Rule of Three? 422. Give the Rule for solving examples by proportion? How do you cancel? 423. What is Compound Proportion? 425. Give the Rule for solving examples in Compound Proportion. How do you cancel ?

INVOLUTION.

427. A Power of a number is the product obtained by taking the number any number of times as a factor.

428. Involution is the process of raising a number to a power.

The number involved is the first power of itself. It is also the root of the other powers.

429. The Index or Exponent of a power is a figure placed at the right and a little above the root to show how many times it is used as a factor; thus,

4 X 4 =

=

1642, or, the 2d power, or square of 4.

4 X 4X4 6448, or, the 3d power, or cube, of 4. 4x4x4x4 = 25644, or, the 4th power of 4. 4 X 4X4 X 4 x 4 = 1024 = 45, or, the 5th power of 4. Hence,

430. To involve a number to any required power,

Rule.

Take the number as a factor as many times as there are units in the index of the required power.

NOTE 1. The number of decimal places in the power of a decimal is equal to the number of decimal places in the root multiplied by the index of the power (Art. 161).

NOTE 2. The powers of a number greater than unity are greater than the number itself, and the powers of a number less than unity are less than the number itself; thus, the cube of 2 is 8, which is greater than 2; and the square ofis, which is greater than ; but the square of is, which is less than .

EVOLUTION.

432. A Root of a number is one of the equal factors whose continued product is that number.

433. Evolution is the resolving of a quantity into as many equal factors as there are units in the index of the root.

434. There are two methods of indicating a root, one by means of the radical sign, √, and the other by means of a fractional index.

The figure placed over the radical sign is the index of the root, and is always the same as the denominator of the fractional index; thus, the cube root of 8 is $8, or 8.

The square root of the cube of 4, or the cube of the square root of 4, is √43, or 4%.

If no number is over the radical sign, 2 is understood. 435. Evolution is the reverse of Involution.

In Involution the root is given and the power required. In Evolution the power is given and the root required. 436. A perfect power is a number whose root can be found. A perfect square is a number whose square root can