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65. By his will a father disposed of his property as follows, namely, to his oldest son he gave 100 dollars of the property and a tenth part of the residue; to the second, 200 dollars and a tenth part of the residue; to the third, 300 dollars and a tenth part of the residue; and so on to the last, always increasing the sum first paid out by 100 dollars. It appeared that the portions of all the children were alike. Required the value of the property, the number of children, and the portion of each child.

Ans. The estate was 8100 dollars, the children 9, and the portion of each 900 dollars.

66. Of a battalion of soldiers, are on duty, sick, and % of the residue absent; and there are 48 officers: what is the number of persons in the battalion ?

Ans. 800.

67. A and B have the same income; A is extravagant and contracts an annual debt amounting to of his income; but B lives upon of his; at the end of 10 years, B lends A money enough to pay off his debts, and has 160l. to spare: what is their income? Ans. 2801.

68. A person passed of his age in childhood,

in youth, and 5 years besides in matrimony, at the end of which time. he had a son, who died 4 years before his father, and reached only half his father's age: at what age did the father die?

Ans. 84.

69. A shepherd, driving a flock of sheep in time of war, meets with a company of soldiers, who plunder him of half his flock and half a sheep over; and he receives the same treatment from a second, third, and fourth company, each succeeding company plundering him of half the flock the last had left and half a sheep beside, insomuch that in the end he had only 8 sheep left: how many sheep had he in the beginning?

Ans. 143.

70. A person, fifteen years after he was married, being asked the age of himself and of his wife at the time of their marriage, replied, that he was then thrice as old as his wife, but that now he was only twice as old: what were their ages ?

Ans. He was 45 and she 15.

71. It is required to find two numbers such that the first added to half the second shall make 20, and the second added to one third the first shall also make 20. Ans. 12 and 16.

72. Two travellers, distant 154 miles, set out at the same time to meet each other, the one proceeding at the rate of 3 miles in 2 hours, and the other at the rate of 5 miles in 4 hours: how long and how far did each travel before they met ?

Ans. The time was 56 hours, the spaces travelled 84 and 70 miles.

IX. Formation of Powers and Extraction of Roots.

73. What is the square root of 9 x2? (Art. 122.)

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75. What is the square root of a +4 a3x+6a2x2 + 4 ax3

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77. What is the third power of 8 x2 y3 ? (Art. 127.)

Ans. - 512 x y3.

78. What is the fifth root of 32 x y10? (Art. 129.)

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79. What is the fourth power of x-a? (Art. 141.) Ans. x44x3 a + 6 x2 u2 - 4 x a3 + a*.

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80. What is the square of a2 + 2 ax + x2? (Art. 145.) Ans. a4a3 x + 6 a3 x2 + 4 a x3 + x*.

81. What is the cube root of a +15 x2-6x+1? (Art. 155.)

82. What is the fifth root of 32 x5 +10x-1?

20 x3

Ans. x-2x+1.

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30 x 80 x3

- 40 x2

Ans. 2x-1.

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