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Ex. 6. What principal at 6 per cent. compound interest wil amount to $101 in 4 years?

Ans. $80.

Ex. 7. At what rate will $10 amount to $16 in 16 years? Ans. Three per cent Ex. 8. What will $300 amount to in 10 years at compound interest semi-annually, the yearly rate being 6 per cent.? Ex. 9. In what time will a sum of money double at 6 per cent. compound interest?

Ans. 11.89 years.

Ex. 10. In what time will a sum of money triple itself at 4 per cent. compound interest?

Ans. 28.01 years.

(355.) The natural increase of population in a country may be computed in the same way as compound interest. Knowing the population at two different dates, we compute the rate of increase by formula (3), and from this we may compute the population at any future time on the supposition of a uniform rate of increase.

EXAMPLES.

Ex. 1. The number of the inhabitants of the United States in 1790 was 3,900,000, and in 1840, 17,000,000. What was

the average increase for every ten years?

Ans. 34 per cent.

Ex. 2. Suppose the rate of increase to remain the same for the next ten years, what would be the number of inhabitants in 1850?

Ans. 22,800,000.

Ex. 3. At the same rate, in what time would the number in 1840 be doubled?

Ans. 23.54 years.

Ex. 4. At the same rate, what was the population in 1780 Ans. 2,900,000.

Ex. 5. At the same rate, in what time would the number in

1840 be tripled?

Ans. 37.31 years

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x+y+z+t+w=27, to find the values of x,

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2√y−x+2√a−x=5√a-x, of x and y.

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Ex. 11. Given x2+xy=ay, to find the values of x and y.

x2y+ y3=bx,

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Ex. 16. Given √5x+5y+√x+√y= 10,

to find the values of

=275, * and Ans. x=9, y=4.

y.

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Ex. 18. Given (x+y)3+x+y=30, to find the values of ≈

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Ex. 19. Given x1—4x3+7x2-6x=18, to find the values of z

by a quadratic equation.

1.

Ans. x=3, or -i. Ex. 20. Given (x +y ) (x y +1)= 18x y, to find the values (x2+y') (x2y2+1)=208x'y', } of x and y.

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Ans. x=2±√3, y=7±4√3.

Ex. 21. Given (x2+y')xy=13090, }

Ex 21. Given (x2+y2)xy=13090, to find the values of x

x+y =18,

and y.

Ans. x=7, or 11,

y=11, or 7.

Ex. 22. Given 5(x2+y2)+4xy=356, | to find the values of

x2+y2+x+y=62,

}

x and y.

Ans. x=4, y=6.

Ex. 23. Given (x2+y3)xy=300, to find the values of x and

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Ex. 24. Given (x2+y3) (x3+y3)=455, to find the values of

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x and y.

Ans. x=3, y=2.

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Ans. x=2, or -3; y=3, or −2.

Ex. 27. Given (x+y)xy =30, to find the values of z

(x2+y3)x3y2=468,

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and y.

Ans. x=2, y=3.

Ex. 28. The sum of two numbers is a, and the sum of their reciprocals is b. Required the numbers.

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Ex. 29. In the composition of a certain quantity of gunpowder, the nitre was ten pounds more than two thirds of the whole; the sulphur was four and a half pounds less than one sixth of the whole; and the charcoal was two pounds less than one seventh of the nitre. How many pounds of gunpowder were there?

Ans. 69 pounds.

Ex. 30. Find three numbers such that if six be subtracted from the first and second, the remainders will be in the ratio of 2:3; if thirty be added to the first and third, the sums will be in the ratio of 3: 4; but if ten be subtracted from the second and third, the remainders will be as 4: 5.

Ans. 30, 42, 50.

Ex. 31. Divide the number 165 into five such parts that the first increased by one, the second increased by two, the third diminished by three, the fourth multiplied by 4, and the fifth divided by five, mav all be equal.

Ans 19, 18, 23, 5, and 100

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