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and the cost per yard is either 10s. for the one, and 5s. for the other, or 12s. 6d. for the one, and 2s. 6d. for the other.

4. A and B start at the same time from two distant towns. At the end of 7 days, A is nearer to the half-way house than B is by 5 miles more than A's day's journey. At the end of 10 days they have passed the half-way house, and are distant from each other 100 miles. Now it will take B three days longer to perform the whole journey than it will A. Required the distance of the towns and the rate of walking of A and B. Let 2x be the distance of the towns:

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the second, x 5y+5z-50 (7z+5)+5z-50; and the third is 2x(y − z) = 3yz, which thus becomes

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55

The solution z = 25 gives y = 30 and x = 225 hence the distance of the towns is 450 miles; and A and B walk at the rate of 30 and 25 miles a day respectively.

Ex. 27.

1. Two thirds the sum of two numbers exceeds threefourths their difference by 9; and one-fifth the sum of their squares is less by 2 than half the difference of their squares. Required the numbers.

2. The difference of two numbers is of their sum; and the difference of their squares is double their sum. Required the numbers.

3. The difference of two numbers is the sum of their squares, and the sum the difference of their squares. Required the numbers.

4. The sum of two numbers is equal to the square of one of them, and their difference is equal to one-sixth their product. Required the numbers.

5. Two women took to market 100 lbs. of apples between them. They returned with equal sums. If each of them had sold her apples at the same price that the other actually did, the one would have returned with 15s., and the other with 6s. 8d. At what price per pound did they sell respectively, and how many pounds had each ?

6. The difference of two numbers is equal to of their product; and their sum is equal to one-twelfth of the difference of their squares. Required the numbers.

7. The product of two numbers is 77, and the difference of their squares 72. Required the numbers.

8. The product of two numbers is 30, and the sum of their cubes is 341. Required the numbers.

This is one of the class of equations in which the unknown quantities may be held to be xy and x+y.

9. The product of two numbers is 21, and the difference of their cubes 316. Required the numbers.

10. The sum of two fractions is 1%, and the sum of their cubes 8. Required the fractions.

11. A person buys two bales of cloth, each containing 80 yards, for £60. By selling the first at a gain of as much per cent as the second cost him; and the second at twice the loss per cent, he finds that he has sustained a loss of £12 on the whole. Required the cost price per yard of each bale.

12. A person bought a number of sheep for £13:10s. Reserving a portion of them, he sold the remainder for the same sum that the whole had cost him, and had a profit per head on those sold of as many shillings as the number of the sheep he had reserved. Now he found that had he reserved one sheep less, his profit per head would have been 1s. 101d. Required the number of sheep bought and the number reserved.

13. A wine merchant sold 12 dozen of wine and 5 dozen of brandy for £46: 16s. He sold 3 more bottles of wine for 28s. than of brandy for 24s. Required the price per dozen of each. 14. The fore-wheel of a coach makes 94 more revolutions per mile than the hind-wheel does also the sum of their circumferences is 27 feet. Required the circumference of each.

15. To find three numbers such that the product of each by the sum of the other two is given = a, b, c, respectively.

16. Find two equal rectangles the sum of whose bases is a ; and which are such that if the first had the altitude of the second, its area would be b; and if the second had the altitude of the first, its area would be c.

17. Two candles of equal intensity are placed at a given distance a from each other-to find a point in this line a, at which the sum of the illuminations from the two candles shall be a given quantity-it being assumed that the intensity of illumination varies as the square of the reciprocal of the distance from the candle.

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18. Shew that the last problem is impossible, when the

illumination at the given point is required to be less than at the middle point between the two candles.

19. To divide three given straight lines, a, b, c, each into two parts such that the rectangle by a part of the first and a part of the second may be =p; the rectangle by the other part of the second, and a part of the third q; and the rectangle by

the remaining parts of the first and third = r.

20. Find two numbers such that 41 their product shall be equal to the sum of their squares, and 15 times their quotient shall be equal to the difference of their squares.

Here

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Also

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× 15x=xy' and y = x2(x2- × 15)

17 x2 + y2

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-y2
60 x2 — y2

Substitute for y2.

and b for 15.

21. In the previous question put a for

The reduction of the results requires the extraction of the

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CHAPTER VI.

ON EQUATIONS.

PART II.-ON THE INTERPRETATION OF THE SOLUTION OF AN EQUATION-CUBIC AND BIQUADRATIC EQUATIONS- -DIOPHAN

TINE ANALYSIS-INDETERMINATE PROBLEMS-FORMATION OF

EQUATIONS.

ON THE INTERPRETATION OF THE SOLUTION OF AN EQUATION.

105. There are certain peculiarities in the solutions of involved equations to which it is well now to direct attention.

1o A solution may be inverted; or, what is nearly the same, may invert the conditions of the problem.

Of this class of solution we have already given an example (Art. 26); and something of the same kind will occur again under the head of superfluous solutions. A single additional example will suffice.

Ex. 1. Divide the number 13 into two such parts that three times the first shall exceed half the second by as much as the first exceeds 4.

Let x be the first part, 13 x the second;

3x=(13x)+x-4

then

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and the parts required are 1 and 12.

But on examination, it will appear that the conditions of the question are not complied with: three times the first part does

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