.. x = ± 5y, the negative value not answering the conditions

of the problem.

Now from the second proportion 5y : 4y :: y: 4;

... there are 25 gallons of rum, and 5 of brandy.

24. What two numbers are those, whose difference being multiplied by the greater, and the product divided by the less, quotes 24; but if their difference be multiplied by the less, and the product divided by the greater, the quotient is 6?

and.. in the first case, (x - y =) y = 12, and x = 24;
but if x =
2y;
зу х - 2 = 24;
.. y = 4, and x = 8.

25. It is required to find two numbers such, that the product of the greater and square root of the less may be equal to 48, and the product of the less and square root of the greater may be 36.

26. Find two numbers such, that the square of the greater multiplied by the less may be equal to 448, and the square of the less multiplied by the greater may be 392.

Let x = the greater, and y = the less;

then x2y = 448, and xy2 = 392;

27. A and B carried 100 eggs between them to market, and each received the same sum. If A had carried as many as B, he would have received 18 pence for them, and if B had taken only as many as A, he would have received only 8 pence. How many had each ?

• 3x = ± 2y, the negative value of which will not answer the conditions of the problem.

28. What two numbers are those, which being both multiplied by 27 the first product is a square, and the second the root of that square: but being both multiplied by 3, the first product is a cube, and the second the root of that cube?

29. It is required to find the three sides of a right-angled triangle from the following data. The number of square feet in the area is equal to the number of feet in the hypothenuse + the sum in the other two sides; and the square described upon the hypothenuse is less than the square described upon a line equal in length to the two sides, by half the product of the numbers representing the base and area.

Let x = the number of feet in the altitude,
and y = the number in the base;

√(x2 + y2) = the number in the hypothenuse, (Eucl. B. 1. p. 47.)

∴xy = √(x2 + y2) + x + y;

also x2 + y2 = {(x + y)2 - xy2 = } x2 + 2xy + y2 - xy2;

... by transposition, xy2 = 2xy,

and y = 8;

hence from the first equation, 4x = √(x2 + 64) + x + 8,

2

and by transposition, 3x - 8 = √(x2 + 64);

9x2 - 48х + 64 = x2 + 64,

whence the hypothenuse = ✓ (64 + 36) = 10;
... the sides are 6, 8, and 10 feet, respectively.

30. A Farmer has 2 cubical stacks of hay. The side of one is 3 yards longer than the side of the other; and the difference of their contents is 117 solid yards. Required the side of each.

cubing the latter equation, æ3 - 3x2y + 3xy2 - y3 = 27;

and the sides of the stacks are 5, and 2 yards, respectively.

31. When a parish was enclosed, the allotment of one of the proprietors consisted of two pieces of ground; one of which was in the form of a right-angled triangle; the other was a rectangle, one of the sides of which was equal to the hypothenuse of the triangle, the other, to half the greater side; but wishing to have his land in one

Q