or 50y-xy+50−x+20=2xy;

.. 3xy=50y+70-x ... (1).

Again, after 10 years, A's capital and interest = 4x

.. 80x+8xy=250y5xy +2750-55x;

by transposition, 13xy=250y+2750-135x. (2) but from (1) 15xy=250y+350-5x;

.. by subtraction, 2xy=130x2400 . . . (3) multiplying (1) by 13, and (2) by 3;

.. 650y+910—13x=(39xy=) 750y + 8250 – 405 x ; by transposition, 100y=392x - 7340,

or 50y=196x-3670, which being multiplied by x, 50xy = 196x2 — 3670x; but (3) being multiplied by 25, 50xy S250x-60000;

=

.. 196x3670x=3250x-60000;

by transposition, 1962-6920x=-60000;

.. A's money=4x=80 pounds, and B's 120 pounds,

.. A's rate of interest was 5 per cent., and B's 6 per

cent.

a

44. When the price of brandy was three times the price of British spirit, a merchant made two mixtures of brandy and British spirit, and the prices per gallon were in the ratio of 9 to 10. He afterwards mixed twice as much brandy with the same quantity of British spirit in each case, and the relative price was the same as before. Required the ratio of the quantities mixed.

Suppose at first x gallons of British spirit were mixed

with one gallon of brandy, in one case; and y gallons of British spirit with one gallon of brandy in the other case;

and .. xy+x+3y+3: xy +3x+y+3 :: 9 : 10, divdo. xy+x+3y+3: 2x-2y:9: 1;

:: xy + x + 3y+3=18x

18y;

by transposition, xy-17x+21y+3=0;

also xy+2x+6y+12: xy+6x+2y+12 :: 9 :: 10; divdo. xy+2x+6y+12: 4x-4y 9: 1;

::

xy+2x+6y+12=36x − 36y;

by transposition, xy-34x+42y+12=0;
but xy-17x+21y+ 3=0;

.. by subtraction, 17x-21y-9=0;

also 2xy-34x+42y+6=0,

extracting the root, y+

+1

.. the first mixtures were in the ratio of 3 to 1, and 2 to 1; and the second in the ratio of 3 to 2, and of equality.

268

SECT. X.

Examples of the Solution of Problems in Arithmetical and Geometrical Progressions.

1. A PERSON bought 7 books, the particular prices of which (in shillings) were in arithmetical progression. The price of the next above the cheapest was 8 shillings, and the price of the dearest 23 shillings. What was the price of each book?

Let

the price of the cheapest,

and y=the common difference;

then x+y=the price of the second = 8,

and r+6y=the price of the dearest=23;

.. by subtraction, 5y=15,

and y = 3;

.. x=8-y=8-3=5,

and.. the prices are 5, 8, 11, 14, 17, 20, 23 shillings, respectively.

2. A number consists of 3 digits, which are in arithmetical progression; and this number divided by the sum of its digits is equal to 26; but if 198 be added to it, the digits will be inverted. Required the number.