Elements of Algebra

CHAP. VI. OF ALGEBRA. 227 be $2x^2 + 2x$; lastly, if we divide by the difference 2, the quotient will be $x^2 + x$, which is the sum of the progression; so that we have $x^2 + x = 110$; therefore $x^2 = -x + 110$, and $x = -\frac{1}{2} + \sqrt{(\frac{1}{4} + 110)} = -\frac{1}{2} + \frac{21}{2} = 10$. And hence the number of pieces of cloth is 10. 652. *Question 7.* A person bought several pieces of cloth for 180 crowns; and if he had received for the same sum 3 pieces more, he would have paid 3 crowns less for each piece. How many pieces did he buy? Let us represent the number sought by $x$; then each piece will have cost him $\frac{180}{x}$ crowns. Now, if the purchaser had had $x+3$ pieces for 180 crowns, each piece would have cost $\frac{180}{x+3}$ crowns; and, since this price is less than the real price by three crowns, we have this equation, $\frac{180}{x+3} = \frac{180}{x} - 3$. Multiplying by $x$, we obtain $\frac{180x}{x+3} = 180 - 3x$; dividing by 3, we have $\frac{60x}{x+3} = 60 - x$; and again, multiplying by $x+3$, gives $60x = 180 + 57x - x^2$; therefore adding $x^2$, we shall have $x^2 + 60x = 180 + 57x$; and subtracting $60x$, we shall have $x^2 = -3x + 180$. The rule consequently gives, $x = -\frac{3}{2} + \sqrt{(\frac{9}{4} + 180)}$, or $x = -\frac{3}{2} + \frac{27}{2} = 12$. He therefore bought, for 180 crowns, 12 pieces of cloth at 15 crowns the piece; and if he had got 3 pieces more, namely, 15 pieces for 180 crowns, each piece would have cost only 12 crowns; that is to say, 3 crowns less. 653. *Question 8.* Two merchants enter into partnership with a stock of 100 pounds; one leaves his money in the partnership for three months, the other leaves his for two months, and each takes out 99 pounds of capital and profit. What proportion of the stock did they separately furnish? Suppose the first partner contributed $x$ pounds, the other will have contributed $100 - x$. Now, the former receiving $99l$., his profit is $99 - x$, which he has gained in three months with the principal $x$; and since the second receives also $99l$., his profit is $x - 1$, which he has gained in two months with the principal $100 - x$; it is evident also, that the profit of this second partner would have been Q 2