Elements of Algebra

226 ELEMENTS SECT. IV. $x = -\frac{3}{4} \pm \sqrt{(\frac{9}{16} + 45)} = -\frac{3}{4} \pm \frac{27}{4}$. Consequently $x=6$, or $x = -7\frac{1}{2}$. 650. Question 5. A horse-dealer bought a horse for a certain number of crowns, and sold it again for 119 crowns, by which means his profit was as much per cent as the horse cost him; what was his first purchase? Suppose the horse cost $x$ crowns; then, as the dealer gains $x$ per cent, we have this proportion: As $100 : x :: x : \frac{x^2}{100}$; since therefore he has gained $\frac{x^2}{100}$, and the horse originally cost him $x$ crowns, he must have sold it for $x + \frac{x^2}{100}$; therefore $x + \frac{x^2}{100} = 119$; and subtracting $x$, we have $\frac{x^2}{100} = -x + 119$; then multiplying by 100, we obtain $x^2 = -100x + 11900$. Whence, by the rule, we find $x = -50 \pm \sqrt{(2500 + 11900)} = -50 \pm \sqrt{14400} = -50 \pm 120 = 70$. The horse therefore cost 70 crowns, and since the horse-dealer gained 70 per cent when he sold it again, the profit must have been 49 crowns. So that the horse must have been sold again for $70+49$, that is to say, for 119 crowns. 651. Question 6. A person buys a certain number of pieces of cloth: he pays for the first 2 crowns, for the second 4 crowns, for the third 6 crowns, and in the same manner always 2 crowns more for each following piece. Now, all the pieces together cost him 110 crowns: how many pieces had he? Let the number sought be $x$; then, by the question, the purchaser paid for the different pieces of cloth in the following manner: for the $1, 2, 3, 4, 5 \dots x$ pieces he pays $2, 4, 6, 8, 10 \dots 2x$ crowns. It is therefore required to find the sum of the arithmetical progression $2+4+6+8+\dots+2x$, which consists of $x$ terms, that we may deduce from it the price of all the pieces of cloth taken together. The rule which we have already given for this operation requires us to add the last term to the first; and the sum is $2x+2$; which must be multiplied by the number of terms $x$, and the product will