Elements of Algebra

CHAP. VIII. OF ALGEBRA. 243 6. The sum of two numbers is $a$, and the sum of their squares $b$. Required the numbers. Ans. $\frac{a}{2} \pm \sqrt{(\frac{2b-a^2}{4})}$, and $\frac{a}{2} \mp \sqrt{(\frac{2b-a^2}{4})}$. 7. To divide 36 into three such parts, that the second may exceed the first by 4, and that the sum of all their squares may be 464. Ans. 8, 12, 16. 8. A person buying 120 pounds of pepper, and as many of ginger, finds that for a crown he has one pound more of ginger than of pepper. Now, the whole price of the pepper exceeded that of the ginger by six crowns: how many pounds of each had he for a crown? Ans. 4 of pepper, and 5 of ginger. 9. Required three numbers in continual proportion, 60 being the middle term, and the sum of the extremes being equal to 125. Ans. 45, 60, 80. 10. A person bought a certain number of oxen for 80 guineas: if he had received 4 more for the same money, he would have paid one guinea less for each head. What was the number of oxen? Ans. 16. 11. To divide the number 10 into two such parts, that their product being added to the sum of their squares, may make 76. Ans. 4 and 6. 12. Two travellers A and B set out from two places, $\Gamma$ and $\Delta$, and at the same time; A from $\Gamma$ with a design to pass through $\Delta$, and B from $\Delta$ to travel the same way: after A had overtaken B, they found on computing their travels, that they had both together travelled 80 miles; that A had passed through $\Delta$ four days before, and that B, at his rate of travelling, was a journey of nine days distant from $\Gamma$. Required the distance between the places $\Gamma$ and $\Delta$. Ans. 6 miles. R 2