and the price of an orange was to the number bought as 1 to 4. How many oranges did he buy, and what did he give apiece? 8. A merchant sold a quantity of flour for a certain sum, and at such a rate, that the price of a barrel was to the number of barrels as 4 to 5: if he had received 45 dollars more for the same quantity, the price of a barrel would have been to the number of barrels as 5 to 4. How many barrels did he sell, and at what price? 9. A gentleman exchanges a field, 81 rods long and 64 rods wide, for an equal quantity of land in the form of a square. What was the side of the square? 10. How long and wide is a rectangular field containing 864 rods, the width of which is equal to of the length? 11. A certain street contains 144 rods of land; and if the length of the street be divided by its width, the quotient will be 16. How long and wide is the street? 12. A trader sold two pieces of broadcloth, which together measured 18 yards; and he received as many dollars a yard for each piece as it contained yards. Now, the sums received for the two were to each other as 25 to 16. How many yards were there in each piece? Again, the whole price of one, and y2 the price of the other. Therefore, a2 = 257, by the question; 5y 16 and x = by, by evolution. 4 18-y, by comparing the values of x. ANS. 10 yards; 8 yards. 9 13. A man divided 14 dollars between his son and daughter in such a manner, that the quotient of the daughter's part divided by the son's, was of the son's part divided by the daughter's. What was the share of each? 14. A house contains two square rooms, the areas of which are to each other in the proportion of 25 to 9; and a side of the larger room exceeds a side of the smaller by 10 feet. What are the dimensions of the rooms? 15. In a certain orchard there are 4 more rows of trees than there are trees in a row; and if the same number of trees were so arranged that there should be 64 added to each row, the number of the rows would be reduced to 4. How many trees are there in the orchard? Let the trees in a row. Then x+4= the number of rows, Also (+64) × 4, 2 or 4x+256, S the number of trees. Then a2+4x= 4 x + 256. ANS. 320 trees 16 When an army was formed in solid column, there were 9 more men in file than in rank; but when it was formed in 9 lines, each rank was increased by 900 men. Of how many men did the army consist? 17. A gentleman has two squares of shrubbery in his grounds, the difference of whose sides is to the side of the greater square as 2 to 9; and the difference of their areas is 128 yards. What are the sides of the squares? 18. Says A to B, "Our ages are the same; but if I were 5 years older, and you were 5 years younger, the product of our ages would be 96." What are their ages? 19. What number is that, which being added to 10 and subtracted from 10, the product of the sum, multiplied by the difference, will be 51 ? 20. There is a rectangular field, whose length is to its breadth in the proportion of 6 to 5. A part of this, equal to of the whole, being an orchard, there remain for tillage 625 square rods. What are the length and breadth of the field? 21. It requires 108 square feet of carpeting to cover a certain entry; and the sum of its length and breadth is equal to twice their difference. How long and wide is it? 22. Required two numbers which are to each other as 1 to 3, and the sum of whose second powers is equal to 5 times the sum of the numbers. 23. The area of an oblong room is 400 square feet: and 'f its width were equal to its length, its area would be greater. room? What are the dimensions of the 24. A charitable person distributed a certain sum among some poor men and women, the numbers of whom were in the proportion of 4 to 5. Each man received as many shillings as there were persons relieved; and each woman received twice as many shillings as there were women more than men. The men received, altogether, 18 shillings more than the How many were there of each? women. 25. A gentleman, being asked the ages of his two sons, replied, that they were to each other as 3 to 4; and that the product of their ages was 48. What were their ages? 26. A gentleman has an oblong garden of such dimensions, that if the difference of the sides be multiplied by the greater side, the product will be 40 square rods; but if the difference be multiplied by the shorter side, the product will be 15 rods. What are the length and width of the garden? and y = the difference of the sides. 40, by the conditions of the Therefore, xy + y2 15 x = and x y = 15, question. By substituting the value of x in the first equation, SECTION II. Affected Equations. As a pure equation of the second degree contains the unknown quantity only in the form of its second power, all the terms in which it appears can be united in one term, whose root, as we have seen, can be readily found. An affected equation of the second degree contains not only the square of the unknown quantity in one term, but also the unknown quantity itself in another term. Thus, x2+4x=77 is an affected equation of the second degree, in which the unknown quantity appears in two terms; for x2 and x cannot be actually added together so as to make but one term. When an equation of this sort is formed, it may contain the unknown quantity in any number of terms, provided it be only in the first and second powers; for, in this case, the terms may all be reduced to two. Thus, if we have the equation 5 x2 + 8 x 24 + x2 - 3 x = 4x2 + x + 48, by transposition we obtain and, by adding the similar terms, 2x2 + 4 x = 72; and, by dividing all the terms by 2, Affected and pure equations of the second degree |