29. A and B together could have completed a piece of work in 15 days, but after laboring together 6 days, A was left to finish it alone, which he did in 30 days. In how many days could each have performed the work alone? Let x represent the number of days in which A could do it, and y the number of days in which B could do it; then 1 is the part A could do in 1 day; The first Equation will then be 1 1 1 + Now, since is the part done by A and B together in 6 days, is the part which A did in 30 days. days in which A alone could do the work. Hence, 30 is of the number of · The second Equation will then be 9x =30. 15 These two Equations will give x=50, and y=214 days. 30. On comparing two sums of money, it is found that of the first is $96 less than 3 of the second, and that of the second is as much as of the first. What are the sums? Let x represent the first sum, and y the second. Then, by the conditions of the problem, the Equations will be 2x 3y -96; and 5 4 бу 4x 8 9 which will give x=$720, and y=$512. = 31. A privateer, running at the rate of 10 miles an hour, discovers a ship 18 miles off, making way at the rate of 8 miles an hour. In how many hours will the ship be overtaken ? Let x represent the number of hours; then 10x represents the number of miles the privateer runs from the commencement of the pursuit; and 8x represents the number of miles the ship runs from the commencement of the pursuit. But as the ship was 18 miles ahead at the commencement, 8x+18, also represents the number of miles the privateer runs before overtaking the ship. The Equation will then be 10x=8x+18; which will give x=9 hours. 32. In a composition of copper, tin, and lead, 1⁄2 of the whole minus 16 pounds was copper, of the whole minus 12 pounds was tin, and 1 of the whole plus 4 pounds was lead. What quantity of each was there in the composition? Let x represent the copper, y the tin, and z the lead. Then, by the conditions of the problem, the Equations will be -16=x; x + y + z x + y + z x + y + z 4 -12=y; and +4=2. These Equations will give x=128, y=84, and z=76 pounds. 33. The sum of $660 was raised for a certain purpose by four persons, the first giving as much as the second, the third as much as the first and second, and the fourth as much as the second and third. What were the several shares contributed? y. x= ; 2 Let x, y, z, and w, respectively represent the several shares. x+y+z+w=660; z=x+y; and w= y + z. These Equations will give x= $60, y=$120, z=180, and w=$300. 34. Two pedestrians start from the same point, and travel in the same direction; the first steps twice as far as the second, but the second makes 3 steps while the first is making 2. How far has each gone when the first is 300 feet in advance of the second? Let a represent the number of feet traveled by the first, and y the number traveled by the second. Now, from the problem, it is evident that the first takes as many steps as the second; and, since the first steps twice as far as the second, the distance traveled by the first must be twice, or of that traveled by the second; the first Equation is therefore and And since the first is 300 feet in advance of the second, the second Equation will be x=y+300. These Equations will give x= =1200, and y=900 feet. 35. A merchant has cloth at $3 a yard, and another kind at $5 a yard. How many yards of each kind must he sell to make 100 yards which shall bring him $450? Let x represent the number of yards of the 1st kind, and Y the number of the 2d kind; then 3 represents the number of dollars received for the 1st kind; 5y represents the number of dollars received for the 2d kind. The Equations will then be x+y=100; and 3x+5y=450. From these Equations we shall find x=25, and y=75 yards. 36. In the composition of a quantity of gunpowder, the nitre was 10 pounds more than of the whole, the sulphur 4 pounds less than of the nitre. 1 of the whole, and the charcoal 2 pounds less than What was the amount of gunpowder? Let a represent the number of pounds; then 2x 3 +10 represents the number of pounds of nitre; of charcoal. x -4 represents the number of pounds of sulphur; and 6 (+10) represents the number of pounds The Equation will then be 2x +10+56 -2, or 2.x 4 3 which will give x=69 pounds. x= x 4 2x 21 37. Four places are situated in the order of the letters A, B, C, D. The distance from A to D is 34 miles; the distance from A to B is of the distance from C to D; and of the distance from A to B, plus of the distance from C to D, is 3 times the distance from B to C. What are the distances between A and B, B and C, C and D ? Let x represent the distance from A to B, y the distance from B to C, and z the distance from C to D. + 4 Then, since the distance from A to D, or the whole distance from the first to the last place is 34 miles, the first Equation will be x+y+2=34. And by the problem, the other Equations will be 2z and 3 2 = x; These three Equations will give x= =3y. =12, y=4, and z=18 miles. 38. A vintner sold at one time 20 dozen of port wine, and 30 of sherry, for $120; and at another time 30 dozen of port, and 25 of sherry, at the same prices as before, for $140. What was the price of a dozen of each sort of wine? Let x represent the price of the port, and y the price of the sherry, per dozen; then 20x is what he received for the port at the 1st sale; and 30y is what he received for the sherry at the 1st sale. The first Equation will then be 20x+30y=120. In a similar manner we shall have 30x+25y=140. These Equations will give x=$3, and y=$2. 39. A person pays at one time, to two creditors, $53, giving to one of them of the sum due to him, and to the other $3 more than of his debt to him. At another time he pays them $42, giving to the first of what remains due to him, and to the other of what remains due to him. What were the debts? Let x represent the sum due to the first, and y the sum due to the second; then 4x 11 represents the 1st sum paid to the first; and y +3 represents the 1st sum paid to the second. 6 The first Equation will then be 4.x 11 6 +2/+3=53. 7x Now, is the sum remaining due to the first; and 11 5y -3 is the sum remaining due to the second; then 6 3/7x represents the 2d sum paid to the first; and 3) represents the 2d sum paid to the second. (-3) The second Equation will then be 3/7x 5y (i)+(-3)=42; 36 3x 5y or + -1=42. 11 18 The two Equations will give x=$121, and y=$36. 40. A farmer has 86 bushels of wheat at 4s. 6d. per bushel, with which he wishes to mix rye at 3s. 6d. per bushel, and barley at 3s. per bushel, so as to make 136 bushels, that shall be worth 4s. a bushel. What quantity of rye and barley must he take? Let x and y represent the number of bushels of rye and barley, respectively. 3 |