80 190 185 iton of the ingot 50 or 5 is gold, or is silver and or 190 60 190 is copper. Therefore, if x gr. be taken from this ingot one takes 5x Sx gr. of gold, gr. of silver, gr. of copper. 6.c 19 Similarly, the second ingot contains 30+ 50 + 70 or 150 gr., of which 30 is gold, 50 silver, and 70 copper; on taking y gr. from this ingot, one takes 74 34 51 15 gr. of silver, gr. of copper. 15 Finally, the third ingot contains 190 gr., of which 35 gr. is gold, 65 silver, and 90 copper; in taking z gr. of this ingot, one takes gr. of gold, gr. of silver, gr. of copper. 13 z 19 And, since the ingot should contain 79 gr. of gold, 118 gr. of silver, and 162 gr. of copper, one has to determine the x, y, z of the three equations which after clearing of fractions become 50x38y + 35% = 15010 36x38y + 39% = 13452 120x133y+ 135z = 46170. On eliminating y from the first two equations one has On eliminating y from the last two equations one has 228 x + 57 z = 34656. On solving equations (4) and (5) for x and z, it follows that x = 133 and z = 76. Now substituting these values for x and z in (I) 4. Three cities have together 532,000 inhabitants. The first and second have together 206,000, the second and third together 200,000 inhabitants. How many inhabitants has each? 5. The different sums that can be formed from three given numbers by taking two at a time in all possible ways are respectively a, b, c. Find the numbers. 6. From the sum of every two of three given numbers the remaining number is subtracted, and the numbers so obtained are a, b, c. Find the given numbers. 7. Three numbers whose sum is 1332 are to each other as 3:4:5. Find the numbers. 8. A father said: "My age now is twice the sum of the ages of my sons Otto and Max. Two years ago I was four times as old as Otto, and four years ago I was six times as old as Max." What was the age of the father and his two sons? 9. A grocer pays $2.50 for 7 lbs. of coffee and 5 lbs. of sugar, $1.50 for 3 lbs. of coffee and 10 lbs. of rice, and $1.50 for 7 lbs. of sugar and 6 lbs. of rice. Find the price per pound paid for each article. 10. A has lent money at 2 per cent interest, B at 2 per cent, and Cat 3 per cent. A and B together receive $1592 interest, B and C receive $1766, C and A receive $1638. Find the number of dollars each has lent out. 11. Three towns form a triangle A B C travelling through B to C along the triangle is The distance from A 82 miles; the distance similarly from B, travelling through C to A is 97 miles; and from C, travelling through A to B, 89 miles. How far are A, B, and C from each other? 12. Divide the number 96 into three parts, such that the first divided by the second gives 2 with a remainder of 3, and the second divided by the third gives 4 with 5 as a remainder. 13. Find five numbers such that the sum of each and four times the sum of the remaining numbers gives respectively, 49, 43, 55, 61, 64 as a result, 14. A man has seven baskets of apples. From the first basket he puts into each of the other baskets as many apples as are contained in them; then from the second he puts into each of the other baskets as many as they then contain, and so on, to the last basket, when he finds that each basket contains 128 apples. Find the number of apples in each basket before the distribution. 15. A and B can build a wall in 12 days, B and C can do the same work in 20 days, A and C can do it in 15 days. How long will it take (1) each one alone, (2) all three together, to build the wall? 16. A miner has three ingots composed of gold, silver, and copper; the first ingot contains 2 kg. of gold, 3 kg. of silver, and 4 kg. of copper; the second contains 3 kg. of gold, 4 kg. of silver, and 5 kg. of copper; the third contains 4 kg. of gold, 3 kg. of silver, and 5 kg. of copper. How many kilogrammes is it necessary to take from each ingot in order to make a fourth ingot which contains 9 kg. of gold, 10 kg. of silver, and 14 kg. of copper? 17. A number is composed of four figures whose sum is 21; the figure in thousands' place is one-half the sum of the other three figures; the figure in units' place is one-half the figure in tens' place; finally, if 3906 be subtracted from the number, the remainder is the required number reversed. What is the number? 18. A merchant bought wheat at the rate of $2.40 for 4 bushels, corn at the rate of $1.60 for 7 bushels, and barley at the rate of $1.10 for 3 bushels. He spent $546.90; the cost of the wheat exceeded that of the corn by $80; the cost of the corn, that of the cost of the barley by $85.10. How many bushels of wheat, corn, and barley did he buy? 19. In one hour 150 persons enter a theater at the first door, 250 at the second door, and 400 at the third; and the receipts were $1625. During the next hour 120 persons enter the first door, 210 the second, and 324 the third; and the receipts were $1329. During the third hour 135 persons entered the first door, 280 the second, and 366 the third; and the receipts were $1606. What is the price of seats at the first, second, and third doors? CHAPTER IX GRAPHICAL REPRESENTATION OF POINTS AND LINES The solution of an equation of one unknown quantity, and solutions of systems of equations of two and three unknown quantities, discussed in the preceding chapters, have very beautiful geometrical interpretations. THE GRAPHICAL REPRESENTATION OF A POINT 243. The first problem in this geometrical discussion is to give a geometrical representation of a point in a plane. Consider two fixed lines, OX and OY, which are drawn at right angles to each other; these lines are called respectively the x-axis and the y-axis, they are lines of reference. To represent a point P in the plane of the paper proceed as follows: Draw PM perpendicular to OX. P will be determined in position by the perpendicular PM and the distance OM of M from O, Fig. 1. The line OM is called the abscissa, and PM the ordinate, of the point P. They are, for the sake of brevity, represented by the letters and y, respectively, which are called the coördinates of the point P. Thus, by definition, As in the case of positive and negative numbers we may lay off positive x's from 0 to the right toward X, and negative x's from O to the left toward X'. Similarly, +y's from the x-axis along the y-axis or parallel to it upward, and -y's from the x-axis downward. |