more had been bought for the same sum, they would have cost a sovereign less each. How many were bought, and what was the price? Let x=No. bought. £5, the 80x76x-x2+320:.x2-4x=320. x2-4x+4=324 x-2=18 x=16 and cost of each. Here the negative value of x is useless in the particular case given in the equation. 2. Bought a number of pigs for £8 8s., if two more had been purchased for the same outlay, each would have cost 4s. less. What was the number. 160x-4x2+336=168x.. 4x2+8x=336 3. The product of two numbers is 72, and their difference 1. What are the numbers? Ans. 9 and 8 4. Divide 112 into two such parts that their product shall be 1280. Let x=One portion, then 112-x=other =56+8√29 4. There are two numbers, the difference between them being 6, and twice their product, minus 32, is 33 times the less. What are the numbers. Ans. 12 and 18. 5. A person bought sheep for £40, and sold them all but 4 for £48, gaining £1 on each. What was the number he bought? Ans. 20. 6. -x:. 48x-40x+160=x2-4x x2-12x=160 x=20 Divide 72 into two such parts that their product may be equal to 160 times their difference. Ans. 40 and 32. 7. Bought oxen for £675, and sold them again for £48 each, gaining the price of a single bullock. How many did I buy? Ans. 15. 8. Bought sheep for £72, and sold them at 3s. 6d. each, gaining the cost price of 4 sheep by the transaction. How many did I buy? Ans. 24. 9. A and B set out on a walk of 200 miles, A travels one mile an hour faster than B, and finishes the journey in 10 hours less time. What was the rate of each person? Ans. 4 and 5 miles per hour. 10. A number consists of 2 digits; their sum is 13, and their product is 34 less than the number. What is that number? Ans. 76. 11. Divide 72 into two such parts that their product may be 1296. Ans. 36 and 36. 12. What two numbers are those whose product is 333 and sum 46? Ans. 37 and 9. Quadratic Equations with two unknown quantities. Of course there must be two equations since there are two unknown quantities. When one of them is a simple equation we can substitute the value it gives for one of the unknown quantities in the equation, which is quadratic, and solve this by the methods already given. Thus 22+3xy+y=79) (1) GEOMETRY.-BOOK I. DEFINITIONS. I. A point has no parts, or no magnitude. III. Points are the extremities of a line. IV. A straight line is that which lies evenly between its extreme points. V. A superficies has only length and breadth. VI. The extremities of a superficies are lines. VII. A plain superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. VIII. A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction. IX. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. An angle is named from three or one letters, as ABC or B. X. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it. XI. An obtuse angle is greater than a right angle. XII. An acute angle is less than a right angle. XIII. A term or boundary is the extremity of any. thing. XIV. A figure is that which is enclosed by one or more boundaries. XV. A circle is a plane figure contained by one line, called the circumference, such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. XVI. This point is called the centre. XVII. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. XVIII. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter. XIX. A segment of a circle is the figure contained by a straight line, and the circumference it cuts off. XX. Rectilineal figures are contained by straight lines. XXI. Trilateral figures, or triangles, by three straight lines. XXII. Quadrilateral, by four straight lines. XXIII. Multilateral figures, or polygons, by more than four straight lines. XXIV. An equilateral triangle has three equal sides. XXV. An isosceles triangle has only two sides equal. XXVI. A scalene triangle has three unequal sides. XXVII. A right-angled triangle has a right angle. XXVIII. An obtuse-angled triangle has an obtuse angle. A |