Ex. 5. If xxx+my; and when x=1, y=2, ≈=3; but when x=2, y=3, z=4; find m. Ex. 6. Assuming that the area of a triangle varies as the altitude and base jointly, and that when the altitude is 4ft. Gin. and base 2ft. 8in. the area is 6 square feet; find the area of a triangle whose base is 6ft. 10in. and altitude 5 ft. 2in. Ex. 7. Assuming that the time of oscillation of a pendulum varies as the square root of its length; if the length of a pendulum which oscillates in a second be 39.2 inches, find the length of one which oscillates 56 times in a minute. Ex. 8. Assuming that the wages vary jointly as the men and the number of hours during which they work; if 13 men earn £7 in 15 days of 8 hours each, what will be the wages of 52 men for 12 days of 9 hours each? In this case wages=Cx men x hours worked, C being a constant. 1. Given that y varies as x, and that when x=1, y=2, what will be the value of y, when x=2? 2. Given that ≈ varies jointly as x and y, and that when x=1 and y=1, z=1; find the value of ≈ when x= 2 2 and y=2. 3. If A vary inversely as B, and when A=2, B=10, what will B become when A=8? 4. If Ac BC, and three corresponding values of A, B, C are 6, 9, 10 respectively, find the value of A when B-5 and C=3. 5. If A & B, shew that A" ∞ B". 6. If A & B, shew that AD ∞ BD, and stant or not. A B whether D be con 7. Shew that the bases of equal parallelograms vary inversely as their altitudes. 8. At what rate of interest will £a bear the same interest as £b bears when interest is at c per cent.? 9. If several slabs of marble have the same solid content, shew that the flat surface varies inversely as the thickness. 10. If x-ax y-b, and when x=m, y=n; find the constant which connects x-a with y-b. 11. z cx+y, and y ∞ x2, and when x=1, y=2, z=3; find the relation between ≈ and x. 12. Shew that if y xx, and x x x2, then y+x ∞ √x. 13. If x mx+y; and if x=3 when x=1 and y=2, but z=5 when x=2 and y=3; find m. 14. Assuming that the area of a triangle varies as the altitude and base jointly, and that when the altitude is 1 ft. 6 in. and base 1 ft. 4 in. the area is 1 square foot, find the area of a triangle whose base is 3 ft. 5 in. and altitude 2ft. 7 in. 15. If the volume of a cone whose height is 12 inches and base 30 inches be 120 cubic inches, find the volume of another whose height is 20 inches and base 1 foot: the volume of a cone varying as the height and base jointly. 16. A locomotive engine without a train can go 24 miles an hour, and its speed is diminished by a quantity which varies as the square root of the number of waggons attached. With four waggons its speed is 20 miles an hour. Find the greatest number of waggons which the engine can move. Ex. LIX. Miscellaneous Questions and Examples in Equations, Ratio, Proportion, and Variation. A. 1. Define an Equation; explain fully what is meant by an Identity, so as to shew clearly the difference between an Identity and an Equation. What is meant by 'solving an Equation"? 3. There are 3 beggars A, B, C, who nevertheless have some little money amongst them; and B has one penny more than A and C together. Upon giving them three pence apiece, A's money becomes doubled, and B's becomes twice as much as at first A and C had together. What had each at first? 4. Define Ratio and Proportion; a ratio of equality; one of greater; and one of less inequality; continued proportion. How is a ratio of greater inequality affected by subtracting the same quantity from each of its terms? 5. (a) If a b c d; prove that (1) ad=bc. (2) a: b :: ma+nc: mb+nd. (B) If a b c d, and also a ee c, then b: d :: a2 : e2. 6. When is one quantity said to vary directly as another; inversely as another; jointly as two others; directly as a second, and inversely as a third ? If xy, shew that x= Cy, where C is some constant quantity. Given that y varies as the sum of two quantities, one of which varies as a directly, and the other as a inversely, and that when x=1, y=4; and when x=2, y=5; find the relation between x and y. B. 1. Define 'the root of an equation.' What are Simultaneous Equations? Distinguish between a pure quadratic, and an adfected quadratic equation. In every quadratic equation, shew that the sum of the roots coefficient of a with its sign changed, and the product of the roots the |