Ex. (2) If xxy and xz, then will x∞ √(yz). Ex. (3) If y vary as x and when x=1, y=2, what will be the value of y when x=2} Here ya a given value of y: corresponding value of x ; ..y:x= 2:1; Hence, when x=2, y=4. ... y=2x. Ex. (4) If A vary inversely as B and when A=2, B=12, what will B become when A=9? Ex. (5) If A vary jointly as B and C, and when A = 6,B=6, and C=15, find the value of A when B=10 and C=3. Here A: BC= a given value of A: corresponding value of BC; :. A: BC=6:6×15; .. 90A 6BC. Ex. (6) If z vary as x directly and y inversely, and if when z=2, x=3 and y=4, what is the value of z when x=15 and y=8? 3. If A ∞ B and C∞ D then will AC∞ BD. 4. If xy and when x=7, y=5, find the value of x when y=12. and when X= 10, y=2, find the value of y when x=4. 6. If xyz and when x=1, y=2, z=3, find the value of y when x=4 and z = 2. 7. If x∞ y 2, and when x=6, y=4 and ≈=3, find the value of x, when y = 5 and z=7. 8. If 3x+5y ∞ 5x+3y, and when x=2, y=5, find the 9. If Aoc B and B3 ∞ C2, express how A varies in respect of C. 10. If z vary conjointly as x and y, and z=4 when x=1 and y=2, what will be the value of x when z=30 and y = 3 ? 11. If AB and when A is 8, B is 12; express A in terms of B. 12. If the square of a vary as the cube of y, and x= = 3 when y=4, find the equation between x and y. 13. If the square of a vary inversely as the cube of y, and x=2 when y=3, find the equation between x and y. 14. If the cube of a vary as the square of y and x = 3 when y=2, find the equation between x and y. 16. Shew that in triangles of equal area, the altitudes vary inversely as the bases. 17. Shew that in parallelograms of equal area, the altitudes vary inversely as the bases. 18. If y=p+q+r, where p is invariable, q varies as x, and r varies as a, find the relation between y and x, supposing that when x=1, y=6; when x=2, y=11; and when x=3, y = 18. 19. The volume of a pyramid varies jointly as the area of its base and its altitude. A pyramid, the base of which is 9 feet square and the height of which is 10 feet, is found to contain 10 cubic yards. What must be the height of a pyramid upon a base 3 feet square in order that it may contain 2 cubic yards? 20. The amount of glass in a window, the panes of which are in every respect equal, varies as the number, length and breadth of the panes jointly. Shew that if their number varies as the square of their breadth inversely, and their length varies as their breadth inversely, the whole area of glass varies as the square of the length of the panes. 372. AN Arithmetical Progression is a series of numbers which increase or decrease by a constant difference. Thus, the following series are Arithmetical Progressions: 2, 4, 6, 8, 10; 9, 7, 5, 3, 1. The Constant Difference being 2 in the first series and -2 in the second. 373. In Algebra we express an Arithmetical Progression thus: taking a to represent the first term and d to represent the constant difference, we shall have as a series of numbers in Arithmetical Progression and so on. a, a+d, a +2d, a+3d, We observe that the terms of the series differ only in the coefficient of d, and that each coefficient of d is always less by 1 than the number of the term in which that particular coefficient stands. Thus Consequently the coefficient of d in the nth term will be Therefore the nth term of the series will be a +(n−1) d. and the last term, the term next before z will clearly be z-d, and the term next before it will be z-2d, and so on. Hence, the series written backwards will be 375. To find the sum of a series of numbers in Arithmetical Progression. the series in the second case being the same as in the first, but written in the reverse order. Therefore, by adding the two series together, we get 28=(a+2)+(a + z) + (a + z) + ......+(a + z) + (a + z)+(a + z); and since on the right-hand side of this equation we have a series of n numbers each equal to a +z, we get This result may be put in another form, because in the place of z we may put a + (n−1)d, by Article 373. n Hencé {a+a+ (n − 1) d}, that is, s=2 {2a+(n−1) d'}. |
