Ex. 9. From the equations 3-y3 : (x − y)3 :: 61: 1, and x: 8: 40 y, find the values of x and y. Ex. 10. Divide 60 into two such parts, that the product shall be to the sum of the squares :: 2 : 5. Ex. 11. A land-tax of £23. 6s. 8d. has to be paid by A and B in proportion to their clear rent; what portion of it must be paid by each, when A has a clear rent of £60, and B a gross rent of £380? .. B pays £20. of the land-tax, and A pays £3. 6s. 8d. Ex. 12. A landlord has two tenants, A and B ; A's rent amounts to £80. clear of rates and insurance, and B's gross rent amounts to £162.1s.: A's rates are £9.8s., and A's and B's insurance together is £8.16s. find B's clear rent. 1. Find a fourth proportional to,, and ; and a mean proportional to '017 and 153. 2. What number has the same proportion to 100 that 9 has to 20? Is it true that 5 is to 4 as 16 is to 20? or that ab: bc :: ad: cd? or that 100: 1 as 50: 2? Is Euclid's definition of Proportion satisfied by the four numbers 22, 10, 33, 16? (7) a2+b2: a2-b2 :: c2 + d2: c2-d2. (8) ma+nc: pa+qc :: mb+nd: pb+qd. (10) ma+b: mb+a :: mc+d: md+c. (11) a+b+c+db+d::c+d: d. (13) a (a+c): c2: b (b+d): d2. (15) (a+b+c+d) (a+d-b−c)=(a+c−b−d) (a+b-c-d). 5. If ad= bc, and bf=ed, then a e::c:f. If a=2, b=6, c=9, e=12, find the values of d and f. (3) c2+d2: e2+f2 :: cd: ef. (4) a-mce: b-md+f:: ma- nc+ qe mb-nd+qf. 7. If ma+nb: mc+nd :: b: d, then also ma + nb: mc+nd :: a: c. 8. If xy :: m2: n2, and m:n :: √p2+x2: √√p2-y2, then p2: xy:: x+y: x-y. 9. If a, b, c, d, e be in continued proportion, then a e::a: 64. 10. If 2a+3b: 4a+5b :: 2x+3y: 4x+5y, then ab::: y. (2) Ny-(y-x)=(20-x), and (y-x): (20-x) :: 3 : 2. (3) x+y:x:: 7:5, and xy + y2=126. (4) 2a: 1 :: x2 + y2: a, and n : m :: x − y : x + y. 12. What number must be added to a and subtracted from b, that the sum may be to the difference as mn? 13. Find two numbers, the greater of which shall be to the less as their sum is to 42, and as their difference is to 6. 14. Divide the number 100 into two such parts that 6 times their product shall be to the sum of their squares :: 24 : 17. 15. Two numbers are in the ratio of 2 to 3; and if 9 be added to each, they are in the ratio of 3 to 4: find the numbers. 16. A house is let for £40 a year subject to the payment by the tenant of the insurance and rates, and another house is let for £80 a year clear of insurance and rates. The insurance on the two houses is £2. 14s. 6d. and the rates on the first one £58. What is the amount of the rates on the second house, and of the insurance on each, supposing the rates and insurance on the two houses respectively to be proportional to their annual value? VARIATION. 143. DEF. If x and y be two variable quantities, and be so related to each other, that when x is changed in any manner, the value of y is changed in the same proportion; then y is said to VARY DIRECTLY as x: and this relation is represented thus y cx; meaning that y varies directly When it is briefly said that y varies as x, it is always meant that y varies directly as x. as x. Ex. If the altitude of a triangle be invariable, the area varies as the base. For it appears from Euclid, Book vi. Prop. 1, that the altitude being the same, in whatever proportion the base of a triangle be altered, the area of the triangle standing upon such base is altered in the same proportion; hence, when the altitude is constant, area ∞ base. 144. If x and y be two variable quantities, so related that y xx, then will y Ca, where C is some constant quantity, that is, some quantity which does not vary. For since when x changes, y also changes in the same proportion; the ratio yx, and therefore the quantity y, which is the measure of the ratio, will be always of the same constant value. Let C denote this constant value. Ex. If the altitude of a triangle be invariable, the area = a constant quantity the base. For if the altitude be constant, then area c base, (Ex. Art. 143); .. area= Cx base. This will readily appear from Euclid, Book 1. Prop. 41. For by that proposition, If ABC be a triangle, and AD be drawn perpendicular to BC, then area of AABC Similarly, if the base be constant, =2C suppose, then area of AABC=C.AD, or area of A= C. altitude. 145. When yox, it is shewn in the last article that y= Cr, where C is some constant quantity. If we know the particular value of y corresponding to any particular value of x, we can determine the constant C. Thus, suppose when x=a, y is known to be = b; then which is the value of C; and the equation connecting x and y will be Ex. A body moves along in such a manner that the space over which it passes varies directly as the square of the time; and when it has been moving 3 minutes, the space described is 27 feet: find the general equation connecting the space and the time. Let y denote the space described in a time x, that is, the number of feet described in x minutes. |