or (5) If a : b=c:d, then a+b: a-b=c+d:c-d. a-6 C-d ; b d b d ... by division, a+b.c+d. a-bc-d a+b:9-b=c+d:c-d, Several other proportions may be proved in a similar way. Example 1. If a:b=c:d=e:f, k; then a=bk, c=dk, e=fl; e a e ae =ka ū * jbf' or = Example 2. If (3a+66+c+20)(30—65c+20) =(3a -6b+c-24) (3a+6b-c-20), prove that a, b, c, d are in proportion. 3a+66+c+2d 3a +6b-6-20 We have [Art. 326.) 3a-6b+0-2d3a-6b-c+20 2 (3a+c) Composition and Division, 2(3-0) 3atc 6b+2d Alternation, 3a-C 66-2à 126 Again, Composition and Division, 2c 4d whence a:b=c:d. X2+x-2 4x2 + 5x – 6 Example 3. Solve the equation X-2 5x -6 22 4x2 Division, 2 whence, dividing by x2, which gives a solution x=0, [Art. 216.] 1 4 ; whence, x= -2; 5x - 6 and therefore the roots are 0, -2. X 5x-6 X-2 a с : a + :c. EXAMPLES XXXV. b. Find a fourth proportional to 1. a, ab, c. 2. a, 2ab, 362. 3. x3, xy, 5x2y. Find a third proportional to 4. a'b, ab. 5. 2., 2.2 6. 3.x, 6xy. 7. 1, a. Find a mean proportional between 8. a, 62 9. 2x3, 8.x. 10. 12ax2, 3a. 11. 27a353, 36. If a, b, c be three proportionals, show that 12. a : a+b=a-6 : a-c. 13. (12+bc+c)(ac-bc+c=14+ac+c! If a : b=c : d, prove that 14. ab+cd : ab- cd=a? +ca : a? – ca. 15. a? +ac+: u? - ac+c=22+bd+d2 : 62 - bd+d%. 16. a : b=1345 +502 : V30++id-. 0 d 17. + р 9 р 2 cd C2 + d2 Solve the equations : 13. 3x - 1 : 6x – 7=70 - 10 : 9x + 10. 20. X – 12 : Y+3=2x – 19 : 5y – 13=5 : 14. x2 – 2x+3 22 – 3x + 5 2.X - 1 21. x +4 22. 2.0 - 3 3x - 5 4+2.0-12++4 23. If (a+b-3c-3d) (2a-2b-c+d) =(20+20 --c-d) (a-6-30+30) prove that a, b, c, d are proportionals. 24. If a, b, c, d are in continued proportion, prove that a :d=a3 + b3 + c3 : 63 +03+d3. 25. If b is a mean proportional between a and c, show that 4a2 – 962 is to 462 – 9c2 in the duplicate ratio of a to b. 26. If a, b, c, d are in continued proportion, prove that b+c is a mean proportional between a +b and c+d. 27. If a+b :b+c=c+d:d+a, prove that a=c, or a+b+c+d=0. a с + : с . VARIATION. 331. DEFINITION. One quantity A is said to vary directly as another B, when the two quantities depend upon each other in such a manner that if B is changed, A is changed in the same ratio. NOTE. The word directly is often omitted, and A is said to vary as B. 332. For instance: if a train moving at a uniform rate travels 40 miles in 60 minutes, it will travel 20 miles in 30 minutes, 80 miles in 120 minutes, and so on; the distance in each case being increased or diminished in the same ratio as the time. This is expressed by saying that when thc velocity is uniform the distance is proportional to the time, or more briefly, the distance varies as the time. 333. The symbol oc is used to denote variation; so that A o B is read “A varies as B.” and so on, 334. If A varies as B, then A is equal to B multiplicd by some constant quantity. For suppose that a,, Ag, Ag..., bj, b2, bz... are corresponding values of A and B. A B А B A B i ; A to ъ, ь, ь, Hence any value of A is always the same; the corresponding value of B A that is, where m is constant. .. A=mB. 335. DEFINITION. One quantity A is said to vary inversely as another B when A varies directly as the reciprocal of B. [See Art. 162.] Thus if A varies inversely as B, A : where m is constant. B The following is an illustration of inverse variation : If 6 men do a certain work in 8 hours, 12 men would do the same work in 4 hours, 2 men in 24 hours; and so on. Thus it appears that when the number of men is increased the time is proportionately decreased; and vice-versâ. mi m 336. DEFINITION. One quantity is said to vary jointly as a number of others when it varies directly as their product. Thus A varies jointly as B and C when A=mBC. For instance, the interest on a sum of money varies jointly as the principal, the time, and the rate per cent. 337. DEFINITION. A is said to vary directly as B and in B versely as C when A varies as C 블 338. Grouping the principles of Arts. 334–337, we have A=mB, if A varies directly as B, A inversely as B. jointly as B and C, directly as B and inversely as C. C' 339. If A varies as B when C is constant, and A varies as c when B is constant, then will A vary as BC when both B and C vary. The variation of A depends partly on that of B and partly on that of C. Suppose these latter variations to take place separately, each in its turn producing its own effect on A ; also let a, b, c be certain simultaneous values of A, B, C. 1. Let C be constant while B changes to b; then A must undergo a partial change and will assume some intermediate where A B ...(1). b 2. Let B be constant, that is, let it retain its value b, while C changes to c; then A must complete its change and pass from its intermediate value a' to its final value a, where a c (2). value a', a с 340. The following are illustrations of the theorem proved in the last article. The amount of work done by a given number of men varies directly as the number of days they work, and the amount of work done in a given time varies directly as the number of men; therefore when the number of days and the number of men are both variable, the amount of work will vary as the product of the number of men and the number of days. Again, in Geometry the area of a triangle varies directly as its base when the height is constant, and directly as the height when the base is constant; and when both the height and base are variable, the area varies as the product of the numbers representing the height and the base. Example 1. If A B, and C D, then will AC OBD. Example 2. If x varies inversely as yo -1, and is equal to 24 when y=10; find x when y=5. m m .. X= By supposition, x=; where m is constant. yo-1 Putting x=24, y=10, we obtain 24= 99' whence m=24 x 99. ; y:-1 hence, putting y=5, we obtain a= =99. Example The volume of a pyramid varies jointly as its height and the area of its base; and when the area of the base is 60 square feet and the height 14 feet the volume is 280 cubic fect. What is the area of the base of a pyramid whose volume is 390 cubic feet and whose height is 26 feet? Let V denote the volume, A the area of the base, and h the height; then V=mAh, where m is constant. Substituting the given values of V, d, h we have 280=m x 60 x 14; 280 1 60 x 14 m. 3 |