110. Algebraic Proportions. If we consider the algebraic fraction (ab)/(ab2) we see (upon dividing both numerator and denominator by ab) that it reduces to a/b. In other words, we have This is an example of an algebraic proportion. Similarly is an algebraic proportion, and it may be written also in the form 111. Principle. Let a/b=c/d be any proportion. By multiplying both sides of this equality by bd (Axiom III, § 9) we obtain Xd=xby, or ad=bc. This result may be stated in words as follows: PRINCIPLE. In any proportion the product of the means (see § 109) equals the product of the extremes. This principle is often useful in testing the correctness of a proportion. Thus, 6:9 14:21 is a true proportion because the product of the means, which is 9×14, is equal to the product of the extremes, which is 6×21; but 6:9-8:15 is not a true proportion because 9X8 is not equal to 6×15. Similarly x3: x2y=x:y because x2y x=x3· y. EXERCISES By means of the principle in § 111 test the correctness of each of the following proportions. 10. (x2 y2): (2x+2y): : (2x-2y): 4. 11. (a2-b2): (a+b)2:: (a−b): (a+b). [HINT. See Formula V, p. 101.] By means of the principle in § 111 find the value which x must have in each of the following proportions. 25. What number bears the same ratio to 2 as 8 does to 3? [HINT. Let x represent the unknown number and form a proportion. Solve for x.] 26. What number bears the same ratio to 7 as 2 does to 3? 27. Divide 35 into two parts whose ratio shall be . [HINT. Let x be one part. Then 35-x will be the other part.] 28. Divide 25 into two parts such that the greater increased by 1 is to the lesser decreased by 1 as 4 is to 1. 29. Two men divide $6300 between them so that the parts are to each other in the ratio 3:4. How much does each receive? 30. A man's income from two investments is $850. The two investments bear interests which are in the ratio of 6 to 8. What income does he receive from each? 31. Concrete for sidewalks is a mixture made of two parts of sand to one part of cement. How much of each is required to make a walk containing 500 cubic feet? 32. Find the number which, when added to each of the numbers 1, 2, 4, and 7, will give four numbers in proportion. 33. Prove that no four consecutive integers, as n, n+1, n+2, n+3, can form a proportion. 34. A bubble of air of volume v units when rising from a depth of d feet below the surface of the water gradually expands until it reaches a volume of V units at the surface such that Whence, find the volume at the surface of a spherical bubble which starts at a depth of 100 feet with a radius of 1 inch. [HINT. See Ex. 28, p. 23.] 35. Solve for d in the formula of Ex. 34 (see § 107) and use your result to answer the following question: From what distance below the surface must a bubble rise in order that its volume may increase from 3 cu. in. to 20 cu. in.? 112. The Lever. If a 2-pound weight be attached to one end of a yardstick and a 1-pound weight to the other end, and the whole be then exactly balanced, as shown in Figure 46, we have an example of a lever. The point (pivot) around which the balance takes place is called the fulcrum. F 24 inches 12 inches 1 lb. FIG. 46. 2 lbs. If this experiment be tried (and it easily can be at home or in the classroom) it will be found that when the balance is exact, the fulcrum is just 24 inches from one end of the stick, and 12 inches from the other end. Thus the ratio of these two distances is 24, or 4, which is therefore just the same as the ratio of the two weights to each other. This experiment illus trates a general law as follows: שן FIG. 47 F D W LAW OF THE LEVER. If two weights W and w are balanced at the ends of any uniform bar at distances D and d respectively from the fulcrum (Fig. 47), we have 1. In the figure above suppose W=4 pounds, w=2 pounds, and D=6 inches. What must be the value of d? [HINT. By the law stated above, we have 4/2 = d/6. Solve for d.] 2. Fill in each of the following question marks (?) in such a way that the balance will be perfect in the above figure. (a) W-9 pounds, w=3 pounds, d= 1 inch, D=? (d) W=3 ounces, w=1 pound, d= 12 centimeters, D= ? 3. Where must we place the fulcrum under a 12-foot plank in order that a 56-pound boy at one end may balance a 112-pound boy at the other end? [HINT. Let x be the distance from one end. Then the distance from the other end will be 12-x.] 4. Two boys balance at seesaw on a 12-foot plank. The fulcrum is 5 feet from the heavier boy, who weighs 105 pounds. How much does the other boy weigh? 5. Sometimes, instead of having two weights balanced, we have a single weight balanced by a force, or, as it is usually called, a power. This may happen in several ways as indicated by the following figures. Note that in the last two figures the fulcrum is at one end of the bar. In all these cases, if we let W represent the weight, p the power, D the distance of the weight from the fulcrum, and d the distance of the power from the fulcrum, we have This is called the general law of the lever. |