dividend when the divisor is given; and inversely as the divisor when the dividend is given. Thus, if we divide 24 and 27 by 3, the quotients 8 and 9 are in the ratio of 24 to 27; that is, 8:9:: 24 : 27. But if we divide 24 first by 3 and then by 6, the quotients 8 and 4 are in the ratio of 6 to 3; that is, 8: 4 :: 6 : 3. 77. Hence, whenever any quantity so depends upon two others, that it is directly as each of them when the other is given, it must vary in the ratio of the product of two numbers taken proportional to those two quantities. Thus, the distance to which a man rides depends upon the time for which he rides, and the speed at which he rides; so as to be directly as either of them when the other is unvaried. If, therefore, A rides for 3 hours and B for 5 hours, and A ride twice as fast as B, the number of miles which A rides must be to the number of miles which B rides as 6 5, the products of the numbers which are proportional to their times and speed. : 78. But whenever any other quantity so depends upon two others, that it is directly as the first when the second is given, and inversely as the second when the first is given, it must vary as the quotient obtained by dividing the first by the second; that is, dividing numbers proportional to these quantities. Now, if I ride a journey, the requisite time so depends on the distance which I have to ride and the speed which I employ. It is directly as the distance, and inversely as the speed. If, therefore, A has to ride 50 miles and B 40, and A ride twice as fast as B, the time in which A performs his journey must be to the time in which B performs his as 50 to 40; that is, 25 to 40, or 5 to 8. 79. Any two products are said to be to each other in a ratio compounded of the ratios of their factors. Thus, the ratio compounded of the ratios of 2 5 and 7: 3 is the ratio of 14: 15. Hence, the ratio compounded of two equal ratios is, the ratio of the squares of the terms of either ratio. Thus, the ratio compounded of the equal ratio 96 and 3: 2 is the ratio of 81: 36, (9×9: 6×6) or 9 4 (3x3 2x2.) For since 9: 6:3: 2, it follows, § 67, that multiplying both antecedents by 3 and both consequents by 2, 27: 12:: 9:4; or multiplying both antecedents by 9 and both consequents by 6, that 81: 36 :: 27 But the ratio 27: 12 is, by definition, the ratio compounded of the ratios 9: 6 and 3 : 2. And thus it appears that, if any four numbers be proportional, their squares are proportional. : 12. Hence also it is evident that the ratio compounded of any ratio and its reciprocal, is a ratio of equality. Thus the ratio compounded of the ratios of 9: 6 and 6:9 is the ratio of 54: 54; that is, a ratio of equality. 80. Again, any ratio being given, we may conceive any number whatsoever interposed between its terms, and the given ratio as compounded of the ratios of the antecedent to the interposed number, and of the interposed number to the consequent. Thus, the ratio of 9 to 6 may be considered as compounded of the ratios of 9: 2 and 2: 6. For 9 is to 6 as twice 9 to twice 6, which is the compounded ratio mentioned. 81. From what has been said it is manifest, that the problem of finding a fourth proportional to three given numbers will frequently admit of an abbreviated solution, by substituting lower numbers. For in the first place, if the first two terms, or terms of the given ratio, admit of being divided evenly by the same number, we may substitute for them the resulting quotients, as being in the same ratio. Thus, if it be required to find a fourth proportional to 27, 63, and 21, solving the problem at large by rules laid down in § 68, we should have to take the product of 63 and 21, and then divide that product 1323 by 27, which gives the quotient 43 as the fourth proportional required. But 3 and 7 being equisubmultiples of 27, and 63 are in the same ratio; and operating with these lower numbers we find the same result. It may be proved, in like manner, that, whenever the first and third terms admit of being evenly divided by the same numbers, we may substitute the resulting quotients; for those equisubmultiples of the given antecedents must be proportional to the given consequent and the consequent sought. 82. Let it be required to find a number, to which a given number shall be in a ratio compounded of two or more given ratios. The ratio compounded of two given ratios is (by definition) the ratio of the products of their respective terms. Therefore this problem resolves itself into that of finding a fourth proportion to three given terms. Thus, if we want to find a number to which 6 shall be in a ratio compounded of 9 : 5 and 15: 36, it is the same thing as if we were required to find a number to which 6 shall be in the ratio 9X 15:5 X 36. But it is plain that both terms of this ratio are divisible by 9 and by 5, and that we may therefore substitute the ratio of the resulting quotients, 3: 4, so that the number sought is 6 X4÷3, or 8. Hence it appears that, in solving this problem, if the antecedent and consequent of either the same or different ratios admit of being evenly divided by the same number, we may substitute the resulting quotients; and that we therefore ought not to take the products of the corresponding terms of the ratios which we want to compound, till we have inspected them for the purpose of ascertaining whether they are capable of being thus reduced; nor till we have compared the antecedents of the given ratios with the given antecedent of the ratio whose consequent we seek. For in the last instance, after reducing the question to that of finding a fourth proportional to 3, 4, and 6, the terms may be reduced still lower by substituting for 3 and 6 their equisubmultiples 1 and 2. And thus a question, at first involving very high numbers and appearing to require a very tedious operation, may frequently admit of a very brief solution. 83. The Rule, § 68, for finding a fourth proportional is commonly called "the Rule of Three," because we have three terms of an analogy given to find a fourth. It may more justly be called the Rule of Proportion. Its very extensive practical application will be shown in Chapter VII. meanwhile the young student may exercise himself in the principles of this chapter by solving the fol lowing questions; and may easily increase the number of the examples at pleasure by substituting any other numbers. 40 Ex. 1. Find a fourth proportional to 15, 40, and 24. Here 40 and 24 are the means 15: 40: 24 -their product must be equal to the product of the extremes; hence, if the product of two numbers, and one of the numbers, be given, the other number is found by dividing that product by the given number, so that the fourth proportional in this example ist 64, and 15: 40 :: 24: 64. 15)960(64 Ans. 60 60 2. The first two and the last terms of an analogy are 17, 9, and 234. What is the third term? 3. From the analogy, 7: 25:: 21: 75, what equality may be derived? Ans. 7×75=25×21. 4. From the equality 12×7=14×6, what analogy may be inferred? Ans. 12 14 :: 6: 7. 5. The first and the last two terms of an analogy are 18, 102, and 17. What is the second term? Ans. 3. 6. What two numbers are in the ratio compounded of the ratios of 7 to 3, 4 to 5, and 11 to 13? Ans. 308: 195. 7. What two numbers are in the ratio compounded of 7 to 3, and 6: 14? Ans. 42 42. 8. What two numbers are in the ratio compounded of 17:3, 3: 14, and 14: 16? Ans. 714: 672, or 17: 16. 9. What is the ratio compounded of 17: 3, and 6 : 34 ? Ans. 102 102. Ans. 18. 10. Find a fourth proportional to 3, 9, and 6. Ans. 560. 12. Find a fourth proportional to 63, 234, and 84. Ans. 312. 13. Find a number to which 10 shall be in a ratio compounded of 6: 12 and 8 24. Ans. 60. 14. Find a number to which 3 shall be in a ratio compounded of the ratios of 4. 40 and 20: 8. Ans. 12. 15. Find a number to which 48 shall be in a ratio compounded of 4: 8 and 12: 16. Ans. 128, Questions. What is meant by ratio? How is the ratio of two quantities written? Which quantity is called the antecedent? which the consequent ? Is the ratio of any two numbers the same as the ratio of any equimultiples of those numbers? How is the equality of two ratios denoted? What is meant by analogy? Having given the first three terms of analogy, how do you find the fourth? What two terms of an analogy are called the extremes? what the means? Is the product of the extremes equal to the product of the means? CHAPTER V. FRACTIONS. On the Nature of Fractions. 84. If we divide any one whole thing, a foot, a yard, a pound, &c. into three equal parts, any one of them is one third of the whole; written thus, . If we take two of them, we take two thirds of the whole, or 3. Such expressions are called fractions; the number above the line is called the numerator of the fraction, and the number below the line the denominator. A proper |