44. Is the ratio of 6×9 to 7×8, a ratio of greater, or less inequality? • 45. Is the ratio of 2×4×16 to 4×32 a ratio of greater, or less inequality? 46. What is the ratio compounded of the ratios of 5 to 3, and 12 to 4? 47. What is the ratio compounded of 8: 10, and 20:16? 48. What is the ratio compounded of 3: 8, and 10:5? 49. What is the ratio compounded of 18: 20, and 30: 40? 50. What is the ratio compounded of 35: 40, and 60:75, and 21 to 19? 51. What is the ratio compounded of 60: 40, and 12:24, and 25:30? 489. In a series of ratios, if the consequent of each preceding couplet is the antecedent of the following one, the ratio of the first antecedent to the last consequent, is equal to that compounded of all the intervening ratios. Thus, in the series of ratios 3:4 4:7 7:16 the ratio of 3 to 16, is equal to that which is compounded of the ratios of 3: 4, of 4:7, and 7:16; for, the compound 3X4X7 3 = or 3:16. 4X7X1616' ratio is 490. If to or from the terms of any couplet, two other numbers having the same ratio be added or subtracted, the sums or remainders will also have the same ratio. (Thomson's Legendre, B. III., Prop. 1, 2.) Thus, the ratio of 12 : 3 is the same as that of 205. And the ratio of the sum of the antecedents 12+20 to the sum of the consequents 3+5, is the same as the ratio of either couplet. That is, 12+20:35::12:3=20:5, or 12+20 12 20 So also the ratio of the difference of the antecedents, to the dif ference of the consequents, is the same. That is, 20-12:5—3::12:3=20:5, or 20-12 12 20 = 5 3 3 4. 491. If in several couplets the ratios are equal, the sum of all the antecedents has the same ratio to the sum of all the consequents, which any one of the antecedents has to its consequent. 12:4=3 Thus, the ratio of 15:5=3 18:6=3 Therefore the ratio of (12+15+18): (4+5+6)=3. OBS. 1. A ratio of greater inequality is diminished by adding the same number to both terms. Thus, the ratio of 8:2, is 4; and the ratio of 8+4:2+ 4 is 2. 2. A ratio of less inequality is increased by adding the same number to both the terms. Thus, the ratio of 2:8 is, and the ratio of 2+16:8+16 is . PROPORTION. 492. PROPORTION is an equality of ratios. Thus, the two ratios 6:3 and 4: 2 form a proportion; for =, the ratio of each being 2. OBS. The terms of the two couplets, that is, the numbers of which the proportion is composed, are called proportionals. 493. Proportion may be expressed in two ways. First, by the sign of equality (=) placed between the two ratios. Second, by four points (::) placed between the two ratios. Thus, each of the expressions, 12:6 4:2, and 12:6::4:2, is a proportion, one being equivalent to the other. The latter expression is read, "the ratio of 12 to 6 equals the ratio of 4 to 2," or simply, "12 is to 6 as 4 is to 2." OBS. The sign (::) is said to be derived from the sign of equality, the four points being merely the extremities of the lines. 494. The number of terms in a proportion must at least be four, for the equality is between the ratios of two couplets, and each couplet must have an antecedent and a consequent. (Art. 476.) There may, however, be a proportion formed from three numbers, for one of the numbers may be repeated so as to form two QUEST.-492. What is Proportion? 493. How many ways is proportion expressed? What is the first? The second? 494. How many terms must there be in a proportion? Why? Can a proportion be formed of three numbers? How? terms. Thus, the numbers 8, 4, and 2, are proportional; for the ratio of 8:44:2. It will be seen that 4 is the consequent in the first couplet, and the antecedent in the last. It is therefore a mean proportional between 8 and 2. OBS. 1. In this case, the number repeated is called the middle term or mean proportional between the other two numbers. The last term is called a third proportional to the other two numbers. Thus 2 is a third proportional to 8 and 4. 2. Care must be taken not to confound proportion with ratio. (Arts. 474, 492.j In a simple ratio there are but two terms, an antecedent and a consequent; whereas in a proportion there must at least be four terms, or two couplets. Again, one ratio may be greater or less than another; the ratio of 9 to 3 is greater than the ratio of 8 to 4, and less than that of 18 to 2. One proportion, on the other hand, cannot be greater or less than another; for equality does not admit of degrees. 495. The first and last terms of a proportion are called the extremes; the other two, the means. OBS, Homologous terms are either the two antecedents, or the two consequents. Analogous terms are the antecedent and consequent of the same couplet. 496. Direct proportion is an equality between two direct ratios. Thus, 12:4:: 9:3 is a direct proportion. OBS. In a direct proportion, the first term has the same ratio to the second, as the third has to the fourth. 497. Inverse or reciprocal proportion is an equality between a direct and a reciprocal ratio. Thus, 8:4:::; or 8 is to 4, reciprocally, as 3 is to 6. OBS. In a reciprocal or inverse proportion, the first term has the same ratio to the second, as the fourth has to the third. 498. If four numbers are proportional, the product of the extremes is equal to the product of the means. Thus, 8:46:3 is a proportion; for =, (Art. 492,) and 8X3=4X6. QUEST. Obs. What is the number repeated called? What is the last term called in such a case? What is the difference between proportion and ratio? 495. Which terms are the extremes? Which the means? Obs. What are homologous terms? Analogous terms? 496. What is direct proportion? Obs. In direct proportion what ratio has the first term to the second? 497. What is inverse proportion? Obs. What ratio has the first term to the second in this case? 498. If four numbers are proportional, what is the product of the extremes equal to ? Again, 12:6::: is a proportion. (Art. 496.) OBS. 1. The truth of this proposition may also be illustrated in the following manner: The numbers 2:3::6:9 are obviously proportional. (Art. 492.) Multiplying each ratio by 27, (the product of the denominators.) The proportion becomes 2×27 6×27 (Art. 21. Ax. 6.) Dividing both the numerator and the denominator of the first couplet by 3, (Art. 191,) or canceling the denominator 3 and the same factor in 27, (Art. 221,) also canceling the 9, and the same factor in 27, we have 2×9=6X3. But 2 and 9 are the extremes of the given proportion, and 3 and 6 are the means; hence, the product of the extremes is equal to the product of the means. 2. Conversely, if the product of the extremes is equal to the product of the means, the four numbers are proportional; and if the products are not equal, the numbers are not proportional. 499. Proportion, in arithmetic, is usually divided into Simple and Compound. SIMPLE PROPORTION. 500. SIMPLE PROPORTION is an equality between two simple ratios. It may be either direct or inverse. (Arts. 479, 496, 497.) The most important application of simple proportion is the solution of that class of examples in which three terms are given to find a fourth. 501. We have seen that, if four numbers are in proportion, the product of the extremes is equal to the product of the means. (Art. 498.) Hence, If the product of the means is divided by one of the extremes, the quotient will be the other extreme; and if the product of the extremes is divided by one of the means, the quotient will be the QUEST.-Obs. If the product of the extremes is equal to the product of the means, what is true of the four numbers? If the products are not equal, what is true of them? 499. How is proportion usually divided? 500. What is simple proportion? What is the most important application of it? 501. If the product of the means is divided by one of the extremes, what will the quotient be? If the product of the extremes is divided by one of the means, what will the quotient be? other mean. For, if the product of two factors is divided by one of them, the quotient will be the other factor. (Art. 156.) Take the proportion 8:4:6:3. Now the product So the product And the product 502. If, therefore, any three terms of a proportion are given, the fourth may be found by dividing the product of two of them by the other term. OBS. Simple Proportion is often called the Rule of Three, from the circumstance that three terms are given to find a fourth. In the older arithmetics, it is also called the Golden Rule. But the fact that these names convey no idea of the nature or object of the rule, seems to be a strong objection to their use, not to say a sufficient reason for discarding them. Ex. 1. If the product of the means is 84, and one of the extremes is 7, what is the other extreme, or term of the proportion? 2. If the product of the means is 54, and one of the extremes is 18, what is the other extreme ? 3. If the product of the means is 720, and one of the extremes is 45, what is the other extreme? 4. If the product of the means is 639, and one of the extremes is 213, what is the other extreme ? 5. If the first three terms of a proportion are 8, 12, and 16, what is the fourth term? Solution.-12×16=192, and 192÷8=24, the fourth term, or number required; that is, 8:12::16:24. 6. It is required to find the fourth term of the proportion, the first three terms of which are 36, 30, and 24. 7. Required the fourth term of the proportion, the first three terms of which are 15, 27, and 31. 8. Required the fourth term of the proportion whose first three terms are 45, 60, and 90. QUEST. Obs. What is simple proportion often called? Do these terms convey an idea of the nature or object of the rule ? |