tor are equal, the value of the fraction is a unit or 1. (Art. 117.) Thus the ratio of 6x2 : 12 is 1; for the value of 13=1. (Art. 121.) Obs. This is called a ratio of equality. 316. If the antecedent of a couplet is greater than the consequent, the ratio is greater than a unit: for, if the numerator is greater than the denominator, the value of the fraction is greater than.). (Art. 117.) Thus the ratio of 12 : 4 is 3. Obs. This is called a ratio of greater inequality. 317. If the antecedent is less than the consequent, the ratio is less than a unit: for, if the numerator is less than the denominator, the value of the fraction is less than 1. (Art. 117.) Thus, the ratio of 3 : 6 is %, or }; for {=} (Art. 120.) Obs. This is called a ratio of less inequality. 11. What is the direct ratio of 3 : 9, expressed in the lowest terms? What the inverse ratio ? Ans. }; and } - =3. (Arts. 308, 309.) 12. What is the inverse ratio of 4 to 12 ? Of 6 to 18? Of 9 to 24 ? Of 21 to 25 ? Of 40 to 56 ? 13. What is the direct ratio of 15s. to £2? Of 13s. 6d. to £1? Of £2, 10s. to £3, 5s. ? 14. What is the direct ratio of 6 inches to 3 feet? 15. What is the direct ratio of 15 oz. to 1 cwt. ? PROPORTION. 318. Proportion is an equality of ratios. Thus, the two ratios 6: 3 and 4 : 2 form a proportion ; for Q=, the ratio of each being 2. QUEST.-315. When the two numbers compared are equal, what is the ratio? Obs. What is it called ? 316. When the antecedent is greater than the consequent, what is the ratio? Obs. What is it called ? 317. If the antecedent is less than the consequent, what is the ratio ? Obs. What is it called ? 318. What is proportion ? Obs. The terms of the two couplets, that is, the numbers of which the proportion is composed, are called proportionals. 319. Proportion may be expressed in two ways. First, by the sign of equality (=) placed between the two ratios. Second, by four points or a double colon (: :) 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. Obs. 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.” 320. 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. 306.) There may, however, be a proportion formed from three numbers, for one of the numbers may be repeated so as to form two terms. Thus the numbers 8, 4, and 2, are proportional; for the ratio of 8 : 4=4:2. 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. 305, 318.) 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. QUEST.-Obs. What are the numbers of which a proportion is composed, called? 319. In how many ways is proportion expressed? What is the first? The second ? 320. How many terms must there be in a proportion? Why? Can a proportion be formed of three numbers ? How? Will there be four terms in it? 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 ! 321. 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. 322, 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. 323. 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. Thus 8 324. If four numbers are in proportion, the product of the extremes is equal to the product of the means. Thus 8:4:: 6:3 is a proportion : for (Art. 318.) Now 8X3=4x6. We have seen that 2:3::6:9. (Art. 318.) I=. (Art. 120.) Now, Quest.–321. Which terms are the extremes ? Which the means? Obs. What are homologous terms? Analogous terms? 222. What is direct proportion? Obs. In direct proportion what ratio has the first term to the second ? 323. What is inverse proportion? Obs. What ratio has the first term to the second in this case ? *324. If four numbers are proportional, what is the product of the extremes equal to? 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 the numbers? Multiplying each ratio by 27, (the product of the denominators) 2X27_6X27 The proportion becomes 9 (Art. 295. Ax. 6.) Dividing both the nimerator and the denominator of the first couplet by 3 ; (Art. 116;) or canceling the denominator 3 and the same factor in 27; (Art. 136 ;) also canceling the 9, and the same factor in 27, we have 2x9=6X3. But 2 and 9 are the extremes of the given proportion, and 3 and 6 are the means; and the product of the extremes 2X9=6X3, 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 prod. ucts are not equal, the numbers are not proportional. 325. It follows from the last Article, that if the product of the means is divided by one of the extremes, the quotient will be the other extreme ; and if the prodduct of the extremes is divided by one of the means, the quotient wil be the other mean. For, if the product is divided by one of the factors, the quotient will be the other factor. (Art. 291.) Take the proportion 8:4::6: 3. And the product 4x6:3=8, the other extreme. Hence, If 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. SIMPLE PROPORTION. 326. Proportion in arithmetic is usually divided into Simple and Compound. Simple Proportion is an equality between two ratios. Its principal object is to find the fourth term of a proportion, when the first three terms are given. (Art. 325.) The QUEST.--225. 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? 226. What is simple proportion? What is its chief object ? resolution of this problem is the most important result in the theory of proportion. 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 first three terms of a proportion are 4, 6, 8, what is the fourth term ? Solution. 6x8=48 and 48-4=12, which is the number required ; that is, 4:6::8:12. Proof. 4x12 is equal to 6 x 8. (Art. 324. Obs. 2.) 2. If 12 bls. of flour cost $72; what will 4 bls. cost, at the same rate ? Solution. It is evident 12 bls. has the same ratio to 4 bls. as the cost of 12 bls. ($72) has to the cost of 4 bls., which is required. That is, 12 bls. : 4 bls. : : $72 is to the cost of 4 bls. Now, 72 x4=288; and 288-12 = $24, the cost required; that is, 12 bls. : 4 bls, ; ; $72 : $24. 3. If 6 men can dig a cellar in 12 days, how many men will it take to dig it in 4 days ? Note. Since the answer is men, we put tho given number of men for the third term. Then, as it will require more men to dig it in 4 days than it will to dig it in twelve days, we put the larger number of days for the second term, and the smaller for the first term. Solution. 4d. : 12d. ; : 6 men to the number of men required. Now, 12 x6=72; and 72-4=18. Ans. 18 men. Do these Quest.-Obs. What is simple proportion often called ? terms convey an idea of the nature or object of the rule ? |