(e.) The numbers which form any ratio are called TERMS of the ratio. The first term is called the ANTECEDENT of the ratio, and the second term is called the CONSEQUENT of the ratio. ILLUSTRATION. - In the ratio 6: 9, 6 is the antecedent and 9 the conse quent. (f.) Ratios, like fractions, may be SIMPLE, COMPLEX, or Coμ POUND. (g.) A SIMPLE RATIO is the ratio of two entire numbers; as, 5:7, 4: 5, or 3, 4. (h.) A COMPLEX RATIO is the ratio of two fractional numbers, or of an entire and a fractional number; as, † : 41, 67 : 3, or 41 3 (i.) A COMPOUND RATIO is the indicated product of two or more ratios; as, (38) × (6 : 4), or § of . (j.) A compound ratio is usually expressed by writing the ratios which compose it under each other. ILLUSTRATION. 7:11 5: 3 the same thing, it means expresses that the product of the two ratios 7: 11 and 5: 3 is to be obtained; or, which is of, or y × 3, or 7 × 5 : 11 × 3. (k.) The principles involved in operations on ratios are similar in every respect to those involved in the corresponding operations on other fractions; hence (1.) Multiplying or dividing both terms of a ratio by the same number does not alter its value. 1. Reduce 9: 12 to its lowest terms. SOLUTION.-Dividing both terms by 3 gives 9 : 12 = 3 : 4, i. e. 12 = §, or 12 = 2. (m.) Reduce each of the following to its lowest terms: 21: 16 7 9 12. 4.5 : .15. 14. 7.5: .05. The given ratio expresses of of 8 21 3 X 8 X 21 NOTE. By the above solution, it appears that, in reducing a compound ratio to a simple one, all its antecedents become factors of the new antecedent, and all its consequents factors of the new consequent. (0.) Reduce the following to simple ratios: (a.) A PROPORTION is an equality of ratios. (b.) Proportions may be either SIMPLE, COMPLEX, or COMPOUND. (c.) A SIMPLE PROPORTION is the equality of two simple ratios. (d.) A COMPLEX PROPORTION is the equality of two complex ratios, or of a complex and a simple ratio. (e.) A COMPOUND PROPORTION is the equality of two ratios, one of which is compound. (f.) A proportion is expressed by writing two ratios one after the other, and placing four dots between them. ILLUSTRATIONS.-4: 6:: 12: 18 is a proportion. It expresses that the ratio of 4 to 6 equals the ratio of 12 to 18, and would be read, "4 is to 6 as 12 is to 18." It means that 6 is the same part of 4 that 18 is of 12, or that 4 is the same part of 6 that 12 is of 18. 4: 7 The compound proportion 16 ::1:2 expresses that the ratio of 4 14: X 14 to 7 X 16 equals the ratio of 1 to 2. It would be read, "4 times 14 is to 7 times 16 as 1 is to 2," and means that 7 times 16 is the same part of 4 times 14, that 2 is of 1, or that 4 times 14 is the same part of 7 times 16 that 1 is of 2. (g.) The sign of equality is sometimes used instead of the four dots. ILLUSTRATION.-Instead of 4: 6 :: 2 : 3, we may have 4: 6 2:3. (h.) Every proportion may be expressed as the equality of two fractions. ILLUSTRATION. -We may express the first of the proportions under f by "= 1," and the second by "of 18. = 2." The outer terms (i. e. the first and fourth) of a proportion are called the EXTREMES; and the inner terms (i. e. the second and third) are called the MEANS of the proportion. ILLUSTRATIONS.-In the first proportion under f, 4 and 18 are the extremes, and 6 and 12 are the means; and, in the second, 4 × 14 and 2 are the extremes, and 7 X 16 and 1 are the means. (i.) If any term of a proportion is omitted, it may easily be supplied; for, from the nature of a proportion, it follows that— 1st. The missing antecedent of any proportion must be the same part of its consequent that the given antecedent is of its consequent. 2d. The missing consequent of any proportion must be the same part of its antecedent that the given consequent is of its antecedent.. 1. What is the missing term of 8: 3 :: 24: - ? SOLUTION. The missing term it is of 24. Hence, 8 : 3 :: 24 : is the same part of 24 that 3 is of 8, i. e. 24 X 3 8 = 9. 2. What is the missing term of 71: 31% ::- :8? SOLUTION. 8 X 14, i. e. it equals = 36. The missing term is the same part of 64 that 7 X 9 is of Find the missing term of each of the following: (j.) From the foregoing explanations and illustrations, we may 1st. Either extreme is equal to the quotient obtained by dividing the product of the means by the other extreme. 2d. Either mean is equal to the quotient obtained by dividing the product of the extremes by the other mean, (k.) Hence, in a proportion, the product of the means is equal to the product of the extremes. 105. Problems in Proportion. NOTE. These problems may be solved by analysis instead of proportion, if the teacher prefers it. (a.) The forming of a proportion from the conditions of a problem is called STATING it. (b.) In stating a proportion, we write the number which is of the same denomination as the answer, for the third term, and arrange the first and second terms so that they may show the ratio of the the third term to the required answer. 1. If 7 cows cost $171.43, how much will 3 cows cost? SOLUTION. As the answer is to be dollars, we write $171.43 for the third term, and as 7 cows cost this sum, 3 cows will cost the same part of this that 3 is of 7, expressed by making 3 the second term, and 7 the first. STATEMENT.-7: 3 :: $171.43 $ NOTE. The preceding process is the most logical, but, practically, it may be more convenient, after having found the third term, to consider that if the conditions of the problem are such as to require an answer greater than the third term, the second term must be greater than the first; but if they are such as to require an answer smaller than the third term, the second term will be smaller than the first. This is illustrated in the 2D SOLUTION.-Since the answer is to be dollars, we write $171.43 for the third term, and as 3 cows will cost less than 7 cows, we make 3 (the smaller number) the second term, and 7 (the larger number) the first, which gives the same statement as before. 2. If a ton of hay will keep 8 cows 2 weeks, how many weeks will it keep 5 cows? 1ST SOLUTION.-Since the answer is to be weeks, we write 2 as the third term, and as the hay will last 8 cows so many weeks, it will last 5 cows the same part of this that 8 is of 5.* Hence the statement, 5: 8 :: 2 : 2× 8 = 4. Ans. 2D SOLUTION.-We make 2 the third term, as before. Then, as the hay will keep 5 cows longer than it will keep 8, we make 8 (the larger number) the second term, and 5 (the smaller number) the first, which gives the same statement as before. 3. If 6 sleighs cost $150, how much will 8 cost? 4. If 9 horses eat 12 tons of hay in a winter, how many tons will 5 horses eat in the same time? 5. If it cost $3.50 to transport 17 cwt. 3 qr. 19 lb. a certain distance, how much will it cost to transport 8 cwt. 3 qr. 22 lb. the same distance? *For it will last 1 cow 8 times as long as it will last 8 cows, and will last 5 cows as long as it will last 1 cow. |