CHAPTER V. PROPORTION AND AVERAGES. RATIO. 374. Ratio is the relation of one number to another of the same kind, expressed by their quotient. (1) The ratio of 6 ft. to 2 ft. is 6 ft. 2 ft., or 3. 375. The sign of ratio is the colon (:), which is equivalent to. (1) 8 4 is read, the ratio of 8 to 4, and is equivalent to 84, or 2. 12: 3 = 4 is read, the ratio of 12 to 3 is equal to 4, or 12 contains 3, 4 times. The terms are the two numbers compared; the antecedent, the first term or dividend; the consequent, the second term or divisor; and a couplet, both terms together. 376. A simple ratio is the ratio of two numbers; and a compound ratio is the product of two or more simple ratios. The compound ratio obtained by multiplying the two simple ratios, 23 and 5 7, is usually written thus, }. Hence, The reciprocal of a ratio, or inverse ratio, is the result of interchanging the places of its terms. (1) The reciprocal or inverse of 2:3 is 3:2, or 3. 377. Since the antecedent corresponds to the numerator, and the consequent to the denominator, changes on the terms of a ratio have the same effect upon its value as like changes have upon the terms of a fraction. Hence, the PRINCIPLES: 1. Multiplying the antecedent, or Dividing the consequent. 2. Dividing the antecedent, or Multiplying the consequent. 3. Multiplying or dividing both terms by the same quantity. } Multiplies the ratio. Divides the ratio. Does not alter the value of the ratio. 378. The ratio, antecedent, and consequent are so related to each other, that if any two of them are given the other may be found. Thus: When the terms are fractions they may be changed into integers by multiplying both by the L. C. M. of their denominators, which does not change the ratio. 25. How much more is 6 2 than 7: 3? 26. How much more is 7: 8 than 6 7? 39. How much less is 3: 5 than (3 + 1) : (5 + 1)? 40. How much more is 7: 5 than (7+ 3): (5 + 3)? 41. If 5 lb. of tea are worth as much as 11 lb. of coffee, and 7 lb. of coffee as much as 15 lb. of sugar, find the ratio of the value of tea to that of sugar. PROPORTION. 379. Proportion is an equality of ratios. (1) 8:46:3, and 12:2: = 30:5, are proportions. 380. The sign of proportion is the double colon (::), which is equivalent to =. (1) 8:4: 6:3 is read, 8 is to 4 as 6 is to 3, and is equivalent to 8:4 6:3, which is read, the ratio of 8 to 4 is equal to the ratio of 6 to 3, or 8 contains 4 as many times as 6 contains 3. = The terms or proportionals are the four numbers compared; the antecedents, the first and third terms; the consequents, the second and fourth terms; the extremes, the first and fourth terms; and the means, the second and third terms. (1) In 5:10 :: 3:6, 5, 10, 3 and 6 are the terms or proportionals; 5 and 3 the antecedents; 10 and 6 the consequents; 5 and 6 the extremes; and 10 and 3 the means. 381. When the second term is equal to the third it is said to be a mean proportional between the extremes. (1) In 4:6: 6:9, 6 is a mean proportional between 4 and 9. 382.-PRINCIPLE. In any proportion, the product of the extremes is equal to the product of the means. For, take any proportion, as. Multiplying both sides by 2× 6.. 3: 29: 6, }=}. 3X6=2X9. Hence, the relation of the four terms of a proportion to each other is such, that if any three of them are given, the other or missing term may be found thus: To find either extreme, Divide the product of the means by the other extreme. To find either mean, Divide the product of the extremes by the other mean. Proportion may be simple or compound. SIMPLE PROPORTION. 383. A simple proportion is an equality of two simple ratios. 384. To find the missing term of a simple proportion. (1) Find the value of x in 6 : 14 :: 9 : x. (2) Find the value of x in :x :: 55: 491. Explanation. Since one of the means is missing I divide the product of the extremes by the given mean, and obtain 3. Operation. x= = 55 3, Ans. Proportion is applied to the solution of problems which involve quantities that increase or decrease in the same ratio. (17) If 28 yd. of cloth cost $3.50, how much will 46 yd. cost? Statement: 28 yd. : 46 yd. :: $3.50: $x. The cost of the cloth increases in the same ratio with the number of yards; hence, the ratio of 28 yd. to 46 yd. is equal to the ratio of $3.50, the cost of 28 yd., to $x, which represents the cost of 46 yd. RULE.-I. Make that term which has the same unit as the answer the third term. II. If the question requires the answer to be greater than the third term, make the greater of the two remaining numbers the second term; but if the answer ought to be less than the third term, make the least of the two numbers the second term. III. Divide the product of the second and third terms by the first, and the quotient will be the fourth term. 18. If 3 yd. of cloth cost $4.50, what will 5 yd. cost? 19. If 160 caps cost $450, what will 840 caps cost? 20. If 4 hats cost $12, how many hats can be bought for $27. 21. How many yards of linen may be bought for $28.50, when 6 yards cost $4.50? |