MISCELLANEOUS WORK 1. At 98 cents a bushel find the value of a load of corn weighing 2980 pounds. (One bushel weighs 70 pounds.) 2. An American merchant bought goods in France for 280,400 francs. What did the goods cost him in U. S. money? (1 franc = $0.193.) 3. Following is the record of 10 loads of wheat: At $1.48 a bushel, find the value of this wheat. 4. At $18.40 a ton what is the value of a load of hay whose net weight is 3190? 5. A can do a piece of work in 7 days, B can do it in 8 days, and C in 9 days. What fraction of the work can they do in one day when all are working together? 6. In one lumber camp 540 bushels of oats are fed to 45 horses in 30 days. At this rate how many bushels should be fed to 117 horses in 115 days? Find the total in each of the following: CHAPTER XVII RATIO AND PROPORTION 237. Uses of Ratio and Proportion. — While ratio and proportion are not of such constant use in business as are the fundamental operations, denominate numbers, or percentage, it is useful in many of the problems of commerce. The problems on page 195 are instances in point. No one can have a complete knowledge of the arithmetic of business who does not understand the elements of ratio and proportion. 238. Ratio. The relative magnitudes of two numbers may be expressed by saying that one is a certain number of times as great as the other, or a certain fraction of the other. Thus, 8 is twice as great as 4, 12 is 3 times as great as 4. Again 8 is of 24. 7 is as great as 42. In each of these the result is obtained by dividing the first number by the second. The ratio of one number to another is the indicated quotient in which the first number is the dividend, and the second the divisor. Since any division may be expressed as a fraction. be indicated as a fraction, every ratio may Thus, the ratio of 8 to 4 is 8 ÷ 4 = , and the ratio of 7 to 21 is 7 ÷21 = 2. A fraction expressing a ratio may be reduced to its simplest form. Hence every ratio may be expressed as an integer or as a fraction reduced to its lowest terms. If, however, a ratio is expressed by an improper fraction not equal to an integer, it should not be reduced to a mixed number. Thus we speak of a ratio and not 13. 239. Antecedents and Consequents. In speaking of the ratio of two numbers, it is important to note the order in which they are given. Thus, the ratio of 2 to 8 is = , while the ratio of 8 to 2 is = 4. The first number of a ratio is called the antecedent, and the second number the consequent. The special sign of ratio is the colon. Thus, the ratio 2 to 8 is written 28. Hence we have: Antecedent: consequent = antecedent numerator = consequent denominator In some respects, the ratio of two numbers is their most important relation. If we know that two numbers differ by 10, this may mean much or little, according to the magnitude of the numbers. Thus, if two streets differ in length by 10 miles, we think of one street as much longer than the other, while two railways which differ by 10 miles may be regarded as of the same length for some practical purposes. On the other hand, if two distances are known to be in the ratio we know a significant relation between them whether they are large or small. 240. Numbers of the Same Kind Compared. - Since only magnitudes of the same kind can be compared, it follows that if either the antecedent or the consequent is a concrete number, the other must be a concrete number of the same kind. Hence a ratio is always abstract. Thus, we speak of the ratio of two distances both expressed in miles, or both in yards, or both in feet, or both in inches. We speak of the ratio of the areas of two fields both expressed in acres, of two quantities of grain, both expressed in bushels, etc. In practice, however, it is best to regard all numbers as abstract for the purpose of computation. See page 193. To find the ratio of one number to another, divide the first by the second. ORAL EXERCISES Express each of the following ratios in its simplest form : EXERCISES Express each of the following ratios in its simplest form: 241. Numbers of Different Kinds Compared. It is frequently convenient to consider the ratio of two numbers which would not be of the same kind if they were considered as concrete numbers. This offers no difficulty, however, since the numbers may always be considered as abstract. Thus, the average yield per acre of a wheat field may be regarded as the ratio of the number of bushels to the number of acres. Again, the per capita expenditure of the government of a certain city may be regarded as the ratio of the number of dollars expended to the number of people. 242. Proportion. The equality of two ratios constitutes a proportion. Thus, the ratios 3:7 and 6: 14 form a proportion if written 3:7 = 6:14, or = 4. = 243. Means and Extremes. — In a proportion such as 3:7 6:14 the first and the last numbers are called the extremes, and the two remaining numbers are the means. When three of the four numbers in a proportion are given, the fourth number may always be found. It is this fact which makes proportion useful in solving problems. In the following example, the numbers to be found are represented by x. 3 Solution. If one seventh of x, or x then x = 7 X ORAL EXERCISES Find the value of x in each of the following: a problem by proportion, the proportion may always be written so that the unknown number is the antecedent of the first ratio. The finding of the unknown number is then a problem like those just solved. Sometimes, however, a proportion may be given, in which the missing number is not the antecedent of the first ratio. 3 = Thus, we may be required to find the value of x in such proportions as 4 4 x 5 8 We will now study a method for finding the value of x = = x 5' 5 3' 7 x in any proportion. The proportion = fo or 2:5 = = 4:10 may be written 2:5 2:5 by reducing the second ratio to simplest form. We then notice that the means and the extremes are the same. This is true of any proportion in which the ratios are reduced to simplest form. Hence, In a proportion the product of the extremes equals the product of the means. This statement holds for any proportion whatever, whether the ratios are reduced to simplest form or not. 1. ORAL EXERCISES Verify the above rule for each of the following proportions: |