RATIO. 357. Ratio is the relation which one number bears te Lother number of the same kind. Ratios are of two kinds, Arithmetical and Geometrical. 358. Arithmetical Ratio is ratio of numbers with respect to their difference; as 6- - 4 = 2. GEOMETRICAL RATIO. 359. Geometrical Ratio is ratio of numbers with respect to their quotient; as 2: :4= , read 2 is to 4, or the ratio of 2 to 4: = ; 6:32, read 6 is to 3, or the ratio of 6 to 3 = 2. 360. The first term of a ratio is called the Antecedent, the second, the Consequent; both together are called a Couplet. What is the antecedent in the first illustration in Article 359? the consequent in the second? the ratio in the first? the consequent in the first? the ratio in the second ? 361. When the terms of a ratio are equal, the ratio is one of equality; when the antecedent is greater than the consequent, it is a ratio of greater inequality; when the antecedent is less than the consequent, it is a ratio of less inequality. 362. It will be readily seen that ratios, being expressions for division, are similar to fractions. They can at any time be written in a fractional form, the antecedent taking the place of the numerator, and the consequent that of the denominator. The principles applicable to fractions apply also to ratio. Hence, 363. Ratios, like fractions, may be simple, complex, or compound. A ratio is simple when each term is a simple number; it is complex when either term contains a fraction; it is compound when it is the indicated product of two or more ratios. 1. Write the ratio of 2 to 3; of 7 to 10; of to ; of 2 X 7 to 5 X 4. 2. Multiply the ratio 3: 4 by 2. 3. Divide the same by 2. 4. Reduce the ratio 6: 8 to lower terms. 5. Write any ratio of equality; of greater inequality; of less inequality. 365. ILL. Ex. Reduce 5 to a simple ratio. 21 OPERATION, 5 21 have 15. Multiplying each term of the ratio : 15 by 3 × 7, we 2 × 3 × 7 15 × 3 × 7 3 7 14:45, Ans. Hence, To reduce a complex ratio to a simple one: Reduce each term to its simplest form, then multiply each by the least common multiple of the denominators, and cancel. PROPORTION. 366, Proportion is an expression of equality between two ratios; thus, 2:3 = 4:6, read 2 is to three as 4 is to 6; that is, 2 is the same part of 3 that 4 is of 6. 2 is of 3, and 4 is 3 of 6. 367. The first and fourth terms of a proportion are called the extremes, and the second and third are called the means. The first ratio called the first couplet, and the second ratio the second couplet. Read the following proportions: Name the extremes of the first proportion; the means of the second; the antecedents of the third; the consequents of the fourth; the sec ond couplet of the first proportion. 368. Inverse Proportion. Four terms are directly proportional when the first is to the second as the third is to the fourth. They are inversely proportional when the first is to the second as the fourth is to the third, or when one ratio is direct and the other inverse. Thus, the amount of work done in any given time is directly proportional to the men employed; i. e., the more men, the more work; but the time occupied in doing a certain work is inversely proportional to the men employed; i. e., the more men, the less time. 369. A compound proportion is an equality between a compound ratio and a simple ratio, or between two compound ratios. 370. Three terms are in proportion when the first is to the second as the second is to the third. The second term is called a mean proportional between the other two; thus, in the proportion, 3:6 6:12, 6 is a mean proportional between 3 and 12. = 371. The performance of arithmetical examples by pro portion depends upon the following important principle: In every proportion the product of the means equals the product of the extremes. ILLUSTRATION. 2 × 3 × 6 3 4 × 3 × 6 2 X 6=4X3 Writing the given proportion in a fractional form, we have. Multiplying each fraction by the product of the denominators, and cancelling, we have 2 X 64 × 3. But 2 and 6 are the extremes, and 4 and 3 the means; hence the product of the extremes equals that of the means. 372. From the above, it follows that whenever an extreme in a proportion is wanting, it can be found by dividing the product of the means by the given extreme; and whenever a mean is wanting, it may be found by dividing the product of the extremes by the given mean. Supply the terms wanting in the following proportions: .. 373. In the proportion 2:4 = 4 : 8, 42 = 2 × 8, 4= √2 × 8 (Arts. 91, 92); hence a mean proportional between twa numbers equals the square root of their product. Supply the mean proportionals between the following numbers and write the proportions: 374. ILL. Ex., I. If 15 boxes of oranges cost $60, what will 17 boxes cost? OPERATION BY ANALYSIS. 4 $60 × 17 13 If 15 boxes cost $60, 1 box will cost of $60, and 17 boxes will cost 17 X of $68, Ans. $60. Cancelling and multiplying, the result is $68. OPERATION BY PROPORTION. 15:17 $60 : Ans. $60, the price of 15 boxes, must bear the same relation to the price of 17 boxes that 15 bears to 17. We have then three terms of a proportion (15:17= $60 :), and can find the fourth by multiplying the second and third together, and dividing the product by the first. Hence we derive the following 4 17 X $60 15 = $68, Ans. RULE FOR SIMPLE PROPORTION. Make the number that is of the same kind as the required answer the third term. If the answer should be greater than the third term, make the larger of the other two numbers, upon which the answer depenas, the second term, and the smaller the first. If it should be less, make the smaller number the second term, and the larger the first. Multiply the means together, and divide their product by the first extreme. NOTE. Analysis is the more natural and philosophical method of solving arithmetical questions; but the principles of Proportion are applicable to certain classes of examples. It is recommended to the pupil to perform the following examples in both ways. He should, at least, perform a sufficient number of them by Proportion to fix the method in his mind. EXAMPLES. 1. If 2 men build 17 rods of wall in a week, how many rods will 100 men build in the same time? We make 17 rods, which is of the same denomination as the required answer, the third term. As 100 men will build more wall than 2 men, we make 100 the second term, and 2 the first term, and the statement is, 2: 100 17: 850. Ans. 850 rods. 2. If 9 lbs. of lead make 150 bullets, how many bullets can be made from 105 lbs.? Ans. 1,750 bullets. 3. If 65 pairs of boots can be made from 75 lbs. of calfskin, how many pairs can be made from 850 lbs.? Ans. 7363 prs, 4. How many tons of hay can be made from 750 acres of land, if 13 tons can be made from 3 acres? 5. If $2000000 will support an army of how many men can be kept for $400000? 6. If $500 purchase 200 hats, how many chased for $87? Ans. 3250 tons. 500000 men a day, Ans. 100,000 men. hats can be pur Ans. 35 hats. 7. If $800 yield $56 of interest in a certain time, what will $390 yield at the same rate? 8. If 16 horses eat a certain quantity of hay in long would the same quantity last 24 horses? Ans. $27.30. 13 weeks, how Ans. 8 weeks 9. What time would be required for 5 men to mow an acre of Land, if 2 men can mow it in 14 days of 10 hours in length? Ans. 6 hours |