CHAPTER X RATIO AND PROPORTION RATIO 172. The ratio of two numbers is the quotient obtained by dividing the first number by the second. Thus the ratio of a and b is for a b. The ratio is also frequently b written ab, the symbol: being a sign of division. (In most European countries this symbol is employed as the usual sign of division.) The ratio of 12: 3 equals 4, 6: 12 = .5, etc. 173. A ratio is used to compare the magnitude of two numbers. 66 Thus, instead of writing is 5 times as large as b," we may write a: b = 5. 174. The first term of a ratio is the antecedent, the second term the consequent. In the ratio a: b, a is the antecedent, b is the consequent. The numerator of any fraction is the antecedent, the denominator the consequent. 176. Since a ratio is a fraction, all principles relating to fractions may be applied to ratios. E.g. a ratio is not changed if its terms are multiplied or divided by the same number, etc. Ex. 1. Simplify the ratio 21: 31. 21:31:10={× == 3:4. 1% A somewhat shorter way would be to multiply each term by 6. 27. 5 abc: 4 x = 15 bc. 28. 16: (x-3)=4. 29. 144: (2x+1) = 16. 30. b:xa — c. 31. 150 (3x+1)= 11. : 32. If 150 ounces of gold cost $3060, and 20 ounces of silver $12, find the ratio of the value of gold to the value of silver. PROPORTION 177. A proportion is a statement expressing the equality of two ratios. ora: bc: d are proportions. 178. The first and fourth terms of a proportion are the extremes, the second and third terms are the means. The last term is the fourth proportional to the first three. In the proportion a: b = c:d, a and d are the extremes, b and c the The last term d is the fourth proportional to a, b, and c. means. 179. If the means of a proportion are equal, either mean is the mean proportional between the first and the last terms, and the last term the third proportional to the first and second terms. In the proportion a:bb: c, b is the mean proportional between a and c, and c is the third proportional to a and b. 180. Quantities of one kind are said to be directly proportional to quantities of another kind, if the ratio of any two of the first kind is equal to the ratio of the corresponding two of the other kind. If 4 ccm. of iron weigh 30 grams, then 6 ccm. of iron weigh 45 grams, or 4 ccm. : 6 ccm. = 30 grams: 45 grams. Hence the weight of a mass of iron is proportional to its volume. NOTE. Instead of "directly proportional" we may say briefly "proportional." Quantities of one kind are said to be inversely proportional to quantities of another kind, if the ratio of any two of the first kind is equal to the inverse ratio of the corresponding two of the other kind. men can do it in If 6 men can do a piece of work in 4 days, then 8 3 days, or 6: 8 equals the inverse ratio of 4: 3, i.e. 3: 4. Hence the number of men required to do some work, and the time necessary to do it, are inversely proportional. 181. In any proportion the product of the means is equal to the product of the extremes. 182. The mean proportional between two numbers is equal to the square root of their product. Let the proportion be a: b=b: c. Then Hence b2ac. (§ 181.) 183. If the product of two numbers is equal to the product of two other numbers, either pair may be made the means, and the other pair the extremes, of a proportion. (Converse of § 181.) If mn = pq, and we divide both members by nq, we have m P. Ex. 1. Find x, if 6: x=12:7.* Hence 12 x 42. (§ 181.) x= 12 = 31. Ex. 2. Determine whether the following proportion is correct 8 × 4 = 35, and 5 × 7 = 35; hence the proportion is correct. I. b: a=d: c. (Frequently called Inversion.) Or IV. a-b:bc-d: d. (Division.) V. a+b: a-b=c+d: c-d. (Composition and Division.) Any of these transformations may be proved by the method of § 169, although in many cases shorter proofs exist. 185. These transformations are used to simplify proportions. I. Change the proportion 4: 5=x: 6 so that x becomes the last term. By inversion 5:46: x. II. Alternation shows that a proportion is not altered when its antecedents or its consequents are multiplied or divided by the same number. = 327x, divide the antecedents by 16, the 3:3=2:x. E.g. to simplify 48:21 consequents by 7, III. To simplify the proportion 5: 6=4−x: x. Apply composition, 11:6 4: x. = IV. To simplify the proportion 8:3=5+x: xX. |