RATIO. 314. RATIO is the relation of one quantity to another of the same kind; or, it is the quotient which arises from dividing one quantity by another of the same kind. 315. Ratio is usually indicated by two dots; thus, 8:4 expresses the ratio of 8 to 4. The two quantities compared are the terms of the ratio; the first term being the antecedent, the second the consequent, and the two terms, collectively, a couplet. 316. Most mathematicians consider the antecedent a dividend, and the consequent a divisor; thus, 8:48÷ 4 = 4 = 2, and 3: 123 ÷ 12 == 4; but others take the antecedent for the divisor, and the consequent for the dividend; NOTE 1. The first method is often called the English method, and the other the French; but there appears to be no good reason for such a distinction. NOTE 2. The first is a direct ratio; the second is an inverse or reciprocal ratio. The first being considered the more simple and natural, is adopted in this work. 317. The antecedent and consequent being a dividend and divisor, it follows that any change in the ANTECEDENT causes a LIKE change in the value of the ratio, and any change in the CONSEQUENT causes an OPPOSITE change in the value of the ratio (Art. 84, 85, and 131). Hence, 1st. Multiplying the antecedent multiplies the ratio; and dividing the antecedent divides the ratio (Art. 83, a and b). 314. What is Ratio? 315. How indicated? What are the terms? The 1st? The 2d? The two collectively? 316. Which term is divisor? Is the custom uniform? Which method is here taken? Why? What is a direct ratio? An inverse ratio? 317. Explain and illustrate Art. 317 fully. 2d. Multiplying the consequent divides the ratio; and dividing the consequent multiplies the ratio (Art. 83, c and d). 3d. Multiplying both antecedent and consequent by the same number, or dividing both by the same number, does not affect the ratio (Art. 84, a and b). 318. The antecedent, consequent, and ratio are so related to each other, that, if either two of them be given, the other may be found; thus, in 12: 3 — 4, we have antecedent antecedent ratio consequent ratio, consequent ratio = antecedent. 319. When there is but one antecedent and one consequent the ratio is said to be simple; thus, 15:5 = 3, is a simple ratio. 320. When the corresponding terms of two or more simple ratios are multiplied together the resulting ratio is said to be compound; thus, by multiplying together the corresponding terms of the simple ratios, pound ratio, 48:4-12 or 480: 124 0. we have the com A compound ratio is always equal to the product of the simple ratios of which it is compounded. NOTE. A compound ratio is not different in its nature from a simple ratio, but it is called compound merely to denote its origin. Ex. 1. What is the ratio of 20 to 4? 2. What is the ratio of 2 to 9? Ans. 20: 4 5. Ans. 2: 9 3. What is the inverse ratio of 20 to 4? 4. What is the inverse ratio of 2 to 9? 5. What is the ratio compounded of 8 to 6 and 9 to 2? 6. Which is the greater, the ratio of 9 to 7 or of 19 to 14? 7. Which is the greater, the ratio of 5 to 4 or of 15 to 13? 818. What of antecedent, consequent, and ratio? 319. What is simple ratio! 320. Compound ratio? Its value? Its nature? Why called compound? PROPORTION. 321. PROPORTION is an equality of ratios. Two ratios, and.. 4 terms, are required to form a proportion. 322. Proportion is indicated by means of dots; thus, 8:46:3, which is read, 8 is to 4 as 6 is to 3; or, as 8 is to 4 so is 6 to 3; or it may be indicated thus, which is read, the ratio of 8 to 4 equals the ratio of 6 to 3. Any 4 numbers are in proportion, and may be written and read in like manner, if the quotient of the 1st divided by the 2d is equal to the quotient of the 3d divided by the 4th. 323. The 1st and 4th terms are called extremes, and the 2d and 3d, means. The 1st and 3d are the antecedents of the two ratios, and the 2d and 4th are the consequents. The product of the extremes is always equal to the product of the means; thus, in the proportion 8: 4 :: 6 : 3, we have 8 × 3 = 4 × 6. 324. Since the product of the extremes is equal to the product of the means, any one term may be found when the other three are given; for the product of the extremes divided by either mean will give the other mean, and the product of the means divided by either extreme will give the other extreme. Fill the blank in each of the following proportions: 321. What is Proportion? 322. How indicated? Proportion, how read? When are four numbers in proportion? 323. What are the 1st and 4th terms called? 2d and 3d? 1st and 3d? 2d and 4th? The product of the extremes equals what? 324. How many terms must be given? How can the other be found? 325. It follows from Art. 317, that if the 1st and 2d, or 3d and 4th, or 1st and 3d, or 2d and 4th, or all four terms of a proportion are multiplied or divided by the same number, the resulting numbers will be in proportion. 326. If 4 numbers are proportional they will be in proportion in 8 different orders; thus, (5) Inverting (1) and transposing couplets 3: 6:4:8 NOTE. These 4 numbers may be written in 16 other orders, but none of rem will be in proportion. 327. When the means of a proportion are alike, the term repeated is a mean proportional between the other two, and the last term is a third proportional to the 1st and 2d; thus, in 4:6: 69, 6 is a mean proportional between 4 and 9, and 9 is a third proportional to 4 and 6. 328. A mean proportional between two numbers may be found by multiplying the two given numbers together, and then resolving the product into two equal factors; thus, the mean proportional between 2 and 8 is 4, for 2 × 8 = 16 = 4 X 4; .. 2:44:8. 329. A third proportional to two numbers may be found by dividing the square of the 2d by the 1st. The third proportional to 5 and 10 is 20; for 102 ÷ 5 = 20; .. 5 : 10 :: 10:20. 330. In all examples in Simple Proportion there are three 325. What terms may be multiplied without destroying the proportion? What divided? 326. In how many orders may four proportional numbers be in proportion? In how many not in proportion? 327. What is a mean proportional? A third proportional? 328. How is a mean proportional found? 329. A third proportional? numbers given to find a fourth; .. Proportion is often called the Rule of Three. Two of the three given numbers must be of the same kind, and the other is of the same kind as the answer. Ex. 1. If 3 men build 6 rods of wall in a day, how many rods will 5 men build? This example may be analyzed as follows: If 3 men build 6 rods, 1 man will build of 6 rods, i. e. 2 rods; and if one man build 2 rods, 5 men will build 5 times 2 rods, i. e. 10 rods, Ans. ; but to solve it by proportion, we say, that 3 men ́ have to 5 men the same ratio that the given number of rods has to the required number of rods; thus, 3 men: 5 men :: 6 rods: required number of rods. Now, since the means and 1st extreme are given, we find the 2d extreme by dividing the product of the means by the given extreme (Art. 324); thus, 6 × 530 and 30 ÷ 3 = 10, Ans. as before. Hence, 331. To solve an example in Simple Proportion, RULE. Write that given number which is of the same kind as the required answer for the third term; consider whether the nature of the question requires the answer to be greater or less than the third term; if greater, write the greater of the two remaining numbers for the second term and the less for the first; but if less, write the less for the second and the greater for the first; in either case, divide the product of the second and third terms by the first, and the quotient will be the term sought. NOTE 1. If the first and second terms are in different denominations, they should be reduced to the same before stating the question. REMARK. Every one who intelligently solves an example by proportion, does, in effect, solve it by analysis; but the teacher should use much care on this point, since the scholar learns much faster when he analyzes a question than when he merely follows 330. Of what kind must two of the three given numbers be? What the other? 331. Rule for solving an example in proportion? Note 1? Remark? |