1st. To multiply the consequent, or divide the antecedent, multiplies the ratio. Arts. 131 and 133. Thus, The ratio of 4 to 12 is 3; of 4 to 12X5, is 3x5; and The ratio of 4-2 to 12, is 6, which is equal to 3×2. 2d. To divide the consequent or multiply the antecedent, divides the ratio. Arts. 132 and 134. Thus, The ratio of 3 to 24 is 8; of 3 to 24÷2, is 4,8÷÷2; and The ratio of 3x2 to 24, is 4, which is equal to 8÷÷2. 3d. To multiply or divide both consequent and antecedent by the same number, does not alter the ratio. Arts. 134, 135. Thus, The ratio of 6 to 18, is 3; of 6×2 to 18×2, is 3; and The ratio of 62 to 18÷2, is 3. ART. 195. A single ratio, as 2 to 6, is a simple ratio. A compound ratio is the product of two or more simple ratios. Thus, The ratio 10 multiplied by the ratio §, is X-18=4. In this case, 3 multiplied by 5, is said to have to 10×6, the ratio compounded of the ratios of 3 to 10 and 5 to 6. ART. 196. Ratios may be compared with each other, by reducing to a common denominator the fractions by which they are expressed: thus, To find the greater of the two ratios, 2 to 5, and 3 to 8, we have and, which, reduced to a common denominator, are 15 and 16; and, as 15 is less than 16, the ratio of 2 to 5, is less than the ratio of 3 to 8. XIV. PROPORTION. ART. 197. Proportion is an equality of ratios. Four numbers are proportional, when the first has the same ratio to the second that the third has to the fourth. REVIEW.-194. Ilow is a ratio affected by multiplying the consequent, or dividing the antecedent? By dividing the consequent, or multiplying the antecedent? By multiplying or dividing both consequent and antecedent by the same number? Why? Illustrate each. Thus, the two ratios, 2: 4 and 3: 6, form a proportion, since, each being equal to 2. ART. 198. PROPORTION IS WRITTEN IN TWO WAYS: 2d. By placing the sign of equality between them. The first is read, 2 is to 4 as 3 is to 6; or, 2 has the same ratio to 4, that 3 has to 6. The second is read, the ratio of 2 to 4 equals the ratio of 3 to 6. REM.-1. The least number of terms that can form a proportion is four, since there are two ratios each containing two terms. But, one of the terms in each ratio, may be the same; thus, 2:4::4:8. The number repeated is called a MEAN proportional between the other two terms. 2. The terms ratio and proportion are often confounded with each other. Two quantities having the same ratio as 3 to 4, are improperly said to be in the proportion of 3 to 4. A ratio subsists between two quantities; a proportion only between four. ART. 199. The first and last terms of a proportion are called the extremes; the second and third, the means. Thus, in the proportion 2 : 3 :: 4 : 6, 2 and 6 are the extremes, and 3 and 4 the means. ART. 200. In every proportion, the product of the means is equal to the product of the extremes. ILLUSTRATIONS.-If we have 3: 4 :: 6 : 8, the ratios of each couplet being equal (Art. 206), we must have. Reducing these fractions to a common denominator (Art. 155), gives 8 4×6 3 X 8 18 18 REVIEW.-195. What is a simple ratio? A compound ratio? Give examples of each. 196. How compare ratios with each other? 197. What is proportion? When are four numbers proportional? Give examples. 198. How is proportion written? How is the first read? The second? REM. 1. What the least number of terms that can form a proportion? What of the terms ratio and proportion? The denominators being the same, and the values of the fractions being equal, the numerators must be equal; that is, 4×6, the product of the means, must=3×8, the product of the extremes. REM. The preceding shows that four numbers are not in proportion, when the product of the extremes is unequal to the product of the means. Thus, 2, 3, 5, and 8, are not in proportion, for 3X5 is not equal to 2×8. Proportions have numerous properties, the full discussion of which belongs to Algebra. See "Ray's Algebra, First Book." PROPORTION is divided into Simple and Compound. ART. 201. SIMPLE PROPORTION. Simple Proportion contains only simple ratios, Art. 195. It is sometimes called the RULE OF THREE, as three terms are given to find a fourth. REM. Some authors divide Proportion into direct and inverse; a distinction of no utility, and always embarrassing to the learner. ART. 202. Since the product of the means equals the product of the extremes, and the product of two factors divided by either of them, gives the other, Art. 37, Therefore, If the product of the means be divided by one of the extremes, the quotient will be the other extreme. Or, If the product of the extremes be divided by one of the means, the quotient will be the other mean. Thus, in the proportion 2 : 3 :: 4 : 6, 3X426, one of the extremes. REVIEW.-199. What are the first and last terms called? The second and third? 200. If four numbers are proportional, to what is the product of the means equal? REM. When are four numbers not in proportion? How is proportion divided? 201. What does Simple Proportion contain? What is it called? Why? 202. When 3 terms of a proportion are given, how find the fourth? Hence, If any three terms of a proportion are given, the fourth may be found by multiplying together the terms of the same name, and dividing their product by the other given term. 1. The first three terms of a proportion are 2, 8, and 6: what is the fourth term? SOLUTION. The preceding shows that the 4th term will be found by taking the product of the 2d and 3d terms, and dividing by the 1st. Or, by the nature of proportion, Art. 197, the ratio of the 3d term to the 4th the ratio of the 1st to the 2d. OPERATION. 8 X 6 2 2:8:: 6 = 24. Ans. Or, 84, and 6 X 4 = 24. Hence, If the third term be multiplied by the ratio of the first to the second, the product will be the fourth term. EXAMPLES TO BE SOLVED BY EITHER METHOD. Ans. 14. 2. The first three terms of a proportion are 5, 7, and 10: what is the fourth term? 3. The last three terms are 8, 6, and 16: what is the first? Ans. 3. 4. The first, third, and fourth terms, are 5, 6, and 12: what is the second? Ans. 10. 5. The first, second, and fourth terms, are 3, 7, and 14: what is the third? Ans. 6. Ans. 18. 6. Seven is to 14, as 9 is to what number? ART. 203. 1. If 2 lb. of tea cost $4, at the same rate, what will be the cost of 6 lb.? SOLUTION. Two pounds have the same ratio to 6 lb., that the cost of 2 lb. ($4), has to the cost of 6 lb. Therefore, the first three terms are given, to find the fourth (Art. 202). To find the result, multiply the 3d term by the 2d, and divide by the 1st; or, multiply the 3d term by the ratio (3) of the first to the 2d. OPERATION. lb. lb. $ 2: 6:4 2)24 Ans. $12. In stating this question, (arranging the terms,) we REVIEW.-202. If the third term of a proportion be multiplied by the ratio of the first to the second, what will be the product? may, with equal propriety, place the $4 as the first term. Thus, $4 2d term :: 21b. : 6 lb. The operation of finding the 2d term in this arrangement, is like that of finding the 4th in the preceding. It is more convenient to arrange the terms so that the required term shall be the fourth; then, Since a ratio can subsist only between quantities of the same denomination, Art. 191, Illus. 1, The third term of a proportion must be of the same denomi nation as that in which the answer is required. 2. If 3 men can dig a cellar in 10 days. in how many days will 5 men dig it? SOLUTION. Since the answer is to be days, write 10 days for the 3D term. The result will be found by multiplying the 3D term by the 2D, and dividing by the 1st (Art. 201); and, Since it will require 5 men a less number of days than 3 men, place the less number of men for the 2D term, and the greater for the 1ST. OPERATION. m. m. da. 5: 3 :: 10 3 5)30 Ans. 6. 3. If 3 men can dig a cellar in 10 days, how many men will dig it in 6 days? SOLUTION. Since the answer is to be men, write 3 men for the third term. Since it will require a greater number of men in 6 days than in 10 days, place the greater number of days for the second term, and the less for the first. *4. If 3 yd. cloth cost $8, what cost 6 yd.? *5. If 5 bl. flour cost $30, what cost 3 bl.? STATEMENT. da. da. m. 6:10:3 Ans. 5 men. Ans. $16. Ans. $18. GENERAL RULE FOR SIMPLE PROPORTION. 1. Write for the third term that number which is of the denomination required in the answer. 2. If, from the nature of the question, the answer should be greater than the third term, place the greater of the other two numbers for the second term, and the less for the first; but, |