X. RATIO, PROPORTION, RULE OF THREE. RATIO is the mutual relation of two numbers to one another. By finding how many times one number is contained in another, or what part one number is of another, we obtain their ratio. Thus, the ratio of 2 to 4 is 2, because 2 is contained 2 times in 4; and the inverse ratio is 2, because 2 is 2 of 4. Both these expressions of the ratio of 2 to 4 amount to the same thing, which is, that one of the numbers is twice as great as the other. A ratio is denoted by two dots, similar to a colon: thus, 3:9 expresses the ratio of 3 to 9. The former term of a ratio is called the antecedent, and the latter the consequent. Thus, 6 12 expresses the ratio of 6 to 12, in which 6 is the antecedent, and 12 the consequent. Since a ratio indicates how many times one number is contained in another, or what part one number is of another, it is a quotient, resulting from the division of one of the terms of the ratio by the other, and may be expressed in the form of a fraction: thus, the ratio 6.3 may be expressed by the fraction, or conversely §. The equality of two ratios is called A PROPORTION; and the terms are called proportionals. Thus, 2:43:6 express a proportion, signifying, that the ratio of 2 to 4 is equal to the ratio of 3 to 6. In a proportion, the first and fourth terms, that is, the antecedent of the first ratio and the consequent of the second, are called the extreme terms; and the second and third terms, that is, the consequent of the first ratio and the antecedent of the second, are called the mean terms. Thus, in the proportion 3: 94: 12, 3 and 12 are the extreme terms, 9 and 4 the mean terms. It is to be observed that, if four numbers be in proportion, the product of the extreme terms is equal to the product of the mean terms. Since the product of the extremes in every proportion is equal to the product of the means, one product may be taken for the other. Now, if we divide the product of the extremes by one extreme, the quotient is the other extreme; therefore, if we divide the product of the means by one extreme, the quotient is the other extreme. To apply these principles to practice, let it be askedIf 64 yards of cloth cost 304 dollars, what will 36 yards cost? In the first place, the ratio of the two pieces of cloth is 64:36; and secondly, the prices are in the same ratio; that is, 304 dollars must have the same ratio to the price of 36 yards, that 64 yards have to 36 yards. Now, if we put A instead of the answer, we shall have the following proportion, 64:36=304: A. Here, the product of the means is 10944, which, divided by 64, one of the extremes, gives the quotient 171, the other extreme, which was the term sought, and the answer. Of the four numbers in a proportion, two are of one kind, and two of another. In the preceding example, two of the terms are yards, and two are dollars. From the principles of ratio and proportion, we deduce THE RULE OF THREE-an ancient rule, by the operation of which, having three numbers given, we find a fourth, which has the same ratio to the third that the second has to the first. RULE OF THREE. Make the number, which is of the same kind with the answer, the third term. And if, from the nature of the question, the fourth term or answer must be greater than the third term, make the greater of the two remaining terms the second term, and the smaller the first; but, if the fourth term must be less than the third, make the less of the two remaining terms the second term, and the greater the first. Multiply the second and third terms together, and divide the product by the first term: the quotient will be the fourth term, or answer. If there are different denominations in the first two terms, they must both be reduced to the lowest denomination in either of them; and the third term must be reduced to the lowest denomination mentioned in it. Operations corresponding to the Rule of Three have already been taught, in Relations of Numbers, Chap. VI. To show the correspondence, suppose it to be asked-If 3 yards of cloth cost 4 dollars, what will 9 yards cost? In Relations of Numbers, the question stands thusWhat is 9 times of 4? 3)4 In the Rule of Three, the question stands thus 3:94: what number? 3:94: A 9 9 12 Ans. 3)36 12 Ans. 1. If I buy 871 yards of cotton cloth for 78 dollars 39 cents, what is the price of 29 yards of the same? The statements of this question may be read thus -The ratio of 871 to 29 is equal to the ratio of 78.39 to the answer. Or 871)2273.31(2.61 Ans. thus-As 871 yd. is to 29 1742 5313 5226 871 871 yd., so is $78.39 to the answer. The operation amounts to nothing more than the multiplication of 78.39 by 871 29 2. If 13 yard of cotton cloth cost 42 cents, what will 87 yards cost, at the same price per yard? 1.75 87.5= .42: A 3. If I can buy 1 yard of cotton cloth for 64 pence, how many yards can I buy for £10 6s. 8d.? 6d. 1qr.: £10 6s. 8d.=1yd. 1qr.: A 4. If I buy 54 barrels of flour for 297 dollars, what must I give for 73 barrels, at the same rate? 5. If 7 workmen can do a piece of work in 12 days, how many can do the same work in 3 days? 6. If 20 horses eat 70 bushels of oats in 3 weeks, how many bushels will 6 horses eat in the same time? 7. If a piece of cloth containing 76 yards cost 136 dolars 80 cents, what is that per ell English? 8. If a staff 4 feet long cast a shadow 7 feet in length, on level ground, what is the height of a steeple, whose shadow at the same time measures 198 feet ? 9. How many yards of paper, 2 feet wide, will hang a room, that is 20 yards in circuit, and 9 feet high? 10. A certain work having been accomplished in 12 days, by working 4 hours a day, in what time might it have been done by working 6 hours a day? 11. If 12 gallons of wine are worth 30 dollars, what is the value of a cask of wine, containing 31 gallons? 12. If 83 yards of cloth cost 4 dollars 20 cents, what will 13 yards cost, at the same rate? 13. How many yards of cloth yard wide, are equal to 30 yards 14 yard wide? 14. If 7 pounds of sugar cost 75 cents, how many pounds can I buy for 6 dollars? 15. If 2 pounds of sugar cost 25 cents, and 8 pounds of sugar are worth 5 pounds of coffee, what wil 100 pounds of coffee cost? 16. A merchant owning of a vessel, sold of his share, (,) for 957 dollars. What was the vessel worth, at that rate? 17. A merchant failing in trade, owes 62936 doliars 39 cents; but his property amounts to only 38793 dollars 96 cents, which his creditors agreed to accept, and discharge him. How much does the creditor receive, to whom he owes 2778 dollars 63 cents? 18. Bought 3 tons of oil, for 503 dollars 25 cents; 85 gallons of which having leaked out. I wish to know at what price per gallon I must sell the residue, that I may neither gain nor lose by the bargain. 19. If, when the price of wheat is 6s. 3d. a bushel, the penny loaf weighs 9 oz., what ought it to weigh, when wheat is at 8s. 21 d. a bushel? 20. If 15 yards of cloth yard wide cost 6 dollars 25 cents, what will 40 yards, being yard wide, cost? 21. Borrowed of a friend 250 dollars for 7 months; and then, to repay him for his kindness, I loaned him 300 dollars. How long must he keep the 300 dollars, to balance the previous favor? 22. If 4 cwt. be carried 36 miles for $51, how many pounds can be sent 20 miles for the same money ? 23. A person owning of a coal mine, sells of his share for 570 dollars. What is the whole mine worth? 24. If the discount on $106, for year, be $6, what is the discount on $477, for the same time? XI. MEASUREMENT OF SURFACES, SOLIDS AND CAPACITIES. It has already been taught, that surfaces are measured in squares, and, that solid bodies are measured in cubes. A SQUARE is a figure, that has four equal sides, and four equal angles. Its angles are called right angles: angles more pointed are called acute angles; and those less pointed, obtuse angles. To find the area of a square, in smaller squares-Multiply one side into itself. 1. How many square feet are there in a table that measures 4 feet on every side? How many square inches? A PARALLELOGRAM is a four-sided figure, having opposite sides equal, and having four right angles. To find the area of a parallelogram-Multiply the length into the breadth. 2. How many square rods in a garden measuring 4 rods in length, and 3 in breadth? How many square feet? A TRIANGLE is a figure, that has three sides and three angles. A triangle, which has one right angle, is called a RIGHT-ANGLED TRIANGLE. To find the b area of a right-angled triangle-Multiply the base by half the perpendicular. 3. How many square rods are there in a right-angled triangular field, measuring 98 rods on the base, and 75 rods on the perpendicular? How many acres? |