Examples. 1. Multiply 9 ft. 4 in. by 8 ft. 3 in. 2. How many cords and cord feet in a pile of wood 24 feet long, 4 feet wide, and 3 feet 6 inches high? 3. Multiply 9 ft. 2 in. by 9 ft. 6 in. 4. How many square feet are there in a board 17 feet 6 inches in length, and 1 foot 7 inches in width? 5. Multiply 24 feet 10 inches by 6 feet 8 inches. 6. What is the number of cubic feet in a granite pillar 3 feet 9 inches in width, 2 feet 3 inches in thickness, and 12 feet 6 inches in length? 7. Multiply 70 feet 9 inches by 12 feet 3 inches. 8. There is a certain pile of wood, measuring 24 feet in length, 16 feet 9 inches high, and 12 feet 6 inches in width. How many cords are there in the pile? 9. How many square yards in the walls of a room, 14 feet 8 inches long, 11 feet 6 inches wide, and 7 feet 11 inches high? 10. If a load of wood be 8 feet long, 3 feet 9 inches wide, and 6 feet 6 inches high, how much does it contain? 11. How many cubic yards of earth were dug from a cellar which measured 42 feet 10 inches long, 12 feet 6 inches wide, and 8 feet deep? 12. What will it cost to plaster a room 20 feet 6′ long, 16 feet wide, 9 feet 6' high, at 18 cents per square yard; and making allowance for a door that is 6 feet 6 in. long, by 3 feet 3' wide? 13. How many feet of boards, 1 inch thick, can be cut from a plank 18 feet 9 in. long, 1 foot 6 in. wide and 3 in. thick, if there is no waste in sawing? 14. What will be the cost of building a stone wall, 45 ft. 6 in. long, 1 ft. 6' thick, and 37 ft. 9' high, at $3.91 per cubic yard? 15. How many loads of earth must be taken out in digging a cellar that is to be 45 ft. 6 in. long, 25 ft. wide, and 10 ft. 9 in. deep, allowing 1 cubic yard of earth to 2 loads? RATIO AND PROPORTION. 229. A RATIO is the quotient obtained by dividing one number by another. 230. The TERMS of a ratio are the divisor and dividend: hence, every ratio has two terms. 231. The divisor is called the ANTECEDENT. 232. The dividend is called the CONSEQUENT. 233. The ratio of one number to another is expressed in two ways: 1st. By a colon; thus, 3 : 12; and is read, 3 is to 12, or 12 divided by 3; 2d. In a fractional form; as,, 12 or 12 divided by 3. 234. The terms of a ratio, taken together, are called a COUPLET. 235. A SIMPLE RATIO is when both terms of a couplet are simple numbers. Thus, 6 18, is a simple ratio. : 236. A COMPOUND RATIO, is one which arises from the mul tiplication of two simple ratios: thus, in the simple ratios, 3: 7, and 6: 8, if we multiply the corresponding terms together, we have 3 x 67 x 8, which is compounded of the ratio of 3 to 7, and of 6 to 8. The factors, 3 and 6, are called ELEMENTS of the first term; and the factors, 7 and 8, are ELEMENTS of the second term. The elements of a term are generally written in a column, thus, ; and read, 3 multiplied by 6, to 7 multiplied by 8. 229. What is a ratio ?-230. What are the terms of a ratio? How many terms has every ratio ?-231. What is the divisor called? 232. What is the dividend called? 233. In how many ways is a ratio expressed? What are they? 234. What are the terms of a ratio, taken together, called ? 235. What is a simple ratio?-236. What is a compound ratio? 237. When the antecedent is less than the consequent, the ratio shows how many times the consequent is as great as the antecedent. 238. When the antecedent is greater than the consequent, the ratio shows what part the consequent is of the antecedent. NOTES.-1. Only numbers having the same unit value, can be compared with each other: hence, all numbers compared, must be reduced to the same unit. 2. The ratio, is always an abstract number. 239. To measure a number, is to find how many times it contains another number of the same kind, which is called, the standard. The unit 1, is the simplest standard of measure, and by this, all numbers, whether integral or fractional, are finally measured. In every ratio, the antecedent is the standard. Examples. 1. What is the ratio of 3 feet to 6 feet? 2. What is the ratio of 10 dollars to 40 dollars ? 3. What is the ratio of the number 9 to 18? 4. What is the compound ratio of 3 x 9 to 9 x 9? 5. What is the compound ratio of 3 × 4 to 12 × 12? 6. What is the compound ratio of 5×3×2 to 6x10×3? 237. What does the ratio show, when the antecedent is less than the consequent ? 238. What does the ratio show, when the antecedent is the greater? 239. What is the operation of measuring a number? What is the measure called? What is the simplest standard for all num bers? What is the standard in any ratio? NOTE. The standard is generally preceded by the word of, and in comparing numbers, may be named second, as in examples 12, 13, 14, 15, and 16; but it must always be used as a divisor, and should be placed first in the statement. is? 19. 4 is what part of 9? 17. What part of 20. is what part of 4? 21. 2.75 is what part of 6.975? 22. 5 is what part of 7.1875? 23. What is the ratio of 2 T. 3 cwt. 2 qr. to 1 T. 11 cwt. 3 qr. 16 lb.? 24. What is the ratio of 1 mi. 6 fur. 8 rd. to 10 mi. 1 fur. 16 rd. 1 yd. 2 ft.? 25. The ratio of two numbers is 3, and the antecedent 16: what is the consequent? ANALYSIS. Since the ratio is equal to the consequent divided by the antecedent, it follows, Ratio 3 = consequent antecedent 1st. That the consequent is equal to the antecedent multiplied by the ratio: 2d. That the antecedent is equal to the consequent divided by the ratio. 26. The ratio of two numbers is 6, and the antecedent 12: what is the consequent ? 27. The ratio of two numbers is 9, and the consequent 108: what is the antecedent? 28. The ratio of two numbers is 5, and the consequent 125 what is the antecedent? 29. The ratio of two numbers is, and the antecedent : what is the consequent ? 30. The ratio of two numbers is, and the consequent : what is the antecedent? 31. The ratio of two numbers is 6, and the consequent 12: what is the antecedent? 32. The antecedent is, and the consequent : what is the ratio? 33. The antecedent is 3 × 6 × 9, and the consequent 1 x 5 x 4 x 2: what is the ratio? SIMPLE PROPORTION. 240. A SIMPLE PROPORTION is the comparison of the terms of two equal simple ratios. Thus, the ratio of 3: 6, is 2; and the ratio of 8: 16, is 2; and we compare the terms by writing a double colon between the couplets; thus, which is read, 3 : 6 :: 8 : 16; 3 is to 6, as 8 to 16. Hence, every proportion has two couplets and four terms. NOTE.-When the ratio of the first couplet is greater than 1, the second term is greater than the first, and the fourth term greater than the third. When the ratio is less than 1, the second term is less than the first, and the fourth term less than the third. 241. The first and fourth terms of a proportion are called the extremes: the second and third terms, the means. Thus, in the proportion, 3 and 24 are the extremes, and 12 and 6 the means. 242. Since the ratio in the first couplet is equal to that in the second, we have, and we shall have, by reducing to a common denominator, 240. What is a Simple Proportion? How is it written? NOTE. When the ratio in the first couplet is greater than 1, what follows? 241. What are the first and fourth terms of a proportion called? The second and third? 242. What is the product of the extremes equal to? |