147. Fractions which can be Reduced to Exact Decimal Equivalents. Fractions whose denominators have no prime factors except 2 and 5 can be reduced to exact decimal equivalents. Thus fractions whose denominators are 2, 4, 5, 8, 10, 16, ... can be so reduced, while fractions whose denominators are 3, 6, 7, 9, 11, 12, 13, 14, 15, 17,... cannot be so reduced. ORAL EXERCISES Which of the following fractions can be reduced exactly to decimals: 1, 1, 3, 1, 1, 1, 4, 5, 8, §, 7, §, 12, 16, 15? 1, 1 3 . 5 5 7 17 15 148. Fractions in Common Use. The following fractions represent the diameters in inches of standard sizes of wire manufactured in the United States: 1, 1, 16, 12, 1, 11, 16, 32, 14, 1, 44, 32, 4, 3, 32, 16, 11, 32, 8, 1, 67, 32, 64, 128, 16, 180, 20, 180, 80, 320, 32, 320, 40, 320, 180, 640, 64, 640, 80, 640, 1230, 320, 1280, 640, 1280, 2580, 180. 5 9 3 5 The following fractions represent the thickness in inches of commercial iron and steel sheets: 800, 280, 580, 125, 100, 280, 500, 84, 580, 30, 500, 10, 250, 32, 280, 25, 280, 20, 200, 16, 200, 100, 10, 25, 200, 180, 20%, 10, 100, 1, 200, 20, 20%, 30, 1, 30, 2%, 4. WRITTEN EXERCISES 13 3 1. Make a list of the fractions in the first group, giving the decimal equivalents and arranging them in the order of their magnitude. If there are any fractions in this list which do not have exact decimal equivalents, give their nearest approximations to five places of decimals. 2. Make a similar list of the fractions in the second group. PROBLEMS INVOLVING DECIMALS 1. American gold coin weighs 25.8 grains per dollar and American silver coin weighs 412.5 grains per dollar. Both coins contain .9 precious metal and .1 alloy. How much pure gold is there in a fivedollar gold piece? How much pure silver is there in a silver dollar? 2. A nautical mile is 6080 feet. Express this as a decimal of the statute mile (5280 ft.). By how many feet does a nautical mile differ from 1.151 statute miles? 3. In her trial trip the fastest merchant ship ever built made 27.3 knots (nautical miles) per hour. How many miles per hour did she make, counting 1.151 miles per knot? 4. The fastest trip ever made across the Atlantic Ocean was made at the rate of 26.02 knots per hour. How many miles per hour was this, counting 1.151 miles per knot? 5. An American bushel contains 2150.42 cubic inches. By how many cubic inches does this differ from the English imperial bushel which contains 2218.192 cubic inches? 6. Multiply the number of cubic inches in an American bushel (2150.42) by .804. By how much does the product differ from 1728 cubic inches (one cubic foot)? 7. A bin is 6 (6.5) feet wide, 8 (8.75) feet long and 51 (5.25) feet deep. How many bushels of grain will it hold if one cubic foot holds .804 of one bushel? 8. Fresh water weighs about 62.5 pounds per cubic foot. Ice weighs .92 times as much as water. Find the weight of a cubic foot of ice. 9. Using the result obtained in problem 8, find by how many pounds 35 cubic feet of ice differs from one ton (2000 lb.). 10. Cork weighs .24 times as much as water. How many cubic feet of cork will weigh one ton? 11. The water of the Dead Sea weighs 1.24 times as much as fresh water (see problem 8). How much would one cubic foot of water from the Dead Sea weigh? 12. If milk weighs 8.6 lb. per gallon, how many gallons are there in 100 lb. of milk? 13. An empty milk can weighs 19.3 pounds, and when filled with milk it weighs 93.8 pounds. How many gallons of milk does it hold? (See problem 12.) 14. A farmer hauled milk to the creamery as follows: Sunday, 474 lb.; Monday, 464 lb.; Tuesday, 507 lb.; Wednesday, 493 lb. ; Thursday, 501 lb.; Friday, 497 lb.; and Saturday, 437 lb. At 17 cents a gallon what was the value of this milk? (See problem 12.) 15. At 85¢ per thousand cubic feet for gas what is the bill of a family consuming 7840 cubic feet of gas? 16. At 95 cents per thousand cubic feet how many cubic feet does a family consume whose gas bill is $6.45? 17. A lot in New York City is sold for $1.27 per square foot. What is the price of the lot if it is 27.4 feet wide and 97.3 feet deep? 18. A lot in Boston is sold for $1.14 per square foot. the area of the lot if it sells for $12,480? What is 19. A school baseball team plays 19 games and wins 11 of them. Reduce the standing of the team to a three-place decimal. 20. It has been found by experimentation that on an average 5.6 pounds of corn fed to hogs cause them to gain 1 pound in weight. A farmer feeds 7806 lb. of corn to his hogs. How much should they gain in weight? 21. In a recent year the population of the state of New York was 9,373,000. This same year 145,654 deaths occurred within the state. Find to the nearest tenth the number of deaths per thousand inhabitants. (This is the usual mode of expressing death rates.) 22. During a recent year the populations and the number of deaths in the states and cities named below were as follows: CHAPTER XIV ALIQUOT PARTS 149. Use of Aliquot Parts in Business. The uses of aliquot parts are illustrated in § 144 and in the exercises of this chapter. They are of very frequent occurrence, and much drill is necessary in order to handle them effectively. Those who wish to enter business as a profession should never lose sight of the fact that their work is that of specialists and that hence they require extensive training which is not needed by people in other walks of life. 150. An aliquot part of a number is contained an integral number of times in the given number. Thus, 2, 2, 3, 5 are aliquot parts of 10, since 10 ÷ 2 = 3 and 10 ÷ 5 = 2. = 5, 10 ÷ 2 = 4, 10 ÷ 3 The number 1 is an aliquot part of any integer and hence is not enumerated among the aliquot parts of numbers. Prices of goods 151. Prices of Goods Aliquot Parts of a Dollar. are frequently fixed as aliquot parts of one dollar, since that is equivalent to a certain number of units per dollar. Thus, 12 is of 100 and hence 123 cents apiece is the same as 8 for one dollar or 2 for a quarter. 152. Aliquot Parts of 100. The following are the more important aliquot parts of 100. ALIQUOT PARTS OF 100 A 16 There are other aliquot parts of 100. of 100; 9 of 100; 7 = less frequently in business. = = 2 = Thus, 14 of 100; 11 153. Multiplication by Aliquot Parts of 100. Since goods are frequently priced at aliquot parts of a dollar, multiplication by aliquot parts of 100 is important. Since 33 is of 100, multiplying by 100 and dividing by 3 is the same as multiplying by 331. Similarly, to multiply by 163, multiply by 100 and divide by 6. Example 1. Multiply 394 by 331. Solution 1. 394 X 33 = Example 2. 280 × 163 394 × 100 = 39400 ÷ 3 = 13133. = = = = 2800064666. 290000 12 = 24166. 79000 ÷ 15 = 5266. = 3980000 ÷ 16 WRITTEN EXERCISES = 248750. |