14. In 1880, 17% of the persons in the United States were illiterate; in 1890, 13.3%; in 1900, 10.7 %. Express these per cents as ratios having 10,000 as a consequent. SIMPLE PROPORTION 502. An equality of two or more ratios is called a proportion. = [2 Thus, 4: 6 : 8 : 12 is a proportion; also ‡ = f1⁄2; also ‡ = 11⁄2 = }· The double colon (::) is often used instead of the sign of equality. 8 = The proportion 4: 68: 12, or 2, or 4: 6:: 8:12 is read "the ratio of 4 to 6 is equal to the ratio of 8 to 12"; or more briefly, "4 is to 6 as 8 is to 12." 503. The first and last terms of a proportion are called the extremes, and the second and third terms the means. In 4:68: 12, the extremes are 4 and 12; the means, 6 and 8. 504. 1. In the proportion 4: 68: 12, what is the product of the extremes? of the means? How does the product of the extremes compare with the product of the means? 2. Compare the product of the extremes with the product of the means in 1: 2=4:8; in 1: 2=5:10; in 2 : 4= 6 : 12. 3. Form other proportions and compare the product of the extremes in each with the product of the means. In the following compare the first ratio with the second to find whether or not they are equal; then compare the product of the extremes in each case with the product of the means. = 4. Does ? 8=5? 8=14? = 12? In any proportion the product of the extremes is equal to the product of the means. THIRD PROG. AR. 19 WRITTEN EXERCISES 505. 1. If 24: 7 = 48 : x, what is the value of x? Reducing 12: 16 to lowest terms, 3:4 = x: 60 Product of means = product of extremes WRITTEN EXERCISES 506. 1. If 4 tons of coal cost $17, how much will 14 tons cost at the same price per ton? SOLUTION The cost of the coal is proportional to the quantity; that is, 2. A man bought an 180-acre farm for $11,000 and sold 30 acres of it at cost. How much did he receive for the part sold? 3. If 55 acres of land yield 1430 bushels of wheat, how many bushels will 132 acres yield at the same rate? 4. If sound travels 825 meters in 2 seconds, how long will it take the report of a cannon to travel 3.3 Km.? 5. If 24 pounds of maple sugar cost $3.60, how much must be paid for 5 pounds? 6. A motor-boat race of 136.5 miles on the Hudson River was run at the rate of 52.5 miles in 2 hours. How long did it take to complete the course? 7. If it costs 66 to have a rug of 81 square yards cleaned, how much will it cost for a rug of 15 square yards? 8. A manufacturing plant turns out 7500 concrete building blocks in 6 days. How large an order for concrete blocks can it fill in 20 days? 9. A series of photographs for a moving picture was taken in the New York Subway at the rate of 500 in 33 seconds. How many photographs were there in the series, if it was completed in 7 minutes? 10. If 10 men can dig a ditch in 12 days, how many days will it take 15 men to dig it? SOLUTION Let x represent the number of days it will take 15 men. Since 15 men can do more work per day than 10 men, they can dig a ditch in less time than 10 men can dig it. Therefore the ratio x days: 12 days is less than 1. The ratio, less than 1, between 10 men and 15 men is 10 men: 15 men. Therefore, x days: 12 days = 10 men : 15 men Test. There are 10 × 12, or 120, days' work to be done. 15 men can do this work in (120 ÷ 15) days, or in 8 days. 11. If 16 men can do a piece of work in 15 days, how long will it take 20 men to do it? 12. If 16 men can do a piece of work in 15 days, how many men will be required to do it in 10 days? 13. If a force of men can do a certain piece of work in 20 days by working 8 hours a day, in how many days can they do the work by working 10 hours a day? 14. It is estimated that it will take 24 men 18 days to repair a portion of Second Street. The work must be done in 16 days. How many extra men must be employed? 15. A contractor agrees to do a piece of work in 60 days and employs 40 men for that purpose. Before beginning, however, the time is extended 15 days. How many men may he discharge? 16. If the interest on a sum of money for 6 months is $33, what will be the interest for 8 months? 17. A man standing on the rear platform of a train observed that the train passed 5 mile posts in 5 minutes 15 seconds. How many miles per hour was the train moving? 18. In a shipment by rail of 450 dozen eggs, 5 eggs out of every 3 dozen were broken. How many dozen eggs were left unbroken? 19. A owned a house assessed at $12,000, on which he paid a tax of $225 one year. The same year B's house, next door, was taxed at $157.50. For how much was B's house assessed? 20. The assessed valuation of the property in a town is $4,650,000, and the tax to be raised is $102,300. If I own property assessed at $23,250, how much tax must I pay? 21. A garrison of 150 soldiers consumed 26 barrels of flour in 9 weeks. At this rate how many days would the same amount of flour last a garrison of 210 soldiers? 22. On a freight train of 18 cars, 4050 barrels of flour are loaded. How many cars must be added so that the train may carry 5175 barrels ? PARTITIVE PROPORTION 507. The process by which a number is separated into parts proportional to two or more given numbers is called partitive proportion. WRITTEN EXERCISES 508. 1. Separate 225 into parts proportional to 2, 3, and 4. SOLUTION Since the parts are proportional to 2, 3, and 4, we may represent them by 2 x, 3 x, and 4 x, respectively. Since the sum of the parts is the number itself, Hence, 2x+3x+4x= 225 .. x = 25 2 x 50, 3 x = 75, and 4 x = 100, the numbers. Test. 50+ 75+100 225; 50:75=2:3; 75: 100 = 3:4. = |