the product of the extremes. Therefore the missing extreme may be found by dividing the product of the means (4 x 5) by the given extreme (10). The missing term then is 2, and the proportion is 2:5=4: 10. 645. From the preceding illustration may be derived the following principles : 1. When four general numbers form a proportion, the product of the means is equal to the product of the extremes. 2. A missing extreme may be found by dividing the product of the means by the given extreme. 3. A missing mean may be found by dividing the product of the extremes by the given mean. 646. Oral Exercises. Supply the missing terms represented by x in the following proportions : a. 3:4= 9: x. d. X : 7 = 8: 9. b. 8:6 e. 27 : 3 = x : 1. c. 12 : x = 15 : 3. f. x: 4 days = $5: $15. NOTE. To find x in Example f, disregard the denominations, and proceed as if the terms were general numbers. 4 = 1}. Then x equals 1} days. =X : 3. 647. Examples for the Slate. Supply the missing terms in the following: (10.) 2 : 100 -- 17 : x. (13.) 750 A. : 3 A. = x : 13 tons. (11.) 9: 150 = 105 : 2. (14.) x : 200 hats = $ 87.50: $ 500. (12.) 65: x = $75: $850. (15.) $800 : $ 56 = $390 : x. ANALYSIS AND PROPORTION. 648. ILLUSTRATIVE EXAMPLE. If 14 slates cost 98 cents, what will 10 slates cost? By Analysis. WRITTEN WORK. Explanation. — If 14 slates cost 98 cents, 7 1 slate will cost i fourteenth of 98 cents, and 98 x 10 10 slates will cost 10 times 1 fourteenth of 98 14 cents, which is 70 cents. Ans. 70 cents. Ans. 70 cents. = 70. By Proportion. WRITTEN WORK. Explanation. — The ratio of 14 slates to 10 14:10 = 98 : X. slates must be the same as the ratio of 98 cents, 7 the cost of 14 slates, to the cost of 10 slates. 98 x 10 We may arrange the terms in any order which 70. 14 will express the equality of these ratios. For convenience, we make 98 cents the third term, Ans. 70 cents. and x, the unknown cost of 10 slates, the fourth term. As the cost of 10 slates will be less than 98 cents, we make 10 the second term and 14 the first. Multiplying 98 by 10 and dividing the product by 14, we have for the fourth or missing term, 70. Ans. 70 cents. 649. Rule. To solve examples by simple proportion: 1. Make the number that is of the same denomination as the required answer the third term. 2. Determine from the statement of the example whether the answer is to be greater or less than the third term. 3. Make the other two numbers in the example the first and second terms of the proportion, taking the greater number for the second term if the answer is to be greater than the third term, and the less number for the second term if the answer is to be less than the third term. 4. Multiply the third term by the second term, and divide the product by the first term. 650. Examples for the Slate. The following examples may be solved by analysis or by proportion, or by both methods, at the option of the teacher. 16. If 4 yards of velvet cost $20, what will 14 yards cost? 17. If 12 bushels of wheat cost $ 8, what will 30 bushels cost? 18. What will 250 sheep cost if 24 sheep cost $ 72 ? 19. What will 75 pounds rf cheese cost if 64 pounds cost $ 6.0 ? 20. How many feet of plank will be required for a bridge 528 feet long, if 17280 feet of plank are required for 288 feet? 21. If 500 bushels of plaster were sufficient for the dressing of 3} acres of land, what would be required for 17} acres of the same kind of soil ? 22. If a building 13 ft. high casts a shadow of 4 ft., what length of shadow will a church spire 3462 ft. high cast at the same time? 23. If crackers can be sold at 10 cents a pound when flour is worth $ 6.50 a barrel, for what can they be sold when flour is worth $ 9.75 a barrel, the cost of making not being considered? 24. If a hind wheel, which is 83 feet in circumference, turns 800 times in a journey, how many times will the fore wheel, which is 6} feet in circumference, turn in the same journey ? 25. If 400 bushels of potatoes were bought for $350.90, and sold for $425.50, what was gained on 25 bushels ? 26. If a 10-cent loaf weighs 1 lb. 2 oz. when flour is worth $7} per bbl., what should it weigh when flour is $6 per bbl. ? 27. If my friend lends me $ 7000 for 15 days, for what time should I lend him $ 4500 to requite the favor ? 28. If my friend lends me money for 4 months when interest is 10 per cent, for what time should I lend him the same sum to requite the favor when interest is 7 per cent ? 29. If 2 lbs. 5 oz. of wool make 1 yd. of cloth 32 inches wide, how much will make a yard of the same quality 14 yards wide ? 30. How many yards of cambric 34 inches wide will be required to line 14} yards of silk which is 22 inches wide ? 31. If 400 lbs. of coal are required to run an engine 12 hours, what number of tons will be required to run three similar engines for 30 days, day and night? 32. A deer, 150 rods before a hound, runs 30 rods a minute; the hound follows at the rate of 42 rods a minute. In what time will the deer be overtaken ? COMPOUND PROPORTION. 651. A compound proporti n is a proportion in which one of the ratios is compound. 652. ILLUSTRATIVE EXAMPLE If it takes a man 5 days of 9 hours each to earn $ 15, how many days of 8 hours each will it take him to earn $ 20 ? WRITTEN WORK. 5 3 74 By Analysis. earn $15, it will take him 1 fifteenth of 5 days 3 x 20 x to earn $1, and 20 times that to earn $ 20. If 15 ~ 8 it takes him this number of days when the days 3 2 are 9 hours long, it will take him 9 times as Ans. 7} days. many days when they are 1 hour long, and 1 eighth of that number when they are 8 hours long, which is 7 days. Ans. 7 days. By Compound Proportion. Explanation. The number of days it 1 1 will take depends, first, on the amount 3 4 of money to be earned, and, secondly, on 15 : 20 5 days : x. the number of hours a day the man works. We might get the answer by 3 using two simple proportions. In the 5 x 3 first we could find the number of days, 2 so far as it depends on the amount of Ans. 71 days. money to be earned ; and then, taking this result as the third term of another proportion, we could find the number of days so far as it depends on the number of hours in a day's work. It will be more convenient, however, to combine the two proportions, thus forming a compound proportion. To do this we make 5 days, which is a number of the same denomi. nation as the required answer, the third term, and then consider the statements of the example in order (1) As $ 20 is a larger sum than $15, it will take a larger number of days to earn it ; that is, the answer, so far as it depends on the amount of money to be earned, will be larger than the third term ; so we make 20 the second term and 15 the first term of the first ratio. (2.) As it will take more days 8 hours long to earn this money than days 9 hours long, the answer, so far as it depends on the length of the days, will be larger than the third term ; so we make 9 the second term and 8 the first term of the second ratio. We now have the compound proportion, 15 : 20 = 5 days : a days. 8 Multiplying 5 days by 20 x 9, and dividing the product by 15 R8, gives 7 days. Ans. 7} days. The work may be shortened, as shown in the written work, by cancelling. 653. Rule. To solve examples by compound proportion: 1. Make the number that is of the same denomination as the answer the third term. 2. Take the two numbers in each separate statement in the example, and consider whether the answer, so far as it depends on them alone, will be greater or less than the third term. Arrange these two numbers accordingly as terms of a ratio. 3. Multiply the third term by the product of the second terms and divide this product by the product of the first terms. 654. Examples for the Slate. 33. If $90 is paid for the work of 20 men 6 days, what should be paid for the work of 5 men 8 days ? 34. If in 84 days 75 men can earn $68.75, in how many days can 90 men earn $ 41.25 ? 35. If it costs $ 30 to paint the front of a building 140 ft. long and 25 ft. high, what will it cost to paint the front of a building 180 ft. long and 20 ft. high? 36. If 450 pounds of merchandise can be carried 26 miles for 304, how many miles can 3 tons be carried for $4 ? |