sold at 98 a bushel. 30. A field of wheat yielded 21 bu. an acre, and it was The cost of cultivation and harvestFind the net gain per acre. ing was $2.02 an acre. 31. In Problem 30, it was estimated that 20% more expended in cultivation would have increased the yield 30%. What would the net gain then have been? 32. If an agent remits his principal $625.38 as the net proceeds from the sale of goods, after deducting $18.72 for freight and 5% for commission, how much was the commission? 33. When an article bought at $25, less discounts of 20%, 10%, and 5%, is sold for $22.23, what is the gain per cent upon the net cost? 34. How much shall a dealer ask for cloth costing $2.40 a yard so that he may make a profit of 10% when selling it at a discount of 331% from the asking price? 35. A boy sold his bicycle for $28 after using it one year. This was 20% less than it cost him. How much did he lose? 36. A man was offered $9000 for a house. He afterwards sold it for but $6800, which was 15% less than it cost him, What per cent would have been made if the first offer had been accepted? 37. When a dealer buys rabbits for $1.50 a dozen and sells them for 35¢ a pair, what per cent is he making? 38. When a dealer buys coal at the rate of $5 for a ton of 2240 lb. and sells it at $6.25 for a ton of 2000 lb., find the gain per cent. 39. If the iceman pays $5 per ton for ice and retails it at 50 per 100 lb., and if delivery costs $2 per ton and the loss is 20% by melting, what per cent does he make? 40. Careful farmers test their seed corn before planting it, in order to select only seed that will grow, and thus increase the yield. From different parts of each of the finest looking ears that can be found, grains are taken and placed in a seed germinator to sprout. Only those ears are used for planting the fields whose grains sprout in the germinator. A man put 540 grains of corn in a germinator. Only 486 of the grains sprouted. What per cent sprouted? 41. By planting only selected seed, all of which grew, he raised 80 bu. per acre. Allowing the per cent found in Problem 40 of grains that would not grow, if he had used seed not selected, what would have been the yield per acre? 42. If he planted 75 acres of corn, how many bushels did he gain by testing the seed? 43. Allowing 60¢ per bushel, how much did he gain on the crop by testing the seed? 44. If he spent only 21 days testing the seed for the entire crop, how much did he get for each day's work? Would it pay for every farmer to test his seed corn? 45. If a man put 7380 grains of corn in his germinator, and only 6790 grains sprouted, what per cent sprouted? 46. If a farmer tested 5220 grains of corn, and only 4594 of the grains sprouted, what per cent sprouted? 47. By testing his seed corn a farmer found that 7% of the grains did not germinate. By planting the selected seed, his yield was 85 bu. per acre. How many bushels per acre did he gain by testing the seed? 48. A farmer found that 81% of the corn tested would not sprout. By planting only selected seed, the yield was How many bushels per acre did he gain by 78 bu. per acre. testing the seed? VII. PROPORTION 65. MEANING OF PROPORTION When two ratios are equal, they form a proportion. Thus, the ratio of the cost of two similar pieces of cloth is the same as the ratio of their length. That is, if 10 yards cost $4, 15 yards will cost $6, or It is read $4 is to $6 as 10 yd. is to 15 yd. When quantities are in proportion they are said to be proportional. 66. SIMILAR FIGURES 1. Similar figures have exactly the same shape. That is, their corresponding sides are proportional. 2. Make two similar triangles. Are their corresponding angles equal? 3. Draw a triangle, ABC, as in the figure. Draw EF parallel to AB. By the use of a protractor compare the angles FEC and BAC. Also angles CFE and CBA. 4. Are triangles ABC and EFC similar; that is, do they have the same shape? E F A B 5. Make a triangle in which AC is 6 inches and CB 3 inches. Mark off CE equal to 4 inches and CF 2 inches. Are the two triangles similar? 6. What is the ratio of CA to CE? Of CB to CF? 7. Cut similar triangles from cardboard. Measure their sides and prove the statement on the next page. In similar triangles the ratios of the corresponding sides are equal, and the ratio of any two sides of one is equal to the ratio of the corresponding sides of the other. C B 8. Inaccessible distances may be found by the principle of similar triangles. Suppose we are to find the distance AB across a small lake. By measuring from A to C, and from B through O to D, making the ratio of OC to OA the same as of OD to OB, we have similar triangles. If OC OA, and CD measures = 20 rods, what is AB? NOTE. D A Make OC any convenient part of OA, and then OD the same part of OB. In the figure OC = OA, and OD = } OB. 9. I wish to find the distance AB. AC is 15 rods and OC is 5 rods. I measure from B through 0 to D. If BO is 8 rods, what shall I make OD? Why? I find DC to be 71⁄2 rods. How far from A to B? 10. A boy, wishing to find the height of a pole CE, made a piece of apparatus which he called his "surveying instrument." It consisted of a right triangle whose two legs were equal. It stood 3 feet from the ground. He moved it along until the point C could just be seen along the hypotenuse of the triangle A D H B F E when the base of the triangle AF was parallel to the ground. A line with a weight (a plumb line) hung from A. If DE was 27 feet, how high was the pole? (Triangles AFH and ABC are similar. Why?) 11. If HF had been twice AF, and DE had been 40 feet, what would CE have been? 12. Make such an instrument, and find the height of trees, telegraph poles, etc. 13. Make one with the triangle having one leg twice the other, and measure the same heights. Do your results check? 14. Two triangles are similar. One has sides 4, 5, and 7 inches, respectively. The long side of the other is 21 inches. What are the other sides? What if the short side of the latter were 2 inches? 15. In this way measure distances on your school lot. the pyramids by the length F Hold a stick, whose length is known, in a vertical position, and mark the end of its shadow. Measure the length of the shadow of the stick and also of the shadow of the tree. From these measurements compute the height of the tree. 17. When a vertical rod 6 feet high casts shadow 9 feet long, a tree casts a shadow of 150 feet. How high is the tree? |