SUMMARY OF THE PROPERTIES OF PROPORTIONS 350. Following are statements, in algebraic symbols, of the properties of proportions: VIII. If a:b = - d: d. c-d: c. c:d, then a + b: a − b = c + d : c − d. IX. If a: b = c:d=e:f, then a + c + e: b + d + f = a: b. = c2: d2, or an: bn = cn; dn. EXERCISE 351. 1. What is meant by transforming a proportion by inversion? by alternation? by composition? by division? by composition and division? (b) Find a fourth proportional to a3b3, a2b2, ab. b3, 11. Transform so that x shall occur only once, 80 12. Transform so that x shall occur only once: 13. A cement block is to be made of Portland cement, sand, and gravel in the proportions 1:2:3. How much of each is there in a block that weighs 300 pounds? 14. The unequal sides of a rectangle are in the ratio of 3 to 5. Find the dimensions if the perimeter is 20 feet. 16. What number added to each of the numbers 1, 3, 19, and 27 will give numbers that form a proportion? 17. A and B do a piece of work for $38. A works 5 days of 8 hours each and B works 4 days of 9 hours each. How should the money be divided ? 18. The angles of a certain triangle are in the ratio 1: 2: 3. Their sum equals 180°. How large is each? 19. The sides of a triangle are in the ratio 3:45. The perimeter is 100 inches. How long is each side? For what value of x does each set of numbers form a true proportion if taken in the order given? HINT. Let n, n + 1, n + 2, n + 3 represent the numbers. 31. Brass consists of 2 parts of copper to 1 part of zinc. How many pounds of each are there in 9 pounds of brass? 32. Gunmetal consists of 9 parts of copper to 1 part of tin. How many pounds of each are there in 20 pounds of gunmetal? 33. It is proved in geometry that if a line is parallel to one side of a triangle, it divides the other two sides into parts that are in proportion. By actual measurement in the figure show that l:m = n : r. 34. If 5 inches, m = m A n m E Dr - 2 inches, and n = 4 inches, find r. 35. If 18 inches, m = 3 inches, and 2.5 inches, find n. = 36. To measure the width of a river, BC, a triangle was laid out as shown in the figure, with BD parallel to EC. By actual measurement AB was found to be 96 feet, AD was 76 feet, and DE was 102 feet. Find BC. n D r E 37. Similar triangles are triangles that have the same shape. It is stated in geometry that their corresponding sides are in proportion. Thus in the two triangles l:p=m:q; also n:r=m: q. Write another proportion involving the sides of the triangles. 38. If 5 inches, m = 3.8 inches, and p = 7 inches, find the l= length of q. Also find the length of n if r = 8 inches. 39. Of two similar triangles (see example 37) the sides of one are 5 inches, 8 inches, 10 inches, and the sides of the other are 7 inches, 12 inches, 15 inches. Show that their perimeters are in the same ratio as two corresponding sides. 40. In the figure, XY is the YZ is the height of the tree; length of the shadow of the tree, BC is a stick set in the ground and AB the length of its shadow. If AB=6 feet, BC=4 feet, and XY=42 feet, find the height of the tree. HINT. The triangles are similar. (See example 37.) 41. The Woolworth building (city of New York), the highest office. building in the world, casts a shadow 625 feet long at the same time that a boy 4.8 feet high casts a shadow 4 feet long. How high is the building? XIII. GRAPHS1 352. The student has seen in his general reading many different graphical, or pictorial, methods of representing data. A series of straight lines can be used to show the relative values of the grain crops, manufactured products, or the wealth of different countries. Pictures of soldiers of different sizes may represent, pictorially, the relative strength of the armies of different nations. In a similar way the strength of navies may be represented by ships. 353. 1. Determine by construction a line representing the sum of three given lines a, b, c. Place the three lines, a, b, and c end to end and the total length is the sum required. с b α α b 2. Represent on a scale of inch to a mile a distance of 10 miles. The line AB is 10 units long, each unit being inch. AB therefore represents a distance of 10 miles on a scale of inch to a mile. AL B 3. Represent a distance of 25 miles on a scale of inch to a mile. 1 The chapter on graphs may be omitted, if desired, without interrupting the sequence of the work. |