The proportion (2) may be obtained by dividing the ratios obtained by applying (vi.) to (1) by the corresponding ratios obtained by applying (vii.) to (1).

(ix.) In a series of equal ratios the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.

f

f

=m: n,

provided that (b + d + ƒ + ··· + n) 9.

Let r denote the value of each of the equal ratios.

By addition, (a+c+e+··· + m) = (b+d+f+····· + n)r.

Then on condition that (b+d+f+

+ n) + 0,

(x.) The products or the quotients of the corresponding terms of two proportions form a proportion.

That is, if a: b = c:d, (1), and x:y=2: W, (2), then ax: by = cz: dw, (3), and also a/x:by=c/z:d/w. (4) Writing (1) and (2) in fractional form, and multiplying corresponding members of the equations, we obtain (3).

Similarly, (4) is obtained by division.

(xi.) Like powers or like principal roots of the terms of a proportion are in proportion.

a:b=c:d, (1)

That is, if then_a":¿"=c":d", (2), and also a: 1⁄4/b=V/c:†ã. (3)

k

k

Wd.

Ex. 1. Find the mean proportional, x, between 4 and 25.

Multiplying the members of (2) and (3) by b and d respectively and adding the corresponding members of the resulting equations, we have

Construct proportions from the following products : 13. x2 = 4·40. 14. (a+b)(a - b) = c2.

12. a2 = bc.

If abcd obtain each of the following proportions:

16. ac : bd = c2 : d2. 17. a+c : b+d=a2d : b2c.

18.

(a + b)2

22. a2 - b2:

4a2-562

= c2 — d2 :

19.

=

(c + d)2

4 c2-5 d2

a+b 23. If a : b = b

c+d

c show that

24. What quantity must be added to the terms of a2c2 to make it

equal to a : c?

25. What expression must be subtracted from each of the following expressions in order that the remainders shall form a proportion?

4a+b+c, 5a+b+c, a + 13b+c, and a + 17b+c.

Solve for in each of the following proportions:

26. 221 = 3 x.

27. 70 x = 14: 2.

28. (m2 n2): (mn) = x 1. 29. (5): 35: 12.

Simplify the following equations by applying the Principles Governing Proportions, and then solve for x:

34. Find three numbers in continued proportion whose sum is 14 and whose product is 64.

35. Divide $42.00 between two men so that their shares shall be in the ratio of 3: 4.

36. Divide 44 into two parts such that the less, increased by one, shall be to the greater, decreased by one, as 5: 6.

37. Two numbers are in the ratio of 4 : 5. If each is increased by 5 the sums will be in the ratio of 5: 6. What are the numbers?

38. What number must be added to the numbers 3, 4, 7, and 9 in order that their sums shall form a proportion?

39. What number must be subtracted from 24, 27, 40, and 55, in order that the remainders shall form a proportion?

40. Find the ratio of the numerator of a fraction to its denominator if the value of the fraction remains unchanged when the numerator is increased by a and the denominator is increased by b?

The areas of two similar plane figures have the same ratio as the squares of any two corresponding dimensions.

41. The area of a triangle is 90 square inches and the base is 12 inches. What is the area of a similar triangle, provided that the base is 16 inches?!

42. The area of the first of two similar polygons is 128 square inches, and the area of the second is 200 square inches. If one side of the first polygon is 8 inches, find the corresponding side of the second polygon.

The volumes of two similar solids have the same ratio as the cubes of any two corresponding dimensions.

43. The diameter of the first of two bottles which are of similar shape is three times that of the second. If the first holds 2 ounces, how much does the second hold?

44. If a sphere which is 2 inches in diameter weighs 5 lbs., what is the weight of a sphere of the same substance which is 3 inches in diameter ?

Problems in Physics

25. The Inclined Plane. If a body rests on a smooth inclined plane, the force (disregarding friction) which must be applied along the plane to hold the body in place against the action of the force of gravity has the same ratio to the weight of the body that the height of the plane has to the length of the plane.

26. If F represents the force applied along an inclined plane, W the weight of the body, h the height of the plane, and the length of the plane, we have

The force represented by F which is applied along the plane is called the component of the weight W which is parallel to the plane.

EXERCISE XXVI. 3

Solve the following problems:

1. Find the force which must be exerted to draw a sled weighing 240 lbs. up a hill which is 300 feet long and 50 feet high.

2. What is the weight of a body if a force of 125 lbs., exerted along a smooth inclined plane which is 80 feet in length and 20 feet in height, prevents the body from sliding down the plane?

3. A boy who is able to exert a maximum force of 80 lbs. is able to keep a barrel from rolling down a plank which is 12 feet in length and the upper end of which is 3 feet from the ground. Find the weight of the barrel.

4. A porter who can exert a maximum force of 200 lbs. undertakes to roll a cask weighing 500 lbs. up a board which is 10 feet long. How high can the upper end of the board be placed without compelling the porter to allow the cask to roll down the board?

5. A car weighing 1200 lbs. is held at rest on a smooth inclined plane by a force of 30 lbs. applied parallel to the plane. If the length of the plane is 800 feet, find the height of the plane.

6. A boy is able to exert a maximum force of 80 lbs. How long an inclined plane must he use to push a truck weighing 320 lbs. up to a doorway which is 34 feet above the ground?

Boyle's Law. The volume of a gas is (approximately) inversely proportional to the pressure, provided that the temperature remains

constant.

That is, representing the pressure by P1 when the volume is V1, and the pressure by P2 when the volume is V2, we have

7. If, when confined with a pressure of 20 lbs. per square inch, a mass of gas occupies a volume of one cubic foot, find the volume of the gas when the pressure becomes 40 lbs. per square inch.

8. A gas bag containing 3 cubic feet of gas under a pressure of 18 lbs. per square inch must be subjected to what pressure to reduce the volume to half a cubic foot?

9. Six cubic feet of gas under a pressure of 45 lbs. per square inch will have what volume if the pressure is reduced to 15 lbs. per square inch? 10. A bladder holds 40 cubic inches of air under a pressure of 15 lbs.