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291. Magnitudes Which Are Inversely Proportional To Other Magnitudes Or To The Squares Ok Other Magnitudes.

Example.

If 5 men do a piece of work in 16 days, how long will it take 8 men to do a similar piece of work?

Operation and Explanation.

It is evident that the time required will be inversely proportional to the number of men employed; that is, if twice as many men are employed, not twice as much, but I as much time will be required. Hence the proportion is not 5 : 8 = 16 : x, but, 5 : 8 = x: 16; hence, 5:8 =10:16.

The interpretation of the above equation is, if 5 men can do a piece of work in 16 days, 8 men can do it in 10 days.

1. If 4 men can do a piece of work in 20 days, how long will it take 5 men to do a similar piece of work?

2. If 8 men can do a piece of work in 12 days, how long will it take 3 men to do a similar piece of work?

It can be shown that the intensity of light upon an object diminishes as the square of the distance between the luminous body and the illuminated object increases; that is, if the distance be twice as great in one case as in another, the intensity is not twice as great, not I as great, but 1 as great; if the distances are as 2 to 3 the intensities are, not as 2 to 3, not as 3 to 2, but as 9 to 4. The intensity at 2 feet is 1 as great as at 3 feet.

3. Object A is 15 feet from an incandescent electric light. Object B is 20 feet from the same light. Object C is 30 feet from the same light. (a) How does the intensity of the light at B compare with the intensity at A? (b) How does the intensity at C compare with the intensity at A?

Algebra.

292. To Find The Missing Term Of A Proportion Without Finding The Ratio.

The first and fourth terms of a proportion are called the extremes, and the second and third terms, the means; thus, in the proportion 12:6 = 8:4, 12 and 4 are the extremes and 6 and 8 are the means.

Observe that in the following proportions the product of the means equals the product of the extremes:

6:3 = 8:4; then 6x4 = 3x8 £: \ = 4 : 2; then $ - x 2 = | x 4 Let a :b = c :d, stand for any proportion.

Then "= £

o a

Clearing of fractions, ad = be

But a and d are the extremes and b and c the means; hence, in any proportion in which abstract numbers are ememployed, the product of the means equals the product of the extremes.

Example I.

30:20 = 18:a;.
30a ; = the product of the extremes.
20 x 18 the product of the means.
Then 30a ; = 20 x 18, or 360,

and x = 12.

Example II.

10: 25 = a;: 50.
Then 25a ; = 10 x 50, or 500,
and * = 20.

Algebra.

Example III.

40: a; = 25: 5.
Then 25a; = 40 x 5, or 200,
and x = 8.

Example IV.

x: 35 =4:28.
Then 28a ; = 35 x 4, or 140,
and x 5.

Find the missing terms:

1. 24:72 = x: 69. 6. f : 4 = x : 30.

2. 45:12 = 75: x. 7. x : f. = 40 : 6.

3. 35 : x = 14 : 40. 8. |: f = x: 8.

4. a: 70 = 3:21. 9. .6 : .8 = 15 : a.

5. 55 : 25 = x : 10. 10. .25 : 5 = x : 40.

11. If 8 acres of land cost $360, how much will 15 acres cost at the same rate? v

8:15 = 360:a;.*

12. If 12 horses consume 3500 lb. of hay per month, how many pounds will 15 horses consume?

13. If 11 cows cost $280.50, how many cows can be bought for $433.50 at the same rate?

* Observe that in the solution of concrete problems by the method here given the numbers must be regarded as abstract. It would be absurd to talk or think of finding the product of 15 acres and 360 dollars and dividing this by 8 acres. It is true, however, that the ratio of 8 acres to 15 acres equals the ratio of 360 dollars to x dollars. It is also true that in the proportion 8: 15 = 360: x, the product of the means is equal to the product of the extremes.

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1. Convince yourself by measurement and by paper cutting that from every trapezoid there may be cut a triangle (or triangles) which when properly adjusted to another part (or parts) of the trapezoid, will convert the trapezoid into a rectangle.

2. Convince yourself that the rectangle made from a trapezoid is not so long as the longer of the parallel sides of the trapezoid, and not so short as the shorter of the parallel sides of the trapezoid—that its length is midumy between the lengths of the two parallel sides of the trapezoid.

Note.Observe that the length of the rectangle thus formed may be found by adding half the difference of the parallel sides of the trapezoid to its shorter side, or by dividing the sum of its parallel sides by 2.

3. To find the area of a trapezoid, find the area of the rectangle to which it is equivalent, or, as the rule is usually given,—"Multiply one half the sum of the parallel sides by the altitude."

4. Find the area of a trapezoid whose parallel sides are 10 inches and 15 inches respectively, and whose altitude is 8 inches.

5. How many acres in a trapezoidal piece of land, the parallel sides being 28 rods and 36 rods respectively, and the breadth (altitude) 25 rods?

294. Miscellaneous Review.

1. If 3 men can build 72 feet of sidewalk in a day, how many feet can 4 men build?

2. If 3 men can do a piece of work in 12 hours, in how many hours can 4 men do an equal amount of Work?

3. If a piece of land 8 rods square is worth $500, how much is a piece of land 16 rods square worth at the same rate?

4. If a ball of yarn 3 inches in diameter is enough for a pair of stockings, how many pairs of stockings can be made from a ball 6 inches in diameter ? *

5. If a grindstone 12 inches in diameter weighs 40 lb., how much will a grindstone 18 inches in diameter weigh. the thickness and quality of material being the same?

6. The opening in an 8-inch drain tile is how many times as large as the opening in a 2-inch drain tile ? f

7. Find the area of a rhomboidal piece of land whose length (base) is 64 rods and whose width (altitude) is 15 rods.

8. Find the area of a trapezoidal piece of land, the length of the parallel sides being 44 rods and 52 rods respectively, and the width (altitude) being 18 rods. J

9. Find the area of a triangular piece of land whose base is 42 rods and whose altitude is 20 rods.

* Compare a 3-inch cube and a 6-inch cube. Remember that a 3-inch sphere is a little more than half of a 3-inch cube, and a 6-inch sphere a little more than one half of a 6-inch cube.

t Compare a 6-inch square with a 2-inch square. Remember that a 2-inch circle is about J of a 2-inch square, and an 8-inch circle about J of an 8-inch square.

t Draw a diagram of the land on a scale of J inch to the rod.

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