Since the ratio of the first couplet is 3, the ratio of the second couplet must be 3, and x must equal 1 third of 48. 1 third of 48 is 16.

The ratio of each couplet is 3; so each consequent must be 1 third its antecedent, and x, 1 third of 36, or 12.

The ratio of each couplet is 4; so each antecedent must be 4 times its consequent, and x, 4 times 12, or 48.

* Since the ratio is 22 (5) the consequent must be of the antecedent.

1. If 75 yd. of cloth cost $115.25, how much will 15 yd. cost at the same rate?

2. If 2 acres of land cost $76.20, how much will 15 acres cost at the same rate?

3. If 7 tons of coal can be bought for $26, how many tons can be bought for $39?

7 tons: x tons :: $26: $39.

4. If 36 lb. coffee can be bought for $7, how many pounds can be bought for $17?

5. If sugar sells at the rate of 18 lb. for $1, how much should 63 lb. of sugar cost?

6. If a post 6 ft. high casts a shadow 4 feet long, how high is that telegraph pole which at the same time and place casts a shadow 20 feet long?

7. If a post 5 feet high casts a shadow 8 feet long, how high is that steeple which casts a shadow 152 feet long?

8. If a train moves 50 miles in 1 hr. 20 min., at the same rate how far would it move in 2 hours?

9. If a boy riding a bicycle at a uniform rate goes 12 miles in 1 hr. 15 min., how far does he travel in 25 minutes?

TO THE TEACHER -After the pupil has solved the above problems by making use of the fact of the equality of the ratios, he should solve them by an analysis somewhat as follows: Prob. 1. Since 75 yd. cost $115.25, 1 yd. costs of $115.25; but 15 yd. cost 15 times as much as 1 yd., so 15 yd. cost 15 times of $115.25.

289. MAGNITUDES WHICH ARE PROPORTIONAL TO THE SQUARES OF OTHER MAGNITUDES.

The areas of two squares are to each other as the squares of their lengths.

The areas of two circles are to each other as the squares of their diameters.

Observe that the ratio of the areas of the above squares is (or 2). But the area of each circle is .78 + of its circumscribed square; so the ratio of the areas of the circles is (or 2). See Book II, p. 256.

PROBLEMS.

1. The diameters of two circles are as 3 to 4. Compare their areas.

2. The area of a 6-inch circle is how many times as great as the area of a 3-inch circle?

3. If a 4-inch circle of brass plate weighs 3 ounces, how much will a 6-inch circle weigh, the thickness being the same in each case?

4. If a piece of rolled dough 1 foot in diameter is enough for 17 cookies, how many cookies can be made from a piece 2 feet in diameter, the thickness of the dough and the size of the cookies being the same in each case?

5. If a piece of wire of an inch in diameter will sustain a weight of 1000 lbs., how many pounds will a wire of an inch in diameter sustain ?

Proportion.

290. MAGNITUDES WHICH ARE PROPORTIONAL TO THE CUBES OF OTHER MAGNITUDES.

The solid contents of two cubes are to

each other as the cubes of their lengths.

The solid contents of two spheres are to each other as the cubes of their diameters.

Observe that the ratio of the solid contents of the above cubes is (or 27). But the solid content of each sphere is .52+ of its circumscribed cube; so the ratio of the solid contents of the spheres is (or 27). See Book II, page 266.

PROBLEMS.

1. The diameters of two spheres are as 3 to 4. Compare their solid contents.

2. The solid content of a 6-inch sphere is how many times as great as the solid content of a 3-inch sphere?

3. If a 4-inch sphere of brass weighs 10 lbs., how many pounds will a 6-inch sphere of brass weigh?

4. If a sphere of dough 1 foot in diameter is enough for 20 loaves of bread, how many loaves can be made from a sphere of dough 2 feet in diameter ?

5. If the half of a solid 8-inch globe weighs 4 lbs., how much will the half of a solid 5-inch globe weigh, the material being of the same quality?

291. MAGNITUDES WHICH ARE INVERSELY PROPORTIONAL TO OTHER MAGNITUDES OR TO THE SQUARES

OF 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 as much time will be required. Hence the proportion is not 5:8 16: x, but, 5 : 8 = : 16; hence 58 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.

PROBLEMS.

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 as great, but 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 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?