4. Illustrate graphically, as in Ex. 9, the point where and time at which the two trains given in the annexed time-table pass each other.

5. A cyclist starts at 7 A.M. from a town and rides 2 hr. at the rate of 10 mi. an hour. He rests 1 hr. and then returns at the rate of 9 mi. an hour. A second cyclist leaves the same place at 8 A.M. and rides at the rate of 6 mi. an hour. When and where will they meet?

6. Two cyclists start from the same place at the same time. The first rides for 2 hr. at the rate of 9 mi. an hour, rests 15 min., and then continues at 6 mi. an hour. The second one rides without stopping at the rate of 7 mi. an hour. Where will the second cyclist overtake the first?

7. The average yield of wheat per acre in the United States for the years from 1893 to 1903 in bushels was as follows: 11.4, 13.2, 13.7, 12.4, 13.4, 15.3, 12.3, 12.3, 15.0, 14.5, 12.9. The highest Chicago cash price per bushel for the same years given in cents was: 64.5, 635, 642, 931, 109, 70, 691, 755, 791, 773, 87. Illustrate graphically, putting the two curves in one figure.

8. The average yield of corn per acre in the United States for the years from 1893 to 1903 in bushels was as follows: 22.5, 19.4, 26.2, 28.2, 23.8, 24.8, 25.3, 25.3, 16.7, 26.8, 26.5. The highest Chicago cash price per bushel for the same years given in cents was: 361, 474, 26, 23, 271, 38, 311, 401, 671, 571, 431. Illustrate graphically, putting the two curves in one figure.

9. The average summer daily temperature in Paris at the foot and top of the Eiffel tower in 1900 was as follows:

2 4 6 8 10 N. 2

6

57.2° 55.4° 58.1° 63.5° 67.8° 69.8° 70.1° 69.8° 68°

8 10 Mt. 62.1° 60.7° 58.9°

57.4° 55.7° 57.2° 58.1° 60.8° 63.5° 63.9° 64° 64.4° 61.2° 60.7° 59.1°

Illustrate graphically, putting the two curves in one figure.

RATIO AND PROPORTION

218. The ratio of one number to another of the same kind is their quotient. The former number is called the antecedent, and the latter the consequent. The terms of the ratio therefore bear the same relation to each other as the terms of a fraction. Thus, the ratio of a to b may be written a : 6 (read the ratio of a to b),

α

forms a : 6, and 2, are generally used.

α

%

b

or a ÷ b. The

The ratio of 3 ft.

to 5 ft. is 3: 5. This may also be expressed by or 0.6.

219. The ratio is always an abstract number, since it is the relation of one number to another of the same kind. There can be no ratio between 5 hr. and $10, nor between 7 lb. and 6 ft. But there can be a ratio between 3 ft. and 6 in., since the quantities are of the same kind. Both terms must, however, be reduced to the same unit. Thus, 3 ft. = 36 in., and 36 in.: 6 in. = 36: = 6.

The ratio is called the inverse or reciprocal of the

ratio

EXERCISE 48

1. How is the value of a ratio affected by multiplying or dividing both terms by the same number?

2. How is the value affected by multiplying or dividing the antecedent? by multiplying or dividing the consequent?

13. What number has to 10 the ratio 2? to 5 the ratio 0.3?

16. Which ratio is the greater, or ? 13 or 14?

17

17. The ratio of the circumference of a circle to its diameter being 3.1416, find the diameter of a circle whose circumference is 125 ft. correct to inches.

18. A map is drawn on the scale of 1 in. to 75 mi. In what ratio are the lengths diminished? In what ratio is

the area diminished?

19. Two rooms are 14 ft. long, 12 ft. wide, and 12 ft. long, 10 ft. wide respectively. What is the ratio of the

cost of carpeting them?

20. What is the ratio of a square field 20 rd. on a side to one 25 rd. on a side?

21. What is the ratio of the circumferences of two circles whose diameters are 2 in. and 4 in.? of two circumferences whose diameters are 5 in. and 7 in.? of two circumferences whose diameters are d and d'? Hence in general the ratio of two circumferences is equal to what?

22. What is the ratio of the areas of two circles whose radii are 3 in. and 5 in.? of the areas of two circles whose radii are 4 in. and 6 in.? of the areas of two circles whose radii are r and r? Hence in general the ratio of the areas of two circles is equal to what?

23. What is the ratio of the volumes of two spheres whose radii are 2 in. and 3 in.? of the volumes of two spheres whose radii are 5 in. and 6 in.? of the volumes of two spheres whose radii are r and r? Hence in general the ratio of the volumes of two spheres is equal to what?

220. Specific Gravity. The specific gravity of a substance is the ratio of its weight to the weight of an equal volume of some other substance taken as a standard.

221. Distilled water at its maximum density, 4° C., is the standard of specific gravity for solids and liquids.

222. Since 1cm3 of water weighs 1 gram, the same number that expresses the weight of any substance in grams will also express its specific gravity. Thus, 1cm3 of water weighs 18; hence, 1 is the specific gravity of water. 1 cm3 of lead weighs 11.358; hence, this being 11.35 heavier than an equal volume of water, the specific gravity of lead is 11.35.

1 cu. ft. of water weighs about 1000 oz., or 62.5 lb.

. 0.845

Ice.

.0.92

. 0.852

Rock Salt

.2.257