18. Find in acres, etc., the area of a field if its length be 100m and breadth 75m.

19. Determine the number of cubic meters in a box 2 yds. long, 3 ft. wide, 23 ft. deep.

20. Determine the number of cubic yards in a box 2m long, 75cm wide, 50cm deep.

21. If a man walk 75m a minute, what is his rate in miles per hour?

22. If cast-iron weigh 7.113% per cubic centimeter, how many pounds does a cubic foot weigh?

23. How many steps 2 ft. 6 in. long will a man take in walking a kilometer?

24. Find the value of a carboy (17 qts.) of sulphuric acid,

of 1.841 specific gravity, at 24 cents a pound.

25. Find the value of a carboy (1711) of nitric acid, of 1.451 specific gravity, at 15 cents a pound.

26. Find the weight in pounds and in kilograms of 31 gals. of the best alcohol, specific gravity .792.

27. If the specific gravity of sea-water be 1.026, and that of olive-oil be .915, what will be the weight of a hektoliter of each in pounds and in kilograms?

28. Find the weight in pounds and in kilograms of the air, specific gravity .00129206, in a room 7m by 5m, and 3.5m high.

29. Find the weight in pounds and in kilograms of the air, specific gravity .00129206, in a room 23 ft. long, 16 ft. wide, and 10 ft. high.

30. If a balloon weigh 2kg, and contain 10,0001 of hydrogen gas, specific gravity .00008929, what is its lifting force in kilograms and in pounds when the air has a specific gravity of .00129206?

NOTE. The lifting force

=

the weight of 10,0001 of air diminished

by the weight of the hydrogen and balloon.

31. If a pile of wood be 1.2m wide, 7m long, and 2m high, how much is it worth, at $4.50 a cord?

32. How many miles will be travelled in 1 hr. 28 min. 21 sec., at the rate of 50km an hour?

33. Find the time of travelling 31 mi. 180 yds. at 1 min. kilometer.

25 sec. per

34. What is the weight of 12 cu. yds. 16 cu. ft. 720 cu. in. of earth of which a cubic meter weighs 1 t. 17 cwt.? 35. Find the weight in grams of a liter of mercury, of which

a cubic inch weighs .4925 of a pound avoirdupois. 36. How many yards of cloth, at $3.12 a meter, should be given in exchange for 15m at $2.75 a yard?

37. If a wine merchant buy 311 of wine for 1600 francs, at what rate, United States money, does he pay a gallon, reckoning 25 francs equal to $4.85 ?

38. A mill-wheel is turned by a stream of water running at the rate of a yard per second in a channel 5 ft. wide and 9 in. deep. Determine the weight of water in metric tons, supplied in 12 hrs., if a cubic foot of water weigh 1000 oz.

CHAPTER XV.

PROPORTION.

323. The relative magnitude of two numbers is called. their ratio, and is expressed by the fraction which the first is of the second.

324. The terms of this fraction are called the terms of the ratio; the numerator is called the antecedent; the denominator is called the consequent.

Thus, the ratio of 2 to 3 is expressed by, of which the numerator 2 is the antecedent, and the denominator 3 is the consequent.

325. The ratio is often written, 2:3.

326. If both terms of a ratio be multiplied or divided by the same number, the ratio is not altered.

Thus, if both terms of the ratio 2:3 be multiplied by 4, the resulting ratio is 8:12, and the ratio 8:12 is equal to the ratio 2:3; for, 12. Again, if the ratio 24: 33 be multiplied by 6, the resulting ratio is 15: 20, and the ratio 21:31 is equal to 15:20; for, -15.

Since 15 reduced to its lowest terms the ratio of 21:33 is 3:4.

21

31

=

the simplest expression for

327. If the numerator and denominator of a fraction be interchanged, the fraction is said to be inverted; likewise, if the antecedent and consequent of a ratio be interchanged, the resulting ratio is called the inverse of the given ratio.

Thus, if the fraction be inverted the resulting fraction is ; and the inverse of the ratio 3:4 is 4:3.

328. If two quantities be expressed in the same unit, their ratio will be the same as the ratio of the two numbers by which they are expressed.

Thus, the quantity $5 is the same fraction of $11 as 5 is of 11.

329. Since ratio is simply relative magnitude, two quantities different in kind cannot form the terms of a ratio.

330. Two quantities the same in kind must be expressed in a common unit before they can form the terms of a ratio.

Thus, no ratio exists between 5 tons and 30 days; and the ratio of 5 tons to 3000 pounds can be expressed only when both quantities are written as tons or pounds.

331. Ratios are mere numbers, and may be compared.

Ex. Which is the greater ratio, 5:7 or 12:18?

Now, and = 11.

As is greater than 11,

the ratio 5:7 is greater than the ratio 12:18.

332. When two ratios are equal the four terms are said to be in proportion, and are called proportionals.

Thus, 5, 3, 15, 9 are proportionals; for 5: 315: 9, since each of these ratios is represented by the fraction §.

333. A proportion is also written by putting a double colon between the ratios. Thus, 5:3 = 15: 9 (read 5 to 3 15 to 9), may be written 5:3::15:9 (read 5 is to 3 as 15 is to 9).

334. The first and last terms of a proportion are called the extremes, and the two middle terms are called the means.

335. Test of a proportion. When four numbers are proportionals, the product of the extremes is equal to the product of the means.

This is seen to be true by expressing the ratios in the form of fractions, and multiplying both by the product of the denominators.

Thus, the proportion 5:3::15:9 may be written

= 15; and if

both be multiplied by 3×9, the result will be 5 × 9 = 3 × 15.

336. Either extreme, therefore, will be equal to the product of the means divided by the other extreme; and either mean will be equal to the product of the extremes divided by the other mean. Hence, if three terms of a proportion. be given, the fourth may be found.

(1) What number is to 4 as 3 is to 6?

Answer, 25.

(3) 18 is to 32 as 45 is to what number?

As these fractions are equal, their reciprocals are equal;