Ex. 155. 1. An express-train runs 40 mi. in 64 min. At the same rate, how many miles will it run in 24 min. ? 2. If 110 A. produce 200 hhds. of sugar, how many hogsheads will 176 A. produce? 3. If 48 reapers cut 20 A. in a given time, how many acres will 156 reapers cut in the same time? 4. If 20 reapers can cut a field in 6 dys., in how many days will 30 reapers do it? 5. The number of copies in the first edition of the "Lady of the Lake" was 2050, and was to the number in the second as 41 to 69. Find the number in the second edition. 6. The length of the steamer-track from Liverpool to Quebec is 2502 mi., and is to that from Liverpool to Boston as 139 is to 155. Find the length of the track from Liverpool to Boston. 7. If a steamer from Liverpool to Portland makes the passage of 2750 mi. in 5ğ dys., in how many days, at the same rate, would the passage of 2980 mi. from Liverpool to New York have been made? 8. If a person can walk 84 mi. in 24 hrs., how many can he walk in 31 hrs.? miles 9. If the shadow of a staff 3 ft. 7 in. high is 4 ft. 9 in., find the height of a steeple whose shadow is 158 ft. 4 in. 10. A train, at the rate of 25 mi. an hour, goes a certain distance in 3 hrs. In how many hours will one at the rate of 241⁄2 mi. an hour go the same distance? 11. The ratio of the diameter to the circumference of a circle was given by Metius as 113: 355. Find the circumference of a fly-wheel 10 ft. in diameter. 12. Find the horse-power of an engine that can raise 11,200 lbs. of coal in an hour from a pit whose depth is 396 ft. NOTE. The labor necessary to raise 1 lb. through 1 ft. is called the unit of work; and a horse can do 33,000 units of work a minute. 396 × 11200 33000 × 60 Therefore one horse-power = 33,000 units of work, and 13. If 1000 sq. yds. of a field produce a load of hay, how many such loads will 25 A. of the field produce? 14. If a train runs 177 mi. 120 rds. in 3 hrs. 56 min., what is the rate per hour? 15. If 136 masons can build a fort in 28 dys., how many men will be required to build it in 8 dys.? 16. There are provisions in a fort sufficient to support 4000 soldiers for 3 mos. How many must be sent away to make them last 8 mos.? 17. A coach travels 7 mi. an hour. How many miles will it go between a quarter past ten A.M. and a quarter to six P.M.? 18. The expense of making the hay on 5 A. 135 sq. rds. is $29.08. What is the expense per acre? 19. If 300 laborers can make an embankment in 48 dys., how many more days would be required if the number of men is diminished by 60? 20. If 2.45 tons of straw cost $22.75, how be bought for $11.70? many tons can COMPOUND PROPORTION. 258. A ratio is said to be compounded of two or more given ratios, when it is expressed by a fraction which is the product of the fractions representing the given ratios. Thus the ratios 2:3 and 7: 11 are represented by the fractions and; and the ratio 14: 33, which is represented by 1 (the product of and), is said to be compounded of the ratios 2: 3 and 7: 11. 259. A proportion which has one of its ratios a compound ratio is called a compound proportion. In stating problems in compound proportion the quantity which corresponds to the answer required is made the third term. Each pair of the remaining quantities is then considered separately with reference to the answer required. The process will be understood by the following example: If 4 men mow 15 A. in 5 dys. of 14 hrs., in how many days of 13 hrs. can 7 men mow 191 A.? As the answer is to be in days, make 5 dys. the third term. I. Will it require more days for 7 men to mow 15 A. than it did for 4 men? Evidently less. Therefore make 7 the first term and 4 the second. II. Will it require more days for the same number of men to mow 19 A. than it did to mow 15 A.? Evidently more. Therefore make 15 the first term and 19 the second. III. Will it require more days of 13 hrs. to mow the same number of acres than it did of 14 hrs.? Evidently more. or Therefore make 13 the first term and 14 the second. Hence the statement is 7:4 15 : 19.5 : : 5 days: what? 4 x 19.5 × 14 × 5 days 7 x 15 x 13 This, simplified by cancellation, gives 4 days. Ex. 156. 1. If 13 bu. of oats serve 3 horses for 11 dys., how many bushels will serve 7 horses for 12 dys.? 2. If a traveller walks 140 mi. in 8 dys., walking 7 hrs. a day, how many miles can he walk in 12 dys. of 8 hrs. each? 3. If 4 masons build 27 yds. of wall in 5 dys., working 9 hrs. a day, in how many days will 32 masons build 81 yds. of a similar wall, if they work 10 hrs. a day? 4. A bootmaker who employs 15 men fills an order for 25 doz. pairs of boots in 4 wks. In how many days can he make 45 pairs if he employs 18 men? 5. If a family, by using 2 gas-burners 7 hrs. a day, pays $6 a quarter when gas is $2.40 per 1000 cu. ft., what will a family using 3 burners 4 hrs. a day pay per quarter when gas is $1.80 per 1000 cu. ft.? 6. If 330 slices of an inch thick are obtained from 12 rounds of beef, how many similar rounds will be required for 495 slices of an inch thick? 7. If 5 horses eat 8 bu. 14 qts. of oats in 9 dys., how many days, at the same rate, will 66 bu. 30 qts. last 17 horses? 8. If a man walks 600 mi. in 25 dys., walking 8 hrs. a day, in how many days will he walk 330 mi., walking 10 hrs. a day? 9. If a pane of glass 18 in. long and 12 in. wide costs 20 cts., what will be the cost, at the same rate, of a pane 221 in. long and 15 in. wide? 10. If 18 men can dig a trench 200 yds. long, 3 yds. wide, and 2 yds. deep, in 6 dys. of 10 hrs. each, in how many days of 8 hrs. each will 10 men dig a trench 100 yds. long, 4 yds. wide, and 3 yds. deep? PROPORTIONAL PARTS. 260. If it be required to divide a quantity into parts proportional to 3, 4, 5, the numbers 3, 4, 5 may be taken as representatives of the parts, and then the whole quantity will be represented by 3+4+5; that is, by 12. (1) Divide $391 into parts proportional to 5, 7, and 11. The whole quantity will be represented by 5 + 7 + 11 = 23. Therefore the respective parts will be 2, 3, 1 of $391. $85, $119, $187. Ans. (2) Divide $248 into parts proportional to, 5, 25. Multiply the fractions by 150, the L.C.M. of their denominators. The results are 15, 10, 6. sented by the numbers 15, 10, 6, Hence the parts will be repreand the whole by 31. Therefore the respective parts will be 1, 1, 1 of $248. $120, $80, $48. Ans. Ex. 157. 1. Divide 1200 into parts proportional to 11, 12, 13, 14. 2. Divide 390 into parts proportional to 1, 1, 1. 3. Divide a profit of $689 among 3 partners, of whom the first owns, the second, and the third joint stock. of the 4. Four men invest $450, $230, $190, $110 respectively in a joint business. Find their respective liabilities in a loss of $313.60. 5. Three partners claim respectively,, and of $1260. Give to each his proportional share. 6. An analysis of dissolved bones gives the following results for every 100 parts. Water, 13.97; organic matter, 15.71; soluble phosphates, 21.63; insoluble phosphates, 11.43; sulphate of lime, 15.83; sulphuric acid, 15.63; alkaline salts, 1.10; silica, etc., the remainder. Find the number of pounds of each in a ton of dissolved bones. |