(1. Let the number. Then or, putting this in the form of a proportion, 号+3 11.3.2: +37 9. III. I. II. III. I. [the product of the extremes. 4..+21, the product of the means being equal to 5. 18-14-4-21, 6. of 21=51, and 7. 2-2 times 5-10 the number. .. 10 the number. If 7 lb. of coffee is equal in value to 5 lb. of tea, and 3 lb. of tea to 13 lb. of sugar, 39 lb. of sugar to 24 lb. of rice, 12 lb. of rice to 7 lb. of butter, 8 lb. of butter to 12 lb. of cheese; how many lb. of coffee are equal in value to 65 lb. of cheese? 1. 7 lb. of coffee=5 lb. of tea, 2. 3 lb. of tea 13 lb. of sugar, 5. 8 lb. of butter=12 lb. of cheese, and 7. 5X13X24X7X12 =39 lb. .. 65 lb. of cheese-39 lb. of coffee. I can keep 10 horses or 15 cows on my farm; how many horses can I keep if I have 9 cows? 15 cows: 9 cows: 10 horses: ?=6 horses. 10 horses-6 horses=4 horses. ... I can keep 4 horses with the 9 cows. I. If 2 oxen or 3 cows eat one ton of hay in 60 days, how long will it last 4 oxen and 5 cows? 2 oxen 4 oxen :: 3 cows: ?=6 cows. If a ton of hay last 3 cows ... 4 oxen eat as mnch as 6 cows. 60 days, it will last 6 cows, which are equal to 4 oxen, cows, or 11 cows, not so long. .. 11 cows: 3 cows:: 60 days: ?=173⁄4 days. and 5 I. If 24 men, by working 8 hours a day, can, in 18 days, dig a ditch 95 rods long, 12 feet wide at the top, 10 feet wide at the bottom, and 9 feet deep; how many men, in 24 days of 12 hours a day, will be required to dig a ditch 380 rods long, 9 feet wide at the top, 5 feet wide at the bottom, and 6 feet deep? 95 rods: 380 rods 24 days 18 days 12 hours: 8 hours 12 feet 9 feet 10 feet: 5 feet 9 feet 6 feet :: 24 men: ?=12 men. L. A Louisville merchant wishes to pay $10000, which he owes in Berlin. He can buy a bill of exchange in Louisville on Berlin at the rate of $.96 for 4 reichmarks; or he is offered a circular bill through London and Paris, brokerage % at each place, at the following rates: £1-$4.90 25.38 francs, and 5 francs-4 reichmarks. What does he gain by direct exchange? 1. $.238-1 mark. 2. $10000 10000-238-42016.807 marks. 3. $.24=1 mark, since this is the rate of exchange. 5. 42016.807 marks (?=$10165.38.) 10. $10165.38-$10084.033-$81.35= gain by direct ex change. III. ... $81.35-gain by direct exchange. I. A wheel has 35 cogs; a smaller wheel working in it, 26 cogs; in how many revolutions of the larger wheel will the smaller one gain 10 revolutions? 1. 35 cogs-26 cogs 9 cogs what the smaller wheel gains on larger in 1 revolution of larger wheel. 2. 26 cogs passed through the point of contact=1 revolution of smaller wheel. 3. 1 cog passed through the point of contact tion of smaller wheel. revolu II.4. 9 cogs passed through the point of contact revolu III. tion of smaller wheel. 2 5. . In 1 revolution of larger wheel the smaller gains revolution of smaller wheel. 6... revolution gained 10 revolutions gained: 1 revolution of larger wheel: ?=288 revolutions of larger wheel. .. The smaller wheel will gain 10 revolutions in 288 revolutions of larger wheel. By analysis and proportion. 26 cogs passed through the point of contact-1 revolution of the smaller wheel. 35 cogs passed through the point of contact 1 revolution of the larger wheel. But when the larger wheel has made 1 revolution, 35 cogs of the smaller wheel have passed through the point of contact. If 26 cogs having passed through the point of contact make 1 revolution of the smaller wheel, how many revolutions will 35 cogs make? By proportion, 26 cogs : 35 cogs 1 rev. ?=1 rev. .. The smaller wheel makes 12 revolutions while the larger wheel makes 1 revolution. .. The smaller gains 1 revolutions. -1 revolution revolution. If the smaller wheel gains revoluion in 1 revolution of the larger wheel to gain 10 revolutions on the larger wheel, the larger wheel must make more revolutions. .. less greater less greater. 2 rev. 10 rev. :: 1 rev. of larger: ?=288 rev. of larger. I. If the velocity of sound be 1142 feet per second, and the number of pulsations in a person 70 per minute, what is the distance of a cloud, if 20 pulsations are counted between the time of seeing the flash and hearing the thunder? II. III. 1. 1142 ft.=distance sound travels in 1 second. 2. 68520 ft.=60×1142 ft.=distance sound travels in 1 min., or the time of 70 pulsations. 3. .. If it travels 68520 feet while 70 pulsations are counted, it will travel not so far while 20 pulsations are counted. 4. .. greater less greater less. [145 yd. 24 ft. 5. 70 pul. 20 pul. :: 68520 ft.: ?=195774 ft.=3 mi. 5 fur. .. The cloud is 3 mi. 5 fur. 145 yd. 24 ft. distant. (R., 3d p., p. 289, prob. 45.) PROBLEMS. 1. If 3 horses, in of a month eat of a ton of hay, how long will of a ton last 5 horses? 2. If a 4-cent loaf weighs 9 oz. when flour is $6 a barrel, how much ought a 5-cent loaf weigh when flour is $8 per barrel? 3. A dog is chasing a hare, which is 46 rods ahead of the dog. The dog runs 19 rods while the hare runs 17; how far must the dog run before he catches the hare? 4. If 52 men can dig a trench 355 feet long, 60 feet wide, and 8 feet deep in 15 days, how long will a trench be that is 45 feet wide and 10 feet deep, which 45 men can dig in 25 days? 5. If of 12 be 3 what will 6. If 3 be of 12, what will of 40 be? of 40 be? Ans. 15. Ans. 6 7. If 18 men or 20 women do a work in 9 days, in what time *can 4 men and 9 women do the same work? Ans. 1347 days. 8. If 5 oxen or 7 cows eat 3 tons of hay in 87 days, in what time will 2 oxen and 3 cows eat the same quantity of hay? Ans. 105 days. g. Divide $600 between three men, so that the second man shall receive one-third more than the first, and the third more than the second. 10. Two men in Boston hire a carriage for $25, to go to Concord, N. H., and back, the distance being 72 miles, with the privilege of taking in three more persons. Having gone 20 miles, they took in A; at Concord they took in B; and when within 30 miles of Boston, they took in C. How much shall each pay? Ans. First man, $7.609103; second, $7.609103; A, $5.873; B, $2,8647; and C, $1.041. 11. Three men purchased 6750 sheep. The number of A's sheep is to the number of B's sheep as is to 33, and 4 times the number of C's sheep is to the number of A's sheep as is to Find the number of sheep each had. 12. Ans. Į A's= B's: C's= If $500 gain $10 in 4 months, what is the rate per cent? Ans. 6%. 13. If 12 men can do as much work as 25 women, and 5 women do as much as 6 boys; how many men would it take to do the work of 75 boys? Ans. 30 men. 14. If 5 experienced compositors in 16 days, 11 hours each, can compose 25 sheets of 24 pages in each sheet, 44 lines on a page, 8 words in a line, and 5 letters to a word; how many inexperienced compositors in 12 days, 10 hours each, will it take to compose a volume (to be printed with the same kind of type), consisting of 36 sheets, 16 pages to a sheet, 112 lines to the page, 5 words to a line, and 8 letters to a word, provided that while composing an inexperienced compositor can do only as much as an experienced compositor, and that the latter work is only as hard as the former? 15. If A can do as much in a day as B, B can do as C, and C can do as much as D, and D can do as E, and E can do as much as F; in what time can F do work as A can do in 28 days? Ans. 16. as much much as as much Ans. 8. 16. A starts on a journey, and travels 27 miles a day; 7 days after, B starts, and travels the same road, 36 miles a day; in how many days will B overtake A? Ans. 21 days. 17. A wheel has 45 cogs; a smaller wheel working in it, 36 cogs; in how many revolutions of the larger wheel will the smaller gain 10 revolutions? Ans. 40. 18. If the velocity of sound be 1142 feet per second, and the number of pulsations in a person 70 per minute, what is the distance of a cloud, if 30 pulsations are counted between the time of seeing a flash of lightning and hearing the thunder? Ans. 5 mi. 108 yd. 14 ft. 19. If William's services are worth $15 a month, when he labors 9 hours a day, what ought he to receive for 4 months, when he labors 12 hours a day? Ans $91.913. 20. If 300 cats kill 300 rats in 300 minutes. how many cats will kill 100 rats in 100 minutes? Ans. 300 cats. CHAPTER XVII. ANALYSIS. 1. Analysis, in mathematics, is the process of solving problems by tracing the relation of the parts. I. What will 7 lb. of sugar cost at 5 cents a pound? Analysis for primary classes. If one ponnd of sugar costs 5 cents, 7 pounds will cost 7 times 5 cents, which are 35 cents. I. If 6 lead pencils cost 30 cents, what will one lead pencil cost? Analysis: If 6 lead pencils cost 30 cents, one lead pencil will cost as many cents as 6 is contained into 30 cents which are 5 cents. I. If 8 cranges cost 48 cents, what will 5 oranges cost? Analysis: If 8 oranges cost 48 cents, one orange will cost as many cents as 8 is contained into 48 cents which are 6 cents; if one orange costs 6 cents 5 oranges will cost 5 times 6 cents, which are 30 cents. I. If a boy 'had 7 apples and ate 2 of them, how many had he left? Analysis: If a boy had 7 apples and ate 2 of them, he had left the difference between 7 apples and 2 apples which are 5 apples. I. If John had 12 cents and found 5 cents, how many cents did he then have? Analysis: If John had 12 cents and found 5 cents, he then had the sum of 12 cents and 5 cents which are 17 cents. |