7. Translate and solve the problem: Χαλκεός ἔιμι λέων, κρούνοι δὲ μοι ὄμματα δοιὰ, Καὶ στόμα, καὶ τὸ θέναρ δεξίτεροῖο πόδος. Καὶ στόμα, καὶ γλῆναι καὶ θέναρ, ειπε πόσοις. XXXIII. 1. If one person take steps of 29 inches each and another person steps of 31 inches; if the latter take 59, and the former 61 in a minute ; what is the difference of time between each person walking a mile? 2. A and B walk to meet each other from two places 100 miles distant. A walks 6 miles an hour and B 4 miles an hour. At what point on the road do they meet, and at what two times are they 50 miles apart from each other? 3. How long will a column of 10,000 men, 4 deep, require to march through a defile of 5 miles at the rate of 75 paces of 2 feet each in one minute, supposing each rank of 4 men to occupy 20 inches in depth? 4. If two trains be moving in contrary directions at the rate of 30 miles an hour, and each of them be 88 yards long, how long will they take to pass each other? But if one of the trains 88 yards long was moving 30 miles an hour, and the other 120 yards long was moving 40 miles an hour, find the time in which they would pass one another, 5. Two trains 100 yards and 150 yards long respectively, are moving in the same direction at the rates of 30 and 40 miles an hour. In what time will one pass the other? 6. If sound travel through air at the rate of 1,130 feet per second, through water at 4,700 feet per second, and through land at 7,000 feet per second, in what times could sound be transmitted a distance of 6 miles through each of these media? 7. A person saw the flash of a gun fired from a ship at sea distant a mile and 480 yards, and 5 seconds afterwards saw the flash of another gun fired from another ship in a line between the first ship and himself, and two seconds still later heard two reports simultaneously. Find the distance between the ships. 8. A person shooting at a target at a distance of 500 yards hears the bullet strike the target 4 seconds after he fired. A spectator equally distant from the target and the shooting station hears the shot strike 21 seconds after he heard the report. Find the velocity of sound. 9. If the velocity of electricity be 288,000 miles per second, what time would it take to travel round the earth, whose circumference is about 24,900 miles? 10. While Roemer was engaged in observing the eclipses of Jupiter's satellites, he found that when Jupiter was in opposition the eclipses happened 8′13′′ earlier than they should according to the astronomical tables, and when Jupiter was in conjunction these eclipses. happened 8′ 13′′ later. Find the velocity of light if the radius of the earth's orbit be 93,000,000 miles. 11. Two cogged wheels, of which one has 16 cogs and the other 30, work in each other. If the first wheel turn 18 times in 10 seconds, how many times will the other turn in 25 seconds? 12. If two bodies move in the circumference of a circle, the swifter making a revolution in 5 hours and the slower in 9, supposing they start from the some point, when will one overtake the other? 13. The periodic times of four bodies being 24, 22, 20, and 18 days respectively, in what times after leaving a conjunction will they all be again in conjunction, and what number of revolutions will each have performed? XXXIV. 1. A clock is set at 12 o'clock on Saturday night, and at noon on Tuesday it is 3 minutes too fast. Supposing its rate regular, what will be the true time when the clock strikes 4 on Thursday afternoon? 2. Find the different times at which the hour and minute hand of a clock are in conjunction, in opposition, and at right angles to one another, between noon and midnight. If the hands were alike, at what times of the day might they be mistaken the one for the other? 3. The seconds hand of a watch revolves about the same axis as the hour and minute hands. Determine all the positions in which the three hands are together, in the same straight line, and at right angles, during one revolution of the hour hand. 9 32 4. If a watch be 4 min. 82 sec. too slow at 9 hrs. 30 min. a.m. on Tuesday, and loses 2 min. 45 sec. daily, what will be the time indicated at 5 hrs. 15 min. p.m. on the following Friday? 5. A watch which is 10 minutes too fast at noon on Monday loses 3 min. 10 sec. daily. What will be the time indicated by the watch at a quarter past 10 on the morning of the following Saturday? 6. What is the magnitude of the angle between the hour and minute hand of a clock at past 11? 4 7. It is between 2 and 3 o'clock, and the hands of the clock are equally inclined to the vertical on opposite sides. In what time will they be inclined to the vertical again on opposite sides? 8. A watch set accurately at 12 o'clock indicates 10 minutes to 5 at 5 o'clock. What is the exact time when the watch indicates 5 o'clock? If it indicated 10 minutes past 5 at 5 o'clock, what would be the exact time when the hands indicated 5 o'clock? 9. One clock gains 2 minutes in 3 days, and another loses 6 minutes in 7 days; if they were set right at 12 noon to-day, when will their times differ by a quarter of an hour? 10. Two clocks begin to strike 12 together; one strikes in 35 seconds the other in 25. What fraction of a minute is there between their seventh strokes ? RESULTS, HINTS, ETC., FOR THE EXERCISES. ABSTRACT FRACTIONS. I. 1. Art. 1. 2. Arts. 1, 9. 3. The improper 3. The improper fractions are given. 4. 7. Art. 13, note. and note. 11. Art. 16. and note. 10. Arts. 15, 16, 12. The numbers which are aliquot parts of 12 are 1, 2, 3, 4, 6. Thus 1 is of 12, 2 is of 12, 3 is of 12, 4 is of 12, 6 is of 12. The other numbers less than 1 4 3 1⁄2 9, 10, 11, can be divided into two or more aliquot parts. 12, namely, 5, 7, 8, It may also be noted that the aliquot parts of 20 and of 100 are much employed, in calculations connected with mercantile transactions. II. The following are the fractions in the lowest terms found by dividing the numerator and denominator of each fraction by their greatest common measure respectively: 11 55 4 55 1309 60 2993 23 19 131 276 8 7 13 61 10996, 20' 64' 59' 8' 360' 77' 3321' 33' 34' 191' 397' 9' 9' 14' 211' 34661 11761 1. 4. 2. 15. 73 5. 1212. 6. 72. 7. 21. 8. 519. 9. 10. 465121. 11. 111. 12. 473. 13. 30541. 14. 3536. 15. 33312. 16. Sig. 17. 5373. 18. 1. 19. 2514. 20. 21. 21. 1%. 23. 43-23. 1260 Note.-The four fractions in example 18 may be more readily added together by resolving the denominators of the fractions into their prime factors. 1. 1. 2. 2. 1. 3. 4. 4. §. 5. 74. 4. 5 5 84 6. 31. 6. 31. 7. 252. 8. 1. 9. 1713379. 10. 24. 11. 3060. 12. 13. 447. 14. 1. 15. 419. 16. 1931. 17. 3460 18. 11547. 19. 1. 1 21 1. Art. 12, note, 1 is greater than 33, and 33 is greater than 11. 18 2. 253, 226, 39. 3.1th part. 4. Their relative magnitudes will be obvious when the fractions are reduced to the same denominator. 5. Art. 7. 6. Art. 12, note. 7. If 3 be added to the numerator and denominator of the proper fraction or that the proper fraction is increased by adding the same number 3 to its numerator and denominator. 2 In a similar way it may be shewn that the proper fraction is diminished by subtracting the same number 3 from its numerator and denominator. The remaining examples 8, 9, 10, 11, 12, offer no difficulties. With respect to Ex. 12 see Art. 7. 26. 4. 143. 5. The sum of the fractions is: and the lowest fraction required. with denominator 1000, will be next greater than, which will be found to be 728 7. 18. 8. 632. 9. 1 excess, 1 of 7. 10. 300. 11. can be subtracted 11 times 3 5 7 1. from 193, and leaves a remainder 10%. 2001 200. 200 12. 3+5+7=15, and X. 2. The difference is the product 1. 3. 4. 4. 10. 4 1 420 6. 11. 7. The quotient is greater than the product. 5. 467 7907 1. 212378. 2. 11830. 3. 1228. 4. 2443. 5. 215. 6. 1404. 7. 907 44100 25 XII. 1913 1800* The operations indicated in the first four expressions offer no difficulty. 23.29 3.37 2.13 5. The expression is equivalent to 3.3.37.23.29 4.11.72.5.13. 6. The left side of the expression can be put under the form 1, 2, 3. Art. 17, and note. 7. 42 and 15. 8. 13. XIII. 4. The direct and converse. 9. 144. 12. 3. 13. The direct and converse. 2769 3020 7 295 378 59 £200 15s. 6d., and 17 or whole estate = £3413 3s. 6d. a sovereign. 10. maining share, and its value £3080. A received £2 5s. 10d., and B £2 1s. 8d. XV. 1. See on Multiplication and Division of Concrete integers. 2. 1 of £1= £1 12s. 11d. 3. The shorter course is to add of the sum to thirteen times the 3 The result is £89 6s. 117d. 4. 1640831. 5. 1243 of £1. 6. £5 ls. 11⁄2d. 7. 2633s. 8. The quotient 91% indicates the number of times the latter sum is con sum. 2440 |