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other 4 parts of water and 3 parts of spirit. When the contents of the two glasses are mixed in a tumbler, find how many parts of the whole mixture are wine and water.

7. Four gallons of wine and one of water are poured into a vessel, and one gallon of the mixture is poured out into a second vessel; then one gallon of water is poured into the first vessel, and the process is performed four times. Determine the quantity of wine in each vessel.

8. If a cubic foot of atmospheric air weigh an ounce and a quarter under the ordinary pressure, what would be the weight of air contained in a room 30} feet long, 20} wide, and 104 high?

9. If the pressure of the atmosphere on a square inch at the surface of the earth be about 14 pounds 11 ounces avoirdupois, what is the pressure on a square foot and on a square yard ?

10. What is the pressure on a square foot at the bottom of a pond 20 feet deep, if the weight of a cubic foot of water be 1,000 ounces? If the pressure of the atmosphere be taken into account, what is the total pressure ?

11. If the weight of a cubic foot of water be 1,000 ounces avoirdupois, and mercury be 13] times as heavy as water, find the height of a column of water which shall be equal to the pressure of mercury in the barometer when it stands at 301 inches.

XXXII. 1. If a vessel of water be emptied in 5 hours through a spout A, and in 3 hours through a spout B, in what time will it be emptied through both spouts together ?

2. If a cistern when full of water can be emptied in 15 minutes by a pipe, and when empty can be filled by another in 20 minutes; if the cistern be full, in what time can it be emptied by both pipes being opened at the same time?

3. If a pipe which conveys 13 gallons in a minute, lower the surface of water in a tank 11 inches in 45 minutes, in what time will a pipe which conveys 4 gallons in minute lower the surface 5 inches?

4. A cistern is filled by two spouts in 20 minutes and in 24 minutes respectively, and emptied by a tap in 30 minutes; if the cistern be empty, what portion of the cistern will be filled in 15 minutes, when all three are opened together?

5. A cistern has 3 pipes, A, B, C; A will fill it in 3 hours, B will fill it in 4, and C will empty it in 1 hour. The cistern being empty, these pipes are opened at 1, 2, and 3 o'clock respectively. At what time will the cistern be full or empty, and which ?

6. In what part of a day will four fountains, being opened together, fill a cistern, which if severally opened, they would each fill it in one day, half a day, the third and the sixth part respectively ? (Lilavati).

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7. Translate and solve the problem:

Χαλκεός έιμι λέων, κρούνοι δέ μοι όμματα δοιά,

Και στόμα, και το θέναρ δεξίτεροϊο πόδoς.
Πλήθει δε κρητηρα δυ' ήμασι δεξίον όμμα,

Και λαιόν τρισσδις, και πισύρεσσι θέναρ:
"Αρκιον έξ ώραις πλήσαι στόμα, ει δ' άμα πάντα,
Και στόμα, και γλήναι και θέναρ, ειπέ 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 2 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.

4. If a watch be 4 min. 8 '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 ?

7. It is between 2 and 3 o'clock, and the hands of the clock arə 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.

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I. 1. Art. 1. 2. Arts. 1, 9. 3. The improper fractions are given. 4. 12+9+6 12 x 9 x 6

1 4+3+2=9, --4X9X6 = 216, and 9 of 216. 5. Art. 3 3

24 11 and note. 6. Art. 12. 7. Art. 13, note. 8. Art. 15 and note. 9. Art. 10 and note. 10. Arts. 15, 16, and note. 11. Art. 16.

12. The numbers which are aliquot parts of 12 are 1, 2, 3, 4, 6. Thus 1 is i of 12, 2 is of 12, 3 is 4 of 12, 4 is į of 12, 6 is 1 of 12. The other numbers less than 12, namely, 5, 7, 8, 9, 10, 11, can be divided into two or more aliquot parts. 5 4 1 1,1

9 6 3 1.1 1 Thus,

12 3

+
4 3

+
12 12 12

of

2
7 6 1 11 1 10 6 4 1 1
+

+
12 12' 12

of
2

2 12 1212273
8 6 2 1 1 1 11 6 3 2 1 1 1
+

of
12

+
12'12
2 12 12 12+12

2 It may also be noted that the aliquot parts of 20 and of 100 are much employed in calculations connected with mercantile transactions.

12 12+

2+2

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4

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' 7' 7' 14' 211' 34661 11761 122387

III. The following are the fractions as required :5 3 8

9 35 1.

168 240

12

20
and

175
and
and

and
15

12 12 105' 105 105 420' 420 420
4680 2457 2912 5544, 2376 2475 1760
and

2880

and
6552' 6552 6552 6552 3960' 3960' 3960 3960
6

8 12 7 20 18 3 5 40 75 35 51 2. and and and

and 9

18 18 18 36' 36' 36 36 100'100' 100 100 160 375 175 16

3

and
1000 1000 1000' 1000 1000
15

28 30 24 25
and

45 42 and

44. 105 98 3.

56

24 and

and 100 100 600' 600 600' 72' 72 72 168 168' 168 168 120 100 32 10 75

:

:

:

15

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:

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:

and
240' 240' 240, 240 240

IV. 1. 2. 15. 3. 731. 4. 557136

6. 771:

7. 23. 1sti. 10. 46534}. 11. 14. 12. 4778b 13. 30541. 14. 3536 i's.

5. 1415

8. 5.1. 9.

15. 33333. IX. 1. Reduce to single fractions and find the sum. 2. The result is 484963. 3..

1303151 16 6 4 6 3 X 139 3

16. 8. il. 17. 51166. 18. 1. 19. 2511. 20. 11. 21. 11. 22. 23. 43116

Note.—The four fractions in example 18 may be more readily added together by resolving the denominators of the fractions into their prime factors. 7 29 1

626 Here

+ 2431

11x13x17x19 589 3446 1 589 3146

45563 And

st
3553' 4199 17 x 19 11

+

13 11x13x17x19

46189 46189 Sum of the four fractions=;

=1. 11x13x17x19 46189

}-11 {

}

V. 1. 11. 2. 23. 3. 136. 4. 1078. 5. 12. 6. 122. 7. og 8. 23841. 9. zus. 10. 7. 11. 42475. 12. 34534. 13. 2174. 14. 234114. 15. 195935..

VI. 1. £. 2. tit. 3. 3. 4. 4. 5. 47. 6. 31. 7. 252. 8. br. 9. 171337927. 10. 1. 11. 3060. 12. z. 13. 417. 14. 1. 15. 421. 16. 1933). 17. 346030. 18. 11586 19. 3:

20. •

21. 1783}. 22. 18887467. 23. 10000200000. 24. 9090909011.

VII. 1. 1. 2. 31. 3. 4. 13. 5. j. 6. 14. 7. 216. 8. 15. 9. 1. 10. 21. 11. 215031. 12. 8727.

VIII.

10

or

or

1. Art. 12, note, jy is greater than zł, and it is greater than . 2. *', 359, 3. 3. Ith 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 1 It becomes 7+3

2
12+3 15
And f is greater or less than the
According as it is greater or less than Ta. Art. 12.
But i, is obviously greater than to
It follows that
7+3

7
is greater than
12+3

12' or that the proper fraction r is increased by adding the same number 3 to its numerator and denominator.

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.

4. złig. 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 1938. 7. 7. 8. 621. 9. 1$ excess, ji of 7. 10. 300. 11. 20% can be subtracted 11 times.

3 5 7 from 187, and leaves a remainder 2006 12. 3+5+7=15, and +

15' 15'15

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