6. Divide 79 bushels 1 peck 7 quarts by 23.

bu. pk. qt. bu. pk. qt. pt.

23)79 1 7 (3 1 7 1

69

10

4

4X10 40, and 1 peck added makes 41, &c.

23)41(1 peck.

23

18

8

23)151(6 quarts.

138

13

2

23)26(1 pint.

23

Rem. 3 pints.

7. A boat load of corn, containing 4927 bushels 3 pecks, is owned equally by 29 persons; what is the share of each?

Ans. 169 bu. 3 pk. 5 qt. 1 pt., and 1 pt. rem. 8. Divide 542 pounds 7 shillings and 10 pence by 97.

Ans. £5 11s. 10d.

9. Divide 123 pounds 11 shillings 24 pence by 127.

Ans. £0 19s. 5d.

10. Divide 330 hundred weight 3 quarters by 14.

Ans. 37 cwt. 3 qrs. 18 lbs.

11. If 35 pieces of cloth, of equal quality, contain 971 yards and Ans. 27 yd. 3 qr.

1 quarter, how many yards in a piece?

12. If 259 acres 1 rood 10 rods of land be divided into 36 equal lots, how much land will be contained in a lot?

Ans. 7 A. 0 R. 32 r. 13. If 56 pounds of butter cost 4 pounds 18 shillings, what is it per pound? Ans. 1s. 9d. 14. Divide 124 pounds 5 shillings and 4 pence into 32 equal parts. Ans. £3 17s. 8d.

15. Divide 336 bushels 3 pecks 4 quarts by 70.

Ans. 4 bu. 3 pk. 2 qt. 16. Divide 336 bushels 3 pecks 4 quarts, by 4 bushels 3 pecks and 2 quarts. Reverse the preceding problems, taking the answers for divisors,

Ans. 70.

and the former divisors will be quotients; but to effect the division, reduce all to the lowest denomination mentioned, and divide as in simple division.

If more clear, we may enunciate the 16th example thus: A farmer has 336 bushels 3 pecks 4 quarts of wheat in his granary, which he wishes to put in casks to send away, each cask containing 4 bushels 3 pecks and 2 quarts; how many casks will be required?

Ans. 70 casks.

17. A certain secret association in Ireland paid a bill of £4 18s., by assessing its members 1s. 9d. apiece; how many members were there? Ans. 56.

N. B. This is but a variation of example 13; and in like manner every example, up to 16, may be varied.

18. I have a pitcher which holds 2 quarts and 1 pint; how many times can it be filled from a barrel of cider, which holds 31 gallons 2 quarts? Ans. 50%. 19. How many times will 13 bushels fill a vessel which holds 1 quart 1 pint and 1 gill? Ans. 256 times. 20. How many suits of clothes, each requiring 3 yards 1 quarter and 2 nails, can be made from a roll of cloth containing 27 yards?

MISCELLANEOUS EXAMPLES,

Applicable to the preceding principles.

1. At 5 cents a quart, what will 5 bushels cost?

Ans. 8.

Ans. $8.

2. At 3 cents a pint, what will 3 pecks and 6 quarts cost?

Ans. 90 cents.

3. At 5 cents a pint, how many gallons can be bought for 10 dollars?

Ans. 25.

N. B. As the pupil has been instructed from the very first of this work, from simple addition, that 10 mills make a cent, 10 cents make a dime, and 10 dimes make a dollar, and that these denominations are the same in order as the order of simple numbers; therefore, the reduction from dollars to cents is to multiply by 100; dollars into mills,

multiply by 1000; and reduction the other way is to divide by these numbers.

Some arithmetical writers have treated Federal Money as compound numbers, and have gone through all the formality of reduction-addition, subtraction, multiplication, and division of federal money-the same as they do the really compound numbers, pounds, shillings, and pence.

But, federal money was purposely adjusted to the scale of simple numbers; and if it is now proper to treat these denominations as compound, we must suppose the design not accomplished. Federal money belongs to whole numbers and decimal fractions; and the subject must be incomplete until we pass decimal fractions.

Example 3, and most of the examples here inserted, should be done by canceling, as explained in article 24.

4. At 8 cents a gill, how many gallons will 12 dollars purchase?

Ans. 41.

5. What will 10 ounces 10 pennyweights cost, at 15 cents a pennyweight? Ans. $31.50. 6. At 20 cents a square rod, what will 2 acres 3 roods of land cost?

7. At $2 a square rod for land, what must be paid 12 rods long and 5 rods wide?

Ans. $88. for a village lot Ans. $132.

8. In 11 bars of gold, each containing 5 pounds 3 ounces 2 pennyweights, how many grains? Ans. 333318. 9. At 5 cents an ounce, what will 10 pounds 4 ounces of copper cost? Ans. $8.20. 10. How many kegs, each holding 4 gallons 2 quarts, can be filled from a hogshead containing 63 gallons? Ans. 14. 11. At 64 cents a quart, what will 6 bushels and 1 peck cost? Ans. $12.50.

12. How much will 4 barrels of molasses cost, at 4 cents a pint.

Ans. $10.08.

13. Divide 2 hours 10 minutes by 5 minutes 5 seconds.

14. How many times is 13° 20' contained in 360° ?

Ans. 255.

Ans. 27 times.

15. If the moon moved 13° 20' in 1 day, how many days would it require to make a revolution of 360°? Ans. 27 days.

16. If Jupiter changed its longitude 5 minutes of a degree, as seen from the sun, how many years, of 360 days each, would it require to make a revolution? Ans. 12. 17. At 8 cents a pint for wine, how many gallons can be bought for 40 dollars?

Ans. 62.

Ans. 10.

18. At 10 cents a nail, how many yards of cloth can be bought for 16 dollars? 19. At 12 cents a quart, how many gallons can be bought for 12 dollars? Ans. 24.

G

20. How many little squares, 3 inches long and 3 inches wide, can be cut from a square yard of paper?

Ans. 144. 21. How many bottles, each containing 1 quart 1 gill, will be required to draw off a barrel of cider containing 31 gallons?

Ans. 112.

22. Divide 421 pounds 14 shillings and 8 pence among 3 men, 5 women, and 7 boys, and give each man double of the sum given to a woman, and each woman 3 times the sum given to a boy; how much is the share of each ?

Ans. Each boy must have £10 10s. 103d. Each woman, : 31 12s. 71d. 53 5s. 22d.

Each man,.

The art of working the preceding problem consists in obtaining the divisor, which is 40.

23. A man bought a chaise, horse, and harness, for 70 pounds. He gave twice as much for the horse as for the harness, and twice as much for the chaise as for the horse; what did he give for each?

Ans. Harness £10. 24. A farmer sold some calves and some sheep for 108 dollars; the calves at 5 dollars, the sheep at 8 dollars apiece; there were twice as many calves as sheep,-what was the number of each sort?

Ans. 6 sheep and 12 calves.

25. The planet Jupiter changes its mean longitude 4° 54′ 18′′ in 59 days; how far will it change in one day? Ans. 4' 59" 2.

26. The moon is observed to move over 197° 38′ 45′′ in 15 days; how far will it move in one day? Ans. 13° 10′ 35′′

27. In 365 days, the planet Saturn will change longitude 12° 15′ 37"; how much will it change in one day? Ans. 2' 0" 91.

28. If the apparent motion of the sun be 59′ 8′′ in one day, how many days will it require to make a revolution of 360°?

Ans. 36534

887

29. The moon changes her longitude 13° 10′ 35′′ in one day; how many days, then, will be required to make one revolution?

Ans. 27 d. 7 h. 43 m. 30. If Venus changed her longitude, (as seen from the sun,) 1o 36' per day, what, then, would be the time of her revolution?

Ans. 225 days.

SECTION III.

FRACTIONS.

(ART. 33.) A part of any one thing is called a fraction.

If an apple, for instance, be divided into 3 equal parts, each part will be one-third, written thus, §.

If it be divided into 4 equal parts, each part will be one-fourth, written 4.

If divided into 5 equal parts, each part will be; two of these parts must be written 2.

Thus, generally, a fraction must be expressed by two numbers one above another. The lower number denotes the number of equal parts into which the unit is divided.

The upper number shows how many of these equal parts are taken.

Hence, as the lower number of a fraction denotes or decides the denomination, whether it be thirds, fourths, fifths, or any other number, it is called the denominator.

The upper number is called the numerator, because it shows the number of parts taken.

(ART. 34.) A fraction may be considered as the result of an impossible division. Thus, 1, one thing, or unity, divided into 5 equal parts, we write I above and 5 under it, or . Three times this is; or, we may consider 3 divided by 5, the quotient is 3.

Hence, the denominator of a fraction may be considered as a divisor, and the numerator as a dividend, and the fraction itself as the result, or quotient, to an example in division.

When the dividend and divisor are equal, the quotient is I; that is, when the numerator and denominator are equal, we have the whole of the thing, the whole as an aggregate or unit.

3

(ART. 35.) When any fraction is before us, as, we judge of its value by comparing its numerator with its denominator; if the numerator is equal to the denominator, as we have just observed, the value of the fraction is 1; if the numerator is nearly equal to the denominator, the value of the fraction is nearly 1; if the numerator is one-half of the denominator, the value of the fraction is one-half. Hence, is the same as .

(ART. 36.) As the value of a fraction depends upon the relation of the numerator to the denominator, and as this relation is not changed by dividing both numbers by the same divisor, we may, there