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3. A farmer has 1000 bushels of apples, which he puts into 350 barrels. How many does each barrel hold? Ans. 2bu. 3pk. 34qt.

4. If it require 1 sheet of paper to print 24 pages of a book, how many reams, allowing 18 quires to the ream, will it take to print 3000 copies, of 250 pages each?

Ans. 72 reams, 6 quires, 2 sheets.

5. An estate worth £2570 is to be divided as follows: the widow has one third of the whole, the remainder is to be divided equally between seven children. How much does the widow receive, and how much does each child have? The widow has £856 13s. 4d.

Ans.

Each child has £244 15s. 2d. 34 far.

6. Divide 100 acres, 3 roods, 8 rods of land, between four persons, A, B, C, and D, so that A shall have one sixth of the whole, B one fourth of the remainder, C one third of what then remains, and D the rest. How much does

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7. A, B, C, and D, having 13cwt. 1qr. 4lb. of sugar, they agree to divide it as follows: A is to have one half of the whole, B is to have one third of the remainder, C is to have one fourth of what then remains, and D is to take what is left. What were their respective portions?

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8. What is the weight of the following coins: 10 guineas, each weighing, 5 pwt. 91 grains; 7 sovereigns, each weighing 1 pwt. 81 grains?

Ans. 3oz. 3pwt. 8 gr. of gold. 9. What is the weight of 13 crowns, each weighing 18 pwt. 4 grains; 14 shillings, each weighing 3 pwt. 15 gr.; 9 sixpences, each weighing 1 pwt. 19, gr.?

Ans. 1lb. 3oz. 3pwt. 15gr. of silver.

10. In one eagle there is 232 grains of pure gold, 12 grains of silver, and 12% grains of copper, and the same proportions of gold, silver and copper, from all other American gold coin. In 10 eagles, 7 half-eagles, 5 quartereagles, how many grains of gold, silver and copper?

3424.95 gr. of gold.

Ans. 190-275 gr. of silver.

190-275 gr. of copper.

11. One pound of pure gold is sufficient for how many

dollars of gold coin, if it require dollar?

23-22 grains for one Ans. 248-062 dollars.

12. One pound of pure silver is sufficient for how many dollars of silver coin, if it require 371-25 grains for one dollar? Ans. 15.515 dollars.

DENOMINATE FRACTIONS.

88. UNDER ART. 64, we defined a denominate number as one whose unit has reference to a particular thing. For a similar reason, a denominate fraction is a part of a unit having reference to a particular thing. Thus, 1⁄2 of a yard is a denominate fraction, expressing a part of the particular unit one yard; of a pound is also a denomi

nate fraction, expressing a part of the particular unit one pound.

We know (by ART. 80,) that denominate numbers may be changed or reduced from one denomination to another without altering their values. By a similar method may denominate fractions be reduced from one denomination to another.

What have we already defined a denominate number to be? What is a denominate fraction? Give some examples. May denominate fractions be changed from one name to another without altering their values?

REDUCTION OF DENOMINATE FRACTIONS.

89. SUPPOSE we wish to reduce of a pound sterling to an equivalent fraction of a farthing, we proceed as follows: since there are 20 shillings in a pound, of a pound is the same as 20 times of a shilling; and this

is the same as 12 times 20 times

in turn, is 4 times 12 times 20 times That is, of a pound sterling

of a penny; which,

of a farthing.

of 20 of 2 of of

a farthing, by calculation, to of a farthing.

Again, let us reduce

pound sterling. In this

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of a farthing to a fraction of a

case, we reverse the preceding

process, and instead of multiplying, divide by the same fractions or what is the same thing, take the reciprocals of the fractions, (ART. 47,) and multiply.

Thus of a farthing of of of of a pound sterling of a pound sterling.

1. Reduce

of an inch to the fraction of a mile.

The increase of denominate value between the inch and the mile, is for the foot 12 times the inch, for the rod 161 or 2 times the foot, for the furlong 40 times the rod,

and for the mile 8 times the furlong. Therefore a compound fraction representing what part of a mile an inch is, would be of (1÷=)2 of of. So that the fraction of an inch, which is to be changed to the fraction of a mile, must be multiplied by the compound fraction just obtained. Consequently we have

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If the question had been the reduction of of a mile to the fraction of an inch, the fraction would have been

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I. When the given fraction is to be reduced to a higher denomination, multiply it by a compound fraction, whose terms are the reciprocals of the numbers that indicate the increase in value of a unit of the successive denominations included between the denomination of the given fraction and the one to which it is to be reduced.

II. When the given fraction is to be reduced to a lower denomination, multiply it by a compound fraction, whose terms have units for their denominators, and for numerators the numbers that indicate the decrease in value of a unit of the successive denominations included between the denomination of the given fraction and the one to which it is to be reduced.

EXAMPLES.

2. Reduce T of a day to the fraction of a second.

In this example, the decrease in value of a unit of the successive denominations between a solar day and a second, are 24 (hours,) 60 (minutes,) and 60 (seconds.) Hence the compound fraction will be 24 of of, which, multiplied by the given fraction becomes

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Cancelling, successively, 60 and 24, factors common to numerator and denominator, we have first

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Finally, cancelling the factor 4, which is common to the numerator 60, and the denominator 8, we have

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We have beer particular to give the complete work of cancelling in these examples, by writing down the whole work at the successive stages of operation. In practice, the expression need not be written more than once. With a little practice the pupil will be able to strike out the common factors with accuracy and despatch.

Reduce of a pipe of wine to an equivalent fraction of a gill.

In this example, the successive denominate values between a pipe and a gill are 2 hogsheads, 63 gallons, 4

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