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. 83gr. 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 23-22 grains for one dollar? 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, of a yard is a denominate fraction, expressing a part of the particular unit one yard; of a pound is also a denomi

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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, ¿ʊ of a pound is the same as 20 times of a shilling; and this is the same as 12 times 20 times of a penny; which, in turn, is 4 times 12 times 20 times of a farthing. That is, of a pound sterling=ʊ of 2o of 12 of ÷ of a farthing, by calculation, to of a farthing.

Again, let us reduce of a farthing to a fraction of a pound sterling. In this 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.

16

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 16 or 32 times the foot, for the furlong 40 times the rod,

nd for the mile 8 times the furlong. Therefore a comound fraction representing what part of a mile an inch is, Tould be of (13) of of. So that the frac ion of an inch, which is to be changed to the fraction f a mile, must be multiplied by the compound fraction ust obtained. Consequently we have

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

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 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 2 of 50 of 50, which, multiplied by the given fraction becomes

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

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

We have been 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 frac tion of a gill.

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

quarts, 2 pints and 4 gills; therefore, our compound fraction is of 3 of 4 of 2 of 4, which, multiplied by the given fraction, becomes of 2 of 530f 4 of 2 of 4; this becomes, after cancelling like factors, 1 gill.

4. Reduce of a yard to a fraction of a mile.

Ans. 4400.

5. Reduce of a gill to the fraction of a gallon. Ans. 96.

6. Reduce 338 of a pound to the fraction of a ton. Ans. 3360.

7. Reduce of a mile to the fraction of a foot.

Ans. 1760 feet.

8. Reduce of of of a yard to the fraction of a mile. Ans. 36960

9. Reduce of 1 of 24 of a gallon to the fraction of a Ans. 4.

gill.

10. Reduce of 4 of a hogshead of wine to the fraction of a gill. Ans. 1792 5971. 11. Reduce of 4 of 44 yards to the fraction of an inch. Ans. 243 344. 12. Reduce of of a farthing to the fraction of a shilling. Ans. 16. 13. Reduce of an ounce to the fraction of a pound avoirdupois. 14. Reduce of of 1 rod to the fraction of an inch,

of a foot, and of a yard.

Ans. 44.

15. Reduce of of 1 hour to the fraction of a month

of 30 days, and to the fraction of a year of 365 days.