damaged, I desire to know how I may sell the remainder per gallon, so as neither to gain nor lose by the bargain? Ans. 4s. 61d. I

25 671

What quantity of water must I add to a pipe of mountain wine, valued at 33l. to reduce the first cost to 4s. 6d. per gallon? Ans. 20 gallons.

If 15 ells of stuff, yard wide, cost 37s. 6d. what will 40 ells of the same stuff cost, being a yard wide? Ans. 6l. 13s. 4d.

Shipped for Barbadoes 500 pairs of stockings at 3s. 6d. per pair, and 1650 yards of baize at is. 3d. per yard, and have received in return 348 gallons of rum at 6s. 8d. per gallon, and 7,50lb. of indigo at 1s. 4d. per lb. what remains due upon my adventure? Ans. 24l. 12s. 6d.

If 100 workmen can finish a piece of work in 12 days, how many are sufficient to do the same in 3 days? Ans. 400 men. How many yards of matting, 2ft. 6in. floor, that is 27ft. long, and 20ft. broad.

How many yards of cloth, 3qrs. wide, ure to 30 yards 5grs. wide?

broad, will cover a

Ans. 72 yards. are equal in measAns. 50 yards.

A borrowed of his friend B 250l. for 7 months, promising to do him the like kindness; sometime after B had occasion for 3001. how long may he keep it to receive full amends for the favor? Ans. 5 months and 25 days. If, when the price of a bushel of wheat is 6s. 3d. the penny loaf weigh 9oz. what ought it to weigh when wheat is at 8s. 21d. per bushel? Ans. 6oz. 13dr. If 4 cwt. can be carried 36 miles for 35 shillings, how many pounds can be carried 20 miles for the same money? Ans. 907lb.. How many yards of canvass, that is an ell wide, will line 20 yards of say, that is Sqrs. wide ? Ans. 12yds.. If 30 men can perform a piece of work in 11 days, how many men will accomplish another piece of work, 4 times as big, in a fifth part of the time? Ans. 600.

A wall that is to be built to the height of 27 feet, was raised 9 feet by 12 men in 6 days; how many men must be employed to finish the wall in 4 days at the same rate of working?

Ans. 36.

If oz. cost 11. what will 1oz. cost?
If of a ship cost 273l. 2s. 6d. what is

At 11. per cwt. what does 33lb. come to ?

Ans. 11. 5s. 8d.

of her worth?

Ans. 2271. 12s. 1d.

Ans. 105d.

If of a gallon cost gl. what will of a tun cost? Ans. 140l. A person, having of a coal mine, sells of his share for 171. what is the whole mine worth?

Ans. 3801. If, when the days are 135 hours long, a traveller perform his journey in 35 days; in how many days will he perform the same journey, when the days are 11

hours long?

Ans. 4061 days.

A regiment of soldiers, consisting of 976 men, are to be new clothed, each coat to contain 24 yards of cloth, that is 1ğyd. wide, and to be lined with shalloon, yd. wide; how many yards of shalloon will line them?

Ans. 4531yds. 1qr. 2 nl.

COMPOUND PROPORTION.

121. PROPORTION is also applied to questions, in which the relation of the quantity required, to the given quantity of the same kind, depends upon several circumstances, combined together; it is then called Compound Proportion, or Double Rule of Three. See some examples.

It is required to find how many leagues a person would go in 17 days, travelling 10 hours a day, when he is known to have travelled 112 leagues, in 29 days, employing only 7 hours a day.

This question may be resolved in two ways, we will first give the one that leads to Compound Proportion.

In each case, the number of leagues passed over depends upon two circumstances, namely, the number of days the man travels, and the number of hours he travels in each day.

We will not at first consider this latter circumstance, but suppose the number of hours be the same in each case; the question then will be; a person in 29 days travels 112 leagues, how many will he travel in 17 days? This will furnish the following proportion;

29: 17:112: x.

The fourth term will be equal to 112 multiplied by 17 and divided by 29, or 1904 leagues.

Now, to take into consideration the number of hours, we mustsay, if a person travelling 7 hours a day, for a certain number of days, has travelled 1904 leagues, how far will he go in the same time, if he travel 10 hours a day? This will lead to the following proportion,

which gives for the fourth term, or answer, 93,793 leagues nearly.

The question may also be resolved by observing, that 29 days travelling, at 7 hours a day, is equal to 203 hours travelling; and that 17 days, at 10 hours a day, amounts to 170 hours; the problem then is reduced to this proportion,

203: 170 :: 112: x,

by which we find the distance he ought to travel in 170 hours, according to what he performed in 203 hours.

We see, by the first mode of resolving the question, that 112 leagues has to the fourth term, or answer, the same proportion, that 29 days has to 17, and that 7 hours has to 10; stating this in the form of a proportion, we have

by which it appears, that 112 is to be multiplied by both 17 and 10, and to be divided by both 29 and 7, that is, 112 is to be multiplied by the product of 17 by 10, and divided by the product of 29 by 7, which is the same as the second method of resolving the question.

122. Again, if 9 labourers, working 8 hours a day, have spent 24 days in digging a ditch 65 yards long, 13 wide, and 5 deep, how many days will it take 71 labourers of equal ability, working 11 hours a day, to dig a ditch 327 yards long, 18 broad, and 7 deep?

Here is a question very complicated in appearance, but which may be resolved by proportion.

If all the conditions of these two cases were alike, except the

number of men and the number of days, the question would consist only in finding in how many days 71 men would perform the work, which 9 men have done in 24 days; we should have then, 71:9:24: x,

but instead of calculating the number of days, we will only indicate, as in article 82, the numbers to be multiplied together, and place as a denominator those by which they are to be divided; we thus have for x days,

24 by 9
71

But as the first labourers worked only 8 hours a day, while the others worked 11, the number of days required by the latter will be less in proportion, which will give

whence we conclude that the number of days, in this case, is

24 by 9 by 8
71 by 11

This number is that of the days necessary for 71 labourers, working 11 hours a day, to dig the first ditch.

The ditches being of unequal length, as many more days will be necessary, as the second is longer than the first, thus we shall have

and the number of days, this new circumstance being considered, will be

24 by 9 by 8 by 327

71 by 11 by 65

Taking into consideration the breadths, which are not alike, we have

and, consequently, the number of days required changes to

Lastly, the depths being different, we have

24 by 9 by 8 by 327 by 18

5:7::

: X,

71 by 11 by 65 by 15

and the number of days, resulting from the concurrence of all these circumstances, is

24 by 9 by 8 by 327 by 18 by 7

71 by 11 by 65 by 13 by 5

Performing the multiplications and divisions, we get the answer required, 21 days 1903931.

123. This number is equal to 24 multiplied by the fractional quantity

but this last quantity, which expresses the relation of the number of days given, to the number of days required, is itself the product of the following fractions ;

Now, going back to the denominations given to these numbers in the statement of the question, we see that these fractions are the ratios of the men and the hours, of the lengths, the breadths and the depths of the two ditches; hence it follows, that the ratio of the number of days given, to the number of days sought, is equal to the product of all the ratios, which result from a comparison of the terms relating to each circumstance of the question.

This may be resolved in a simple manner by first finding the value of each of these ratios; for, by multiplying together the fractions that express them, we form a fraction which represents the ratio of the quantity required to the given quantity of the same kind.

This fraction, which will be the product of all the ratios in the question, will have for its numerator the product of all the antecedents, and for its denominator, that of all the consequents. A ratio resulting, in this manner, from the multiplication of several others, is called a compound ratio.

By means of the fractional expression

9 by 8 by 327 by 18 by 7

71 by 11 by 65 by 13 by 5'

and the given number of days, 24, we make the following proportion,