So also, if the value of the mixture exceeded that of one twice as much as it fell short of that of the other, the proportion of the ingredients would be as one half to one. To mix wine at $2 per gallon with wine at $3, so that the compound shall be worth $2,50, we should take equal quantities, or quantities in the proportion of 1 to 1. If the mixture were required to be worth $2,663, the quantities would be in the proportion of to 1, or of

1

1

to ; and, generally, the nearer the mixture rate is to 6633 331 that of one of the ingredients, the greater must be the quantity of this ingredient with respect to the other, and the reverse; hence, To find the proportion of two ingredients of a given value, necessary to constitute a compound of a required value, make the difference between the value of each ingredient and that of the compound the denominator of a fraction, whose numerator is one, and these fractions will express the proportion required; and being reduced to a common denominator, the numerators will express the same proportion, or show what quantity of each ingredient is to be taken to make the required compound

When the compound is limited to a certain quantity, the proportion of the ingredients, corresponding to it, may be found by saying; as the whole quantity, found as above, is to the quantity required, so is each part, as obtained by the rule, to the required quantity of each.

Let it be required, for example, to mix wine at 5s. per gallon and 8s. per gallon, in such quantities that there may be 60 gallons worth 6s. per gallon. The difference between 6s. and 5s. is 1, and between 6s. and 8s. is 2, giving for the required quantities the ratio of to, or 2 to 1; thus, taking ≈ equal to the quantity at 5s. and y equal to the quantity at 8s. we have these proportions; 3: 60 :: 2 : x, and 3: 60 :: 1: y, giving, for the answer, 40 gallons at 5s. and 20 gallons at 8s. per gallon.

Also, when one of the ingredients is limited, we may say; as the quantity of the ingredient found as above, is to the required quantity of the same, so is the quantity of the other ingredient to the proportional part required.

For example, I would know how many gallons of water at Os. per gallon, I must mix with thirty gallons of wine at 6s. per

gallon, so that the compound may be worth 5s. per gallon. First, the difference between Qs. and 5s. is 5; and the difference between 6s. and 5s. is 1; the quantity of water therefore will be to that of the wine, as to, or as 1 to 5. Then, from this ratio, we institute the proportion, 5 : 30 :: 1: x, which gives 6, for the number of gallons required.

As we have found the proportion of two ingredients necessary to form a compound of a required value, so also we may consider either of these in connexion with a third, with a fourth, and so on, thus making a compound of any required value, consisting of any number whatever of simple ingredients. The two ingredients used, however, must always be, one of a greater and the other of a less value, than that of the compound required.

A grocer would mix teas at 12s. and 10s. with 40lb. at 4s. per pound, in such proportions that the composition shall be worth 85. per lb. If he mix only two kinds, the one at 4s. and the other at 10s, their quantities will be in the ratio of to, or 1: 2; and if he mix the tea at 4s. also with that at 12s. their ratio will be that of to, or of 1 to 1. Adding together the proportions of the ingredient, which is taken with each of the others, we find the several quantities, at 4s. 10s. and 12s. to be as 2, 2, and 1. And taking a for the number of lbs. at 10s. and y for the quantity at 12s. we have the following proportions;

2:40 : 2: x; and 2: 40 :: 1: y;

giving, for the answer, 40lb. at 10s. and 20lb. at 12s. per pound.

The problems of the two last articles are generally distinguished by the names of alligation medial, and alligation alternate. A full explanation of the latter belongs properly to algebra.

Examples.

A composition being made of 5lb. of tea at 7s. per pound, 9lb. at 8s. 6d. per pound, and 141⁄2lb. at 5s. 10d. per pound; what is a pound of it worth? Ans. 6s. 10 d.

How much gold, of 15, of 17, and of 22 carats* fine, must be mixed with 5oz. of 18 carats fine, so that the composition may be 20 carats fine? Ans 5oz. of 15 carats fine, 5 of 17, and 25 of 22.

† A carat is a twenty-fourth part; 22 carats fine means 22 of pure metal. A carat is also divided into four parts, called grains of a carat.

Miscellaneous Questions for practice.

What number, added to the thirty-first part of 3813, will make the sum 200? Ans. 77.

The remainder of a division is 325, the quotient 467, and the divisor is 43 more than the sum of both; what is the dividend? Ans. 390270.

Two persons depart from the same place at the same time; the one travels 30, the other 35 miles a day; how far are they distant at the end of 7 days, if they travel both the same road; and how far, if they travel in contrary directions?

Ans. 35, and 455 miles. A tradesman increased his estate annually by 100l. more than part of it, and at the end of 4 years found that his estate amounted to 103421. 3s. 9d. What had he at first?

Ans. 4000l.

Divide 1200 acres of land among A, B, and C, so that B may have 100 more than A, and C 64 more than B.

Ans. A 312, B 412, and C 476.
Divide 1000 crowns; give A 120 more, and B 95 less, than C.
Ans. 445, B 230, C 325.

What sum of money will amount to 1327. 16s. 3d. in 15 months,
Ans. 1251.

at 5 per cent. per annum, simple interest?

A father divided his fortune among his sons, giving A 4 as often as B 3, and C 5 as often as B 6; what was the whole legacy, supposing A's share 5000%. ? Ans. 118751.

If 1000 men, besieged in a town with provisions for 5 weeks, each man being allowed 16oz. a day, were reinforced with 500 men more; on hearing, that they cannot be relieved till the end of 8 weeks, how many ounces a day must each man have, that the provision may last that time? Ans. 63.

What number is that, to which if of be added, the sum will be 1 ? Ans. §.

A father dying left his son a fortune, of which he spent in 8 months; of the remainder lasted him twelve months longer; after which he had only 410%. left. What did his father bequeath him?

Ans. 9561. 13s. 4d.

'3

A guardian paid his ward 3500l. for 2500l. which he had in
his hands 8 years. What rate of interest did he allow him?
Ans. 5 per cent.

A person, being asked the hour of the day, said, the time past
noon is equal to of the time till midnight. What was the time?
Ans. 20min. past 5.

A person looking on his watch, was asked, what was the time
of the day; he answered, it is between 4 and 5; but a more
particular answer being required, he said, that the hour and
minute hands were then exactly together. What was the time?
Ans. 21 min. past 4.

15 With 12 gallons of Canary, at 6s. 4d. a gallon, I mixed 18
gallons of white wine, at 4s. 10d. a gallon, and 12 gallons, of
cider, at 3s. 1d. a gallon. At what rate must I sell a quart of
this composition, so as to clear 10 per cent.? Ans. 1s. 3 d.
16
What length must be cut off a board, 8 inches broad, to con-
tain a square foot; or as much as 12 inches in length and 12 in
breadth?
Ans. 17 in.
What difference is there between the interest of 350l. at 4 per
cent. for 8 years, and the discount of the same sum, at the same
rate and for the same time?
Ans. 271. 3335.

A father devised of his estate to one of his sons, and of
the residue to another, and the surplus to his relict for life; the
children's legacies were found to be 2571. 3s. 4d. different. What
money did he leave for the widow?
Ans. 6351. 10d.

of %, and to the Ans. 102240

191

What number is that, from which if you take
remainder add of 2%, the sum will be 10?
A man dying left his wife in expectation that a child would
be afterward added to the surviving family; and, making his
will, ordered, that if the child were a son, of his estate should
belong to him, and the remainder to his mother; but if it were
a daughter, he appointed the mother, and the child the remain-
der. But it happened, that the addition was both a son and a
daughter, by which the mother lost in equity 2400l. more than
if it had been only a daughter. What would have been her
dowry, had she had only a son?
Ans. 21007.

3227

21

72

23

A young hare starts 40 rods before a greyhound, and is not perceived by him till she has been up 40 seconds; she scuds away at the rate of 10 miles an hour, and the dog, on view, makes after her at the rate of 18. How long will the course continue, and what will be the length of it from the place, where the dog set out? Ans. 60 seconds, and 535 yards run.

A reservoir for water has two cocks to supply it; by the first alone it may be filled in 40 minutes, by the second in 50 minutes, and it has a discharging cock, by which it may, when full, be emptied in 25 minutes. Now these three cocks being all left open, the influx and efflux of the water being always at the same rate, in what time would the cistern be filled?

Ans. 3 hours 20 mintues.

A sets out from London for Lincoln precisely at the same time, when B at Lincoln sets out for London, distant 100 miles; after 7 hours they met on the road, and it then appeared, that A had ridden 1 mile an hour more than B. At what rate an hour did each of them travel? Ans. A 7, B 611 miles.

What part of 3 pence is a third part of 2 pence? Ans. 3. A has by him 11⁄2cwt. of tea, the prime cost of which was 967. sterling. Now interest being at 5 per cent. it is required to find how he must rate it per pound to B, so that by taking his negotiable note, payable at 3 months, he may clear 20 guineas by the bargain? Ans. 14s. 1d. sterling.

There is an island 73 miles in circumference, and 3 footmen all start together to travel the same way about it; A goes 5 miles a day, B 8, and C 10; when will they all come together again? Ans. 73 days.

A man being asked how many sheep he had in his drove, said, if he had as many more, half as many more, and 7 sheep and a half, he should have 20; how many had he? Ans. 5.

A person left 40s. to 4 poor widows, A, B, C, and D; to A he left, to B, to C, and to D, desiring the whole might be distributed accordingly; what is the proper share of each?

Ans. A's share 14s. d. B's 10s. 6d. C's 8s. 5d. D's 7s. 3d.