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ceive, a clear salary of £511 17 6 annum heing allowed to C as acting partner. Answ. A's £560 6 34, B's £466 18 74, C's £651 19 1.

14. If two persons purchase a house jointly for £2000, and afterwards let it for the yearly rent of £183 6 8; what share of the yearly profit rent is each to receive, the one having contributed £850, and the other £1150 of the purchase money, and the ground rent being £44 8 year? Ans. £59 0 114, and £79 17 8.

COMPOUND FELLOWSHIP.

RULE (1.) Let all the times be of the same denomination, and multiply each stock by the time of its continuance in trade: (2.) Then using the products as stocks, proceed according to either of the rules for Simple Fellowship.

A contributes

Exam. A and B enter into partnership: £600 for 13 months, and B £800 for 10 months. Required the share of each in a gain of £650.

£600 X 13 = £7800
800 X 10 = 8000

15800

Here the products are £7800 and £8000, the sum of which is £15800. Then, as £15800: £650: £7800 :: or, by contracting the first and second terms, as £316: £13:: £7800: £320 17 8, A's share: and as £316: £13:: £8000: £329 2 34, B's share. The sum of these is £650, which proves the correctness of the operation.

In proceeding by RULE II. of Simple Fellowship, we divide 650 with ciphers annexed by 15800, or 13 with ciphers annexed by 316, and we find for quotient 04113924; and multiplying this successively by 7800 and 8000, and cutting eight figures from the products, we obtain £320-886072 and £329-11392, or £320 17 8 and £329 2 34, the same as before.

The reason of this rule will be evident from the consideration, that a stock of £600 for 13 months, would be the same as 13 times £600 for 1 month; and one of £800 for 10 months, the same as 10 times £800 for 1 month. Hence, if these increased stocks be employed, it is evident, that since the times are then to be regarded as equal, the operation will proceed in the same manner as those in Simple Fellowship.

Ex. 1. A's stock £280 for 5 months, B's £266 13 4 for 6 months: whole gain £331 12 6. Answ. A's gain £154 15 2, B's £176 17 4.

2. A's stock £170 for 8 months, B's £280 for 6 months: whole gain £250. Answ. A's £111 16 104, B's £138 3 13.

3. A's stock £248 12 6 for 10 months, B's670 for 3 months, C's £512 7 6 for 6 months: whole gain £439 18 8. Ans. A's £144 9 74, B's £116 16 1, C's £178 12 114.

4. C's stock £178 6 8 for 18 months, D's £237 17 6 for 12 months, E's £536 5 for 10 months: whole gain £370. Answ. C's £103 18 94, D's £92 8 61⁄2, E's £173 12 81.

5. A's stock £485 18 4 for 1 year, B's 279 10 for 9 months, C's £675 11 8 for 8 months: whole gain £386 15. Answ. A's £163 19 11, B's 70 14 111⁄2, C's £152 0 14.

6. A's stock £576 15 for 11 months, B's £365 4 10 for 15 months, C's £582 6 8 for 9 months: whole gain £568 15. Answ. A's £211 9 12, B's £182 12 13, C's £174 13 83.

7. M's stock £1038 13 9 for 5 months, N's £692 9 2 for 9 months, O's £1384 18 4 for 6 months: whole gain £686 1 2 Answ. M's £180 10 10, N's £216 13, O's £288 17 4.

& Three merchants A, B, and C, entered into partnership, and on the 1st of March each contributed £1000. On the 3d of May A took out £300; on the 8th of June B put in £360, and on the 20th of August C withdrew £280. On the 1st of September A put in £450; and on the 16th of October each took out £180. On the 8th of January of the following year, on making up accounts, it is found, that they have gained £1250. How is this gain_to be divided among them? Answ. A's share £384 1 113, B's £512 12 1, C's £353 5 103.

ALLIGATION.

ALLIGATION is a rule which is chiefly employed in calculations respecting the compounding or combining of articles of different kinds.

This rule has its name from a Latin word, which signifies to bind, because in the practical application of the rule, the quantities are usually linked or connected together by lines. It is a rule which is of little practical utility; being principally used in the solution of questions, which are of rare occurrence in real transactions. Besides, every thing that can be effected by this rule can be done in general in a better and easier way by Algebra. Hence this article will be more circumscribed in its limits than it might otherwise have been. The following is the principal problem in this rule, and indeed the only one that belongs to it exclusively.

To find in what proportions, quantities of given values must be taken, to form a compound of a given value.

RULE. (1.) Let the rates of the ingredients, all in the same denomination, be written in a line; and let the mean rate in the same denomination be written above them. (2.) Take two of the rates, one of which is greater, and the other less, than the mean rate, and write the difference between each of them and the mean rate, below the other. (3.) Proceed thus with the rates two by two, if there be

more than two, till one or more differences stand below each. (4.) Then, if only one difference stand below any rate, it will be the quantity required at that rate; but if there be more than one, their sum will be the required quantity.

The connecting or linking of the rates with crooked or curved lines, in the use of this rule, is attended with little advantage. Should that method be preferred, however, it can present no difficulty, as each rate less than the mean rate is to be connected with one greater, and each greater with one less, and the differences are to be set below the rate to which the line directs.

Exam. 1. In what ratios must two kinds of flour, worth 23d. and 33d. Ib. respectively, be taken, to make a mixture worth 34d. ib.?

13

10, 15

2, 3.

Here the mean rate, 13 farthings, is set above the other rates, 10 farthings and 15 farthings. Then, the difference between 10 and the mean is set below 15, and the difference between 15 and the mean below 10. Hence we find, that the quantities must be in the ratio of 2 to 3; that is, for every 2 lbs., or 2 cwts. at 24d. b., 3 lbs., or 3 cwts. at 3d. b. must be taken, to form a compound worth 34d. V M.

The correctness of this operation, and of the principle on which it depends, will appear manifest from the consideration, that in selling 3 lbs. at 34d. per lb., instead of 34d., there is a loss of 14d.; but in selling 2 lbs. at 34d. per lb., which cost. only 24d. per lb., there is a gain of 11d. and therefore, the gain on the one quantity balancing the loss on the other, the value of the compound must be exactly the mean rate.

Exam. 2. In what proportions must wines worth 10/, 14/, 17/, and 18/, gallon respectively, be mixed, so that the compound may be worth 16/gallon?

16

10, 14, 17, 18

1, 2, 6, 2; or 2, 1, 2, 6

Here by setting the difference between 10 and 16 below 17, and the difference between 17 and 16 below 10; and likewise by setting the difference between 14 and 16 below 18, and the difference between 18 and 16 below 14, we find, that for 1 gallon at 10/, we must take 2 at 14/, 6 at 17/, and 2 at 18/. By using 10/ with 18/, and 14/ with 17/, a second answer is obtained, from which it appears, that if 2 gallons at 10/, 1 at 14/, 2 at 17/, and 6 at 18/, be mixed together, the compound will also be worth 16/ gallon. It is scarcely necessary to say, that any quantities in the same ratios will serve the same purpose.

With respect to the reason of the operation, it is obvious from what was said respecting the preceding exercise, that 1 gallon at 10, and 6

at 17/ each, would make a mixture worth 16 per gallon, and likewise, that a mixture of 2 gallons at 14, and 2 at 18/, would be worth the same per gallon; and it is evident, that both mixtures taken together must make a mixture of the same value per gallon also; and in the same way every operation in this rule may be explained.

16

10, 14, 17, 18

From these principles it is also manifest, that if we should multiply or divide 1 and 6 by any number, and 2 and 2 by any number, we should still have results that would satisfy the conditions of the question. Thus, multiplying the former by 3, and dividing the latter by 2, we find for answer 3 gallons at 10/, 1 at 14/, 18 at 17/, and 1 at 18/-Different answers may also be found by connecting the rates differently. Thus, by using 10 and 17 we get 1 and 6, and by connecting 10 and 18, we have 2 and 6; and then by using 14 and 17 we get 1 and 2. After this, by the requisite addition, we find 3, 1, 8, 6 for the required quantities; and it is obvious, that by a still farther application of these principles, different answers might be found without limit.

1 1 6 6

2

3, 1, 8, 6

The correctness of these results is proved by adding together the prices of 3 gallons at 10/, of at 14, of 8 at 17/, and of 6 at 18/. This will be found to be 288, which divided by 3+1+8+6 gallons, gives exactly 16/ for the mean rate; and thus the proof may be conducted in every case, — This method of proof has been generally made a separate case of this rule, and called with no great propriety Alligation Medial. The operation according to the preceding rule, is usually called Alligation Alternate.

Exam. 3. How much linen at 2/ and at 2/5 yard, must be taken with 216 yards at 3/4, that the whole may be worth 2/6 at an average?

30
24, 29, 40

10 10 6

yard

Here by taking the differences, &c. as in the margin, we find that the quantities may be in the ratio of 10, 10, and 7. Then as 7: 10. 216: 3084. It appears therefore, that 3084 yards at 27, the same quantity at 2/5, and 216 yards at 3/4, will compose a parcel worth 2/6 yard, at an average: and various other answers might be found.-This question belongs to what is usually called Alligation Partial.

10, 10, 7

Exam. 4. What quantities of tea, worth 8/, 7/6, and 6/6 lb. respectively, must be mixed together, to form a parcel containing 112lbs. worth 7/4 b.?

88 96, 90,

78

10 10

0000

10

822

Here, in finding the ratios, to make the first two terms different, the differences between 88 and 90, and 88 and 78, are set down twice. In this way it is found, that the quantities may be as 10, 20, and 12, or, by halving each term, as 5, 10, and 6. Hence, by the method of dividing into parts in a given ratio (see page 191,) as 21, the sum of these, :5:: 112 lbs,; or by contracting, as 3:5:: 16 fbs.: 26 lbs.; and as 3 : 10 :: 16 lbs.: 531 lbs.; and lastly, as 3:6 :: 16 lbs. : 32 lbs. It appears therefore, that 263 lbs. at 8/, 534 lbs. at 7/6, and 32 lbs. at 6/6, will form a compound of 112 Ibs., worth 7/4

10, 20, 12

b.-This question belongs to what has generally been called Alligation Total. This name for this particular case of Alligation, as well as those already mentioned for the other cases, is properly falling into disuse.

As many questions in this rule admit of several answers, the pupil should prove his results in working the following exercises, particularly when his answers may differ from those here given.

Ex. 1. In what proportions must sugars, worth 13d., 111⁄2d., and 9d.b. respectively, be compounded, that the mixture may be worth 10d.? Answ. 3, 3, 7; 1, 2, 3 ; &c.

2. How much water must be added to a cask of spirits containing 84 gallons, worth 13/6 gallon, to reduce the value to 11/4 gallon? Answ. 15 gallons.

3. What quantities of three different kinds of raisins worth 11d., 15d., and 22d. b. respectively, must be mixed together, to fill a cask containing 200 lbs., and to be worth 163d. lb. Answ. 61+ lbs.; 61 lbs.; and 773 lbs.,

4. How much land worth 17/6 acre, must be added to a farm containing 51 a. 2 r. 20 p., worth £1 146 acre, to reduce the average value of both together to £1 2 9? Answ. 115 a. 2 r. 63p.

5. A box of linen, containing 1200 yards, worth at an average 3/ yard, consists of two kinds, one worth 2/83, and the other worth 3/91 yard. How much of each kind does it contain? Answ. 876 yards, and 323 yards.

6. How much spirits, at 14/ gallon, must be added to a mixture consisting of 41 gallons at 9/6, and 59 gallons at 10/8, to make the compound worth 11/6 gallon? Answ. 52, gallons. 7. How much first flour worth £1 11 6 per cwt. second flour wort £19, thirg flour worth £17 6, and fourth flour worth 17/6, must be taken, to form a ton worth £25 16 8 ? Ans. 2 c. 1 q. 20 lbs.; 7 c. 1 q. 5 lbs.; 4 c. 3 q. 13 lbs.; and 5 c. 1 q. 17 lbs.

8. A gentleman's labourers consist of men at 1/4, and women at 11 d. per day; and the amount of the wages of the whole is the same as if each of them had 1/23. Required the number of the men, the number of the women being 21. Answ. 49.

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