611. EXAMPLES.

1. If 24 men cut down 124 acres of grass in 34 days, in how many days will 14 men cut down 104 acres?

2. If 12s. 3d. be paid for the carriage of 14 stones for 140 miles, how much ought to be paid for the carriage of 36 stones 160 miles?

3. If the quartern loaf sells at 74d. when wheat is sold at 48s. per quarter, what ought to be the price of six such loaves when wheat is 60s. per quarter.

4. If 16 horses in 18 days eat 48 bushels of oats, how many bushels will serve 36 horses 12 days?

5. If 8 men can mow 112 acres of grass in 14 days, working 12 hours a day, how many men could now 63 acres in 12 days, working 8 hours a day?

6. If 26 men can build a wall 80 feet long, 20 feet high, and 2 feet thick, in 40 days, working 12 hours a day, how many men could build a wall 50 feet long, 15 feet high, 18 inches thick, in 24 days, working 8 hours a day?

7. A farmer, from his experience of former years, calculated that it would take 24 shearers 16 days to cut down his crop ; but after this number of shearers had been at work two days, finding his corn quite ripe, and dreading wet weather, he resolved to have his shearing finished in six days: not being scholar sufficient to calculate how many additional shearers he should engage, he requests the student's assistance. many should he engage?

612. QUESTIONS FOR EXAMINATION

UNDER CHAPTER 20.

1. What is Compound Proportion?

How

2. The student should be exercised in stating Compound Proportion questions until he clearly understands the prin ciple.

CHAPTER XXI.

FELLOWSHIP, OR DISTRIBUTIVE
PROPORTION.

613. Partnership, or Fellowship, shows how to divide the profits or losses of any joint concern, in proportion to the shares or capitals of the several persons interested; how to adjust averages; or how to divide any quantity whatsoever in any number of given proportions.

614. This is effected by Distributive Proportion, that is, by two or more single Rule of Three questions, which are stated thus. As the sum of the given quantities is to each particular quantity, so is the quantity proposed to be apportioned to the several portions required.

615. EXAMPLES.

1. A ship gained by a voyage to France 981. 11s., which it is required to divide between the owners according to their respective shares. A. owned of her and B. §: what is each person's share of the gain?

Note. When the profit or loss is to be divided according to fractional shares, as in the above case, we may take such fractional parts of such profit and loss; thus if 981. 11s, be divided by 8 it will give 121. 68. 4d. as the 4th of 981. 11s.; and 121 68. 4 d., mul iplied by 3, will give the th of the profit, and the same sum, multiplied by 5, will give the gth of it.

2. Three persons enter into partnership, A. advances 5,000%' B. advances 4,000l., and C. advances 3,000l.; in one year they gain 1,5007.; what is each person's share of the gain, in proportion to his capital advanced?

3. Upon a policy of insurance of 4000%., a loss of 3601. 15s. is sustained, which is to be distributed amongst the underwriters. A.'s risk is 1,000%., B.'s 2,500l., C.'s 500l., what will each person have to contribute, and at what rate per cent. ?

Note. Insurances are contrac's by which the insurers agree to pay to the insured the value of the property which may be destroyed, or the amount of damage sustained by accidents against which the insurances are made, in consideration of receiving a certain sum for the risk. These insurances are of various kinds, as fire, life, and sea insurances. The instrument by which the contracting parties become bound to each other is called the policy: every policy must bear a stamp. The amount of remuneration paid to the insurers is called the premium, and is reckoned at so much per cent. on the amount insured.The above question might be performed by four Rule of Three questions, but the above plan is much shorter.

4. A bankrupt's debts are 1,0807 14s., his assets are 6427. 2s., what will the estate pay in the pound? and what will A. receive for his debt of 3227. 2s.

5. A., B., and C. buy a ship for 2,540%.; A has of her, B. has of her, and C. of her; in her first voyage she cleared 3207, what was each persons share of the proceeds?

6. A testator bequeathed to A. 240%., to B. 2207., to C. 1057., to D. 847.; but at his death it was found that the net amount of his property was only 5157.; how much should each legatee receive?

7. A certain common, consisting of 423 acres, is to be divided amongst four individuals, in proportion to the annual value of their respective estates; A.'s estate is valued at 5007. per annum, B.'s at 450l. per annum, C.'s at 230l. per annum, and D.'s at 135l. per annum; what will be each person's share of the common?

8. The rental of a parish amounts to 8,000l., and the expense of maintaining the poor amounts to 6357.; what rate per pound must be laid on the parish to defray this expense, and how much will a person have to pay whose rental is 1347. ?

9. Distribute 367. 18s. by a per centage amongst three persons, according to their respective capitals, viz., A. 1,5007., B. 6847., and C. 1,2807.?

10. Upon a policy of insurance of 5,6007., a loss of 3427. 10s. is sustained, which is to be apportioned amongst the underwriters, according to their respective risks; how much will each have to contribute, A.'s risk being 3601., B.'s 500l., C.'s 6407., D.'s 550., E.'s 5407., F.'s 560l., G.'s 800l., H.'s 1,0007., I.'s 6501.; and how much is the loss per cent. on the amount of policy?

11. Two merchants carrying on business in partnership, gain 3,4067. 12s. 4d., which it is required to divide according to the capital embarked by each; A. advanced 6,000l., and B. advanced 10,500, what is each person's share of the gain?

12. The amount of a bankrupt's debts is 4,7361. 15s., his assets are 2,540 10s., what amount will heay in the pound? and what will A. receive on 7631., B. on 5821. 10s. 10d., C. on 6431. 8s. 6d., D. on 7861. 2s. 6d., E. on 121. 6s. 4d., F. on 31. 6s. 2d., G. on 231. 8s. 4d., H. on 1,4241. 17s. 4d., and I. on 4971. 15s.?

13. A. and B. enter into partnership for a year; A. with 2,0001. and B. with 1,6001. After four months, being ir want of more capital, they admit C. with 1,2001. At the end of the year they had gained 1,5001.; what was each person's share of the proceeds?

Note. In such cases as the above, where the proportion of the several capitals to the aggregate capital is modified by the circumstance of time, the several capitals must be multiplied by the time they are employed; then as the sum of the products is to each particular product, so is the amount to be divided to each required share of it. The reason for this will clearly appear, if we consider that, had their capitals been equal, then their shares of the proceeds would have

been as to the time their capitals were employed, and had the time been equal, then their shares would have been as to the amount of their capitals.

14. Three butchers rented a field from the first of May to the first of November, for which they were to pay 601. A. put in 6 head of cattle for the whole time, viz., 184 days; B. put in 4 head of cattle, and after they had been in 50 days, he increased the number by other 4, and the 8 remained in for the rest of the time. C. at first put in 9 head of cattle, but after they had been in 60 days, he sold 5 head, and the remaining 4 remained in for the remainder of the time: what proportion ought each person to pay of the rent?

616. QUESTIONS FOR EXAMINATION

UNDER CHAPTER 21.

1. For what purposes is the rule of distributive proportion used?

2. May all questions in distributive proportion be performed by two or more Rule of Three questions.

CHAPTER XXII.

ALLIGATION;

Or, Proportion of Ingredients in Compounds.

617. ALLIGATION* is generally divided into Alligation Medial, Alligation Alternate, Alligation Partial, and Älligation Total.

618. ALLIGATION MEDIAL,

Is when the prices and quantities of the several ingredients of which a compound is formed are given, to find the total value of that compound, or any required portion of it.

619. When the total value of the compound is required, multiply the several quantities by their price, and the sum of the products will be the total value; as, for example :

If a tea dealer mixes 20 lbs. of Congou tea, worth 4s. 6d. per lb.; 25 lbs. Congou, worth 4s. 3d. per lb. ; and 30 lbs. Bolea, worth 2s. 6d. per lb.; what is the total value of the mixture?

* Alligation, the act of tying together; from ad, to, and lego, to tie.