19. A grocer mixes two kinds of tea which cost him 3s. 8d. and 48. 4d. per lb. respectively; what must be the selling price of the mixture in order that he may gain 15 per cent. on his outlay?

20. A person has goods worth £30; he sells one-third of them so as to lose 10 per cent.: what must he sell the remainder at so as to gain 20 per cent. on the whole?

21. I buy a house for 500 guineas, and sell it immediately at a profit of 30 per cent.; what do I receive, supposing the expenses of the sale to be 5 per cent.?

22. The prime cost of a 76-gallon cask is £23. 12s. 6d., but 13 gallons are lost by leakage; 9 gallons of water is then mixed with the remainder, and it is sold at 7s. 6d. a gallon. Find the whole gain, and also the gain per cent.

23. A stationer sold quills at 11s. a thousand, by which he cleared of the money; he raises the price to 13s. 6d. What does he clear per cent. by the latter price?

24. A person sold 72 yards of cloth for £8. 14s.; his profit being the cost of 11.52 yards: how much did he gain per cent.?

25. A smuggler buys 6 cwt. of tobacco at 1s. 3d. per lb.; he meets with a revenue-officer, who seizes 3rd of it: at what rate per lb must he sell the remainder, so as, 1st, neither to gain or lose; 2nd, to gain 5 guineas; and 3rd, to gain cent. per cent.?

26. A person expends £3000 in railway shares at 15 per cent. discount, and sells them at par; what does he gain by the transaction, and what per cent.?

27. A wine-merchant bought 143 pipes of wine, which having received damage, he sold for £11201, thereby losing 20 per cent.; find the cost of the wine per pipe, and the selling price of it per gallon.

28. A farm is let for £96 and the value of a certain number of quarters of wheat. When wheat is 38s. a quarter, the whole rent is 15 per cent. lower than when it is 56s. a quarter. Find the number of quarters of wheat which are paid as part of the rent.

29. A man having bought a lot of goods for £150, sells 3rd at a loss of 4 per cent; by what increase per cent. must he raise that selling price, in order that by selling the rest at the increased rate, he may gain 4 per cent. on the whole transaction?

30. A person bought a French watch, bearing a duty of 25 per cent., and sold it at a loss of 5 per cent.; had he sold it for £3 more, he would have cleared 1 per cent. on his bargain. What had the French maker for it?

DIVISION INTO PROPORTIONAL PARTS.

158. To divide a given number into parts which shall be proportional to certain other given numbers.

This is merely an application of the Rule of Three; still it may be well to state a general Rule, by which examples which come under the above head may be worked.

RULE. State thus: "As the sum of the given parts: any one of them :: the entire quantity to be divided: the corresponding part of it."

This statement must be repeated for each of the parts, or at all events for all but the last part, which of course may either be found by the Rule, or by subtracting the sum of the values of the other parts from the entire quantity to be divided.

Ex. 1. Divide 40 guineas among A, B, and C, so that their portions may be as 7, 11, and 14 respectively.

Proceeding according to the Rule given above,

32 7 40 guineas: A's share;

32: 11: 40 guineas: B's share;

whence A's share= £9. 3s. 9d., and B's share=14. 8s. 9d.

C's share may be found from the proportion

32: 14: 40 guineas : C's share;

whence C's share= £18. 78. 6d. ;

or by subtracting £9. 3s. 9d.+£14. 8s. 9d., or £23. 12s. 6d. from £42, which leaves £18. 7s. 6d., as above.

The reason for the above process is clear from the consideration, that 40 guineas is to be divided into 32 equal parts, of which A is to have 7 parts, B 11, and C 14.

Ex. 2. Divide £11000 among 4 persons, A, B, C, D, in the proportions of,,, and .

B's share= £2857. 2s. 103d., C's share= £2142. 17s. 11⁄2d.

D's share £1714. 58 84d.

Ex. 3. Divide £45000 among A, B, C, and D, so that A's share : B's share 12, B's: C's :: 3 : 4, and C's: D's :: 4 : 5.

In this case,

B's share=2 A's share, 3 C's share=4 B's share,
4 D's share=5 C's share;

.. we have C's share=B's share=§ A's share,
and D's share = C's share= 10 A's share;

.. A's share + B's share + C's share + D's share

= A's share (1+2++),

=

=9 A's share ;

.. A's share=£5000, B's = £10000, C's = £13333. 6s. 8d.,
D's £16666. 13s. 4d.

FELLOWSHIP OR PARTNERSHIP.

159. DEF. FELLOWSHIP or PARTNERSHIP is a method by which the respective gains or losses of partners in any mercantile transactions are determined.

Fellowship is divided into SIMPLE and COMPOUND FELLOWSHIP: in the former, the sums of money put in by the several partners continue in the business for the same time; in the latter, for different periods of time.

SIMPLE FELLOWSHIP.

160. Examples in this Rule are merely particular applications of the Rule in Art. (158), and that rule therefore applies.

Ex. 1. Two merchants, A and B, form a joint capital; A puts in £240, and B £360: they gain £80. How ought the gain to be divided between them?

A's share;

£(240+360): £240 :: £80 x (pounds);

whence x = £32, and .. B's share= £48.

The estate of a Bankrupt may be divided among his creditors me method.

. A bankrupt owes three creditors, A, B, and C, £175, £210, , respectively; his property is worth £422. 10s.: what ought to receive?

£650 £175: £4221: A's share;

£650 : £210 :: £4221 : B's share;

whence A's share= £113. 15s., B's share = £136. 10s. ; .. C's share £172. 58.

COMPOUND FELLOWSHIP.

ULE. "Reduce all the times into the same denomination, ly each man's stock by the time of its continuance, and then

sum of all the products; each particular product : the whole be divided the corresponding share."

:

A and B enter into partnership; A contributes £3000 for 9 1 B £2400 for 6 months, they gain £1150: find each man's gain.

ng by the Rule given above,

× 9+2400 × 6) : £(3000 × 9) :: £1150 : A's share of gain,

or £41400 : £27000 :: £1150 : A's share of gain ;

nd £41400 : £14400 :: £1150 : B's share of gain; whence A's share= £750, and B's share = £400.

on for the above process is evident from the consideration, of £3000 for 9 months would be equivalent to a stock of 9 for 1 month; and one of £2400 for 6 months, to one of 6 for 1 month: hence, the increased stocks being considered, hen becomes one of Simple Fellowship.

Ex. 2. There were at a feast 20 men, 30 women, and 15 servants; for every 10s. that a man paid, a woman paid 6s., and a servant 28.; the bill amounted to £41: how much did each man, woman, and servant pay?

=

20 men at 10s. each=200 at 1s., 30 women at 6s. 180 at 1s., and 15 servants at 28.30 at 1s.; and 200+ 180+30=410.

Hence we have

410 : 200 :: £41: 20 men's share (in pounds);
410 180: £41 30 women's share (in pounds);
410 30: £41: 15 servants' share (in pounds);
.. 20 men's shares = £20, 30 women's shares = £18,
and 15 servants' shares = £3;

.. each man paid £1, each woman 12s., and each servant 4s.

EQUATION OF PAYMENTS.

162. DEF. When a person owes another several sums of money, due at different times, the Rule by which we determine the just time when the whole debt may be discharged at one payment, is called the EQUATION OF PAYMENTS.

Note. It is assumed in this Rule that the sum of the interests of the several debts for their respective times equals the interest of the sum of the debts for the equated time.

RULE. "Multiply each debt into the time which will elapse before it becomes due, and then divide the sum of the products by the sum of the debts; the quotient will be the equated time required."

Ex. 1. A owes B £100, whereof £40 is to be paid in 3 months, and £60 in 5 months: find the equated time.

Proceeding according to the Rule given above,

If a be the equated time;

then (40 × 3+60 × 5) = x (40 +60);

whence x= = 4 months.

The reason for the above process, in accordance with our assumption, is clear from the consideration that the sum of the interests of £40 for 3 months, and £60 for 5 months, is the same as the interest of £(120+300), or £420 for 1 month; if therefore A has to pay £100 in one sum, the