the remainder at 7s. per yard, what will be the loss or profit per cent. upon the whole outlay?

(11.) Bought £13 6s. 8d. worth of apples, at 3s. 4d. per bushel, part of which I found to be damaged and worthless. For the rest, which were sold at 50 per cent. profit, I received £16. How many bushels were damaged?

(12.) Bought quills at 4s. 7d. the hundred, and sold them so as to gain of the selling price. What is the selling price? and what is the profit per cent. upon the cost price?

182. To divide a given quantity into two parts which shall have to each other a given ratio.

RULE. Form fractions whose common denominator is the sum of the two terms of the ratio, their numerators being the separate terms themselves. Take these fractions of the given quantity; they will be the parts required.

Ex. Divide 75 into two parts, which shall have the ratio of 2: 3. Here the fractions are and, and the parts required are of 75=30, and of 75-45, which are plainly in the given ratio.

The reason of the Rule is evident, since the sum of the numerators makes up the denominator, and therefore the sum of the fractions makes up unity, and the sum of the parts makes up the whole of the number, while the fractions themselves, having a common denominator, are in the ratio of their numerators, and so, consequently, are the corresponding parts of the whole.

183. Similar reasoning may be applied when the whole is to be divided into more than two parts, having given ratios to each other.

Ex. 1. Gunpowder is composed of 76 parts of nitre, 14 of charcoal, and 10 of sulphur: how much of each of these will be required for a cwt. of powder ?

100 25' 100

7

Here the fractions are 76=19, 145, 100=1, and the parts are 3 qrs. 12 lbs., 157 lbs., and 11 lbs. respectively.

Ex. 2. Divide £1000 among A, B, C, so that A may have half as much again as B, and B as much again as C.

Here, representing C's part by 1, B's is 1, and A's 1+ of 1=2; and therefore the parts are to be as the numbers 2, 13, 1, or 6, 4, 3. Hence the fractions will be 13, 13, ; and the parts required

6 4 3

£461 10s. 9., £307 13s. 10d., £230 15s. 48d.

N.B.-It will be found most convenient, where there are many frac

Here the gain on the prime cost, 5s. 6d., is 11d.: hence we have 5s. 6d. £100 :: 11d.: the Answer,

which is found in the usual way to be £16

£16 13s. 4d.

Ex. 2. If bar-iron, which cost in making £2 1s. 8d. per cwt., be sol at a loss of £5 per cent., what price did it fetch per cwt. ?

Here iron, which cost £100, was sold for £100-£53=£945; bence v

have

£100: £2 1s. 8d. :: £945: the Answer, which=£1 19s. 51⁄2d.

Ex. 3. If 5 per cent. be gained by selling 125 yards of cloth for £9 what was the prime cost per yard?

Here, if the cloth had sold for £105, the prime cost would have bee £100 we have, therefore,

£105: £95 :: £100: the prime cost of 125 yards, which=£9010, and the prime cost of each yard is 14s. 51⁄2d.

Ex. 4. If 4 per cent. be lost by selling linen at 2s. 9d. a yard, at wha price must it be sold to gain £10 per cent.?

[Here, cloth which would have cost £100, would have been sold fo £96 at the first price, and for £110 at the second; we have, therefore, £96: 2s. 9d.:: £110 : second price=3s. 113d. ED.]

Ex. 149.

(1.) Eggs are sold at the rate of five for 2d. At what price must they have been bought to bring a profit of 20 per cent.?

(2.) An article was sold for £5, at a profit of 20 per cent. on the cost price. What did it cost?

(3.) How much per cent. is 23d. on a shilling?

(4.) How much per cent. is 2s. 9d. on the pound?

(5.) I bought goods for £12, and I sold them so as to make a profit of 10 per cent. on the selling price. What profit did I get upon them?

(6.) At what price must tea be sold, which cost 3s. 9d. per lb., so as to make a profit of 20 per cent. upon the cost price? (7.) If cheese, which was bought at 50s. per cwt., be sold at 6d. per lb., what was the gain per cent.?

(8.) If 5 per cent. be gained upon the cost price by selling butter at £5 12s. per cwt., what will be gained per cent. by selling it at 16d. per lb.?

(9.) A grocer mixes two kinds of sugar, at 8d. and 11d., taking two pounds of the first to one of the second. At what price per lb. should he sell the mixture, to gain a profit of 25 per cent.?

yo

(10.) If 336 yards of cambric are bought at 7s. 101d. per 1 sold, one-fourth at 10s. 3d., one-third at 8s. 6d., and

tions with the same den, to find the part corresponding to that den with num unity, and then multiply this successively by the numrs of the different fractions; thus we should find of £1000, and then multiply this by 6, 4, 3, respectively.

Ex. 3. A, B, and C form a joint capital for conducting a business, of which A contributes £500, B £650, and C £700. At the end of a year the profits are £555; what share should each receive?

Their shares should evidently be in the ratio of their contributions of capital, i. e. in the ratio of 500, 650, 700, or of 10, 13, 14; hence the fractions are 19, 13, 14, and since of £555-£15, we have the shares required, £150, £195, £210.

37 37

Ex. 4. A begins business with a capital of £800, and at the end of 3 months takes B into partnership, with a capital of £1000; at the end of another 6 months they divide their profits, £330; what should each receive?

Here A contributes £800 for 9 months, and B £1000 for 6 months: and the interest of £800 for 9 months interest of 9x £800 for 1 mo., and the interest of £1000 for 6 months = interest of 6 x £1000 for 1 month; hence the value of A's and B's outlay may be represented by the products 9 × 800 and 6 × 1000, or 7200 and 6000 respectively, and their shares of the profits must be in this ratio= that of 6:5;

hence A's share of £330=£180, and B's share of £330=£150.

11

N.B.-It appears, as in the above Ex., that the values of sums employed in business, &c., for different times are proportional to the products of the sums by the times, or rather of their numerical values, the sums being expressed in the same den", and so also the times.

Ex. 5. A and B enter into partnership, A contributing £500 and B £300; at the end of 9 months they take in C as partner, who brings into the concern a capital of £1000. The profits, £2000, being divided at the end of another 9 months, what shares did they each receive?

Here, as in Ex. 4, at the end of 18 months, the shares of capital supplied by A, B, C, respectively, may be measured by the numbers 500 × 18, 300 × 18, 1000 × 9, or 5, 3, 5, respectively: hence the fractions will be 5 ; and since of £2000 £153 16s. 11d., their shares of profit will be £769 4s. 7 d., £461 10s. 9d., £769 4s. 7d., respectively.

5 3

13

Ex. 150.

(1.) Divide 441 into parts, which shall be in the ratio of 1, 3, 5.

(2.) Divide 207 into parts, which shall be in the ratio of 1, 1, 1.

(3.) A, B, C, traded together, investing respectively £225, £325, and £425. Their profits were £585. What portion should each receive of this sum?

(4.) Gunpowder is made of 75 parts of nitre, 10 of sulphur, and 15 of charcoal. What weight of each will there be in 5 cwt.?

(5.) Divide £32 18. 8d. among 4 persons, in the proportion of the fractions, 1 1 1

(6.) A, B, C, rent a pasture for £28 10s. A puts in 5 sheep for 4 months, B, 8 sheep for 5 months, C, 9 sheep for 6 months. What must each pay of the rent?

(7.) How much copper and tin will be wanted for a cannon weighing 15 cwt. 3 qrs. 12 lbs., gun-metal being composed of 100 parts of copper, and 11 of tin?

(8.) A works regularly 9 hours a day; B works 8 hours for each of 3 days, and 7 hours for each of the other 3 days of the week. The work they had done between them was worth £2 1s. 3d. What should each receive of this sum?

(9.) A and B rent a field for £35 10s. A puts in six horses for the whole year. B puts in five horses for eleven months, and three more for five months. How much should each pay of the rent?

(10.) The standard silver coin of this realm is made of 37 parts of pure silver, and 3 of copper; and a lb. Troy of this metal yields 66 shillings. What weight of pure silver is there in half-a-crown?

(11.) The rent of a piece of land is £92 14s. A feeds on it eight cows for 4 months, B, five cows for 73 months, C, nine cows for 9 months. What should they each pay of the rent?

(12.) A and B join their capitals, which are as 2:3. At the end of 7 months A withdraws of his capital, and at the end of 9 months B withdraws of his. What portion of the whole profits, £132 12s., should each take at the end of the year?

184. Stock is the name given to money, lent to some trading company, or to our own or some foreign Government, at some given rate of interest.

Thus, if the Government were to borrow to the amount of £5,000,000 at 4 per cent., and A had lent £100 of this sum, A would be said to have £100, 4 per cent. stock, and would be entitled to receive the interest (viz. £4) upon this stock, from year to year, until the Government chose to repay the principal, and put an end to the debt.

The source from which the interest in such a case is paid is called the 'Public Funds,' being, however, only an imaginary property, representing the credit of the country itself, which is pledged to raise taxes from year to year, sufficient to pay the debts contracted by its Government. The interest is paid half-yearly, and the right to receive it may