(12) A puts £2100 into a business and B £1750; at the end of a year each puts in £700 more, and C. joins them with £2500. At the end of 18 months from this time how should a profit of £2166 10s. be divided?

(13) A traveller performed a journey of 126 miles in four days, his second day's journey being half as much again as the first, the third one-third as much again as the second, and the fourth one-fourth as much again as the third. Find the length of each day's journey.

(14) A and B join capitals in the ratio of 7:11; at the end of seven months A. withdraws half of his, and B a third of his, and after eleven months more they divide a profit of £5148 10s. What is the share of each?

(15) Divide £65 98. between three persons so that the first may have as many half-crowns as the second has shillings, and the second as many guineas as the third has pounds.

CHAPTER XIV.

EXCHANGE.

90.-EXCHANGE is the reduction of a sum of money in the coinage of one country into its equivalent value in the coinage of another country at a given rate.

The standard rate of exchange, depending only on the intrinsic value of the coinage of two countries, is called the Par of exchange.

The actual rate of exchange at any time, depending both on the intrinsic value of the coinage and on the state of trade between the two countries at that time, is called the Course of exchange.

The par of exchange may be better understood by an instance of its approximate determination.

Standard English gold contains 22 parts in 24 fine, and a pound Troy is coined into 4628 sovereigns.

Standard French gold contains 90 per cent. fine, and a kilogram is coined into 155 napoleons (1 nap. = 20 francs). Hence taking only the amount of fine gold in each coin,

of exchange between two places by knowing their rates with intermediate countries, as in ex. (3).

Ex. (1) Reduce £271 88. into francs and centimes at the rate of 25 35 francs per pound sterling.

Ex. (2) At the rate of 6 thalers 27 groschens to a pound sterling, how much English money must be remitted to Prussia to pay a bill of 841 thalers 24 groschens (1 thal.= 30 grosch.)?

No. of pounds sterling=841th. 24 gr.÷6th. 27 gr.

=25254 gr.÷207 gr.

=122

Ans. £122.

Ex. (3) If a pound sterling = 16'1 Hamburgh marks, and 143 Hamburgh marks =62 Prussian thalers, and 23 Prussian thalers = 20 Russian rubles, determine the value of an English sovereign in Russian rubles.

A somewhat simpler arrangement of the above reduction, known by the name of Chain rule, is as follows:

£1161 marks

143 marks=62 thalers

23 thalers 20 rubles

•*. £1 × 143 × 23 = 16°1 × 62 × 20 rubles.

•*. £1=16·1×62 X 20

143 × 23 =6.07 rubles.

rubles.

The rule to be observed in writing down the connecting links being this: Begin with one of the extremes (which are to be connected), and let the left hand side of each line be in

the same denomination as the right hand side of the preceding line. Then multiply together all the left hand numbers, and also all the right hand numbers, retaining only in the equation of the products the names of the extreme denominations.

Exercise 72.

(1) Convert £484 into Portuguese money at the rate of 4 milreis 285 reis to £1 sterling (1 milr.=1000 reis).

(2) A merchant in Paris draws upon a merchant in London a bill of 722°15 francs, what will the latter have to pay if exchange is at 25°25 francs per pound sterling.

(3) At the rate of 60°24 piastres to a pound sterling, find the value in Turkish money of £13 148. 2d.

(4) Find the value in English money of 27 Austrian ducats, a ducat being = 5 florins, and a florin =60 kreutzers, and the course of exchange 9 fl. 57 kr. for £1.

(5) Determine the rate of exchange between England and Spain if a pound sterling =25°3 francs, and 11 francs =42 reals.

(6) Exchange 8796 rupees into francs, at the rate of 28. 1 d. per rupee, and 25°44 francs per pound sterling.

(7) Bills on Paris, which are bought at the rate of 25°42 francs per pound sterling, are sold in Lisbon at the rate of 175 reis per franc, determine the rate of exchange between London and Lisbon.

(8) If the course of exchange at Paris upon London be 2583 francs for £1, and at Naples upon Paris 85 lire for 82 francs, how many Italian lire should be paid at Naples for £15?

(9) If a sum of £500 is to be remitted from London to a person in St. Petersburg; will he gain or lose, and how much, by having it sent to Paris, the exchanges being 25°74 francs for £1, 3 rubles for 13 francs, and 5'92 rubles for £I?

(10) If 98 Spanish reals = 25 French francs, 22 francs =9 Frankfort florins, 30 florins =49 Hamburgh marks, and 192 marks =55 Norwegian dollars, how many Spanish reals are equivalent to 100 Norwegian dollars?

CHAPTER XV.

SQUARE AND CUBE ROOT.

91.-THE Square root of a number is defined in (17) to be such a number as will when multiplied by itself produce the given one. Thus 7 is the square root of 49, 13 of

169, &c.

The Cube root of a number is such a number as will when multiplied twice by itself produce the given one. Thus 5 is the cube root of 125, 9 of 729, &c.

The square root of a number which has less than three digits, and which has an exact square root, consists of one digit and must be known on inspection. And so, the cube root of a number which has less than four digits, and which has an exact cube root, consists of one digit and must be known on inspection.

For the extraction of the square and cube roots of other numbers, the following rules may be given, but they cannot be explained without the help of Algebra.

EXTRACTION OF SQUARE ROOT.

92.-RULE.-Divide the given number into periods of two figures each, beginning at the units' place.

Find the highest square number in the first period on the left (which may consist of one or two figures); the corresponding root will be the first figure of the answer. Subtract the square number from the first period, and annex to the remainder the two figures of the next period.

Double the root already found, and prefix it as a divisor to the number last mentioned. Place the quotient both as the second figure of the root, and also as an additional figure in the divisor.

Multiply the divisor so completed by this last figure, subtract, and to the remainder annex the two figures of the next period; and proceed as before.