PROBLEM III.

A general, after a battle, reviews his army; a third has perished, a fourth has been made prisoner, a fifth has fled; there remain only 13000 men. What was the number of

soldiers before the battle?

First supposition. Let 120000 be the number of the soldiers before the battle. We shall have 40000 killed, 30000 made prisoners, 24000 fled.

40000+30000+24000+13000=107000.

But 107000 should be equal to 120000; the error is then 13000 in minus. I write 120000-13000.

Second supposition. Let 360000 be the number of the soldiers; we shall have 120000 killed, 90000 prisoners, 72000 fled. But 120000+90000+72000+13000=295000; the error is then 65000 in minus. I write 360000-65000.

Second supposition 360000-65000.

I operate as above and I find 60000 soldiers, which answers to the conditions of the question.

PROBLEM IV.

A farmer demanded of another how many sheep he had; the other answered, if I had half as many, a fourth, the twothirds, and 5, in addition to what I now have, I should then have 150. How many had he?

1. Let 12 be what he had, to which adding its half 6, its fourth 3, its two-thirds 8, and 5, I find 34 for the sum, the difference of which with 150 is 116, in minus. We then have

12-116.

2. Let 24 be the number, to which adding its half 12, its fourth 6, its two-thirds 16, and 5 more, I find 63 for the sum, the difference with 150 is 87, in minus; we shall then have

12-116 24-87.

I divide the difference of the products 12 x 87, and 24 X 116, that is to say, the difference of the products 1044 and 2784,

which is 1740, by 29, the difference of the errors, and I find 60 for the quotient or answer. Proof.-60+++3×60+5=150.

PROBLEM V.

Alexander and the philosopher Calisthenes reasoned together upon their ages, and those of Clitus and Ephestion, both friends of the conqueror. I am two years older than Ephestion, said Alexander; the age of Clitus is equal to both our ages and four years more, the sum of our three ages is 96. We demand the age of Alexander, of Clitus, and of Ephes

tion.

Let us suppose, first, that Alexander was 28 years, Ephestion would be 26, and Clitus 58; the sum of these is 112; but it should be equal to 96. The error is then 16 years in plus.

28+16.

Let us suppose, secondly, that Alexander was 20 years old, Ephestion would be 18, and Clitus 42; the sum of these is 80; but it should be 96. The error is then 16 in minus. We shall then have

28+16,
20-16.

I divide the sum of the products 28 x 16, 20×16, that is to say, 768, by the sum of the errors, which is 32; I find for the quotient 24; which number fulfils the conditions of the question.

RULE OF CHANGE.

The change is the profit that the bankers and negotiators make of their money; that is to say, that they gain as much as their money would gain them if it were placed at interest.

The difficulty of transporting money from one place to another, as well on account of its weight as the dangers incurred upon the roads, has given rise to the establishment of places which are called places of change: we find such in all great cities. By this means, we can transport any sum of money that we would, by a letter of change from a banker

or negotiator, in paying the sum that we would send and the change of that sum.

For the same reasons, travellers, particularly when they go into very distant countries, are very careful to furnish themselves with letters of change, for the principal cities? which are found in their route.

EXAMPLE I.

A person, going from Paris to Marseilles, requires a banker to enable him to touch 3000 francs neat, in this last city. We demand how much he should give to the banker for the change of 3000 fr., this change being at 3 per cent.

We shall say

100 3: 3000: x=90.

Therefore, in paying 3090 francs, the banker will furnish him with the letter of change which he requires.

EXAMPLE II.

The same person remits 3000 fr. to a banker, the discount being always at 3 per 100. How much will he touch? We shall say, if 103 gives 100, how much will 3000 fr.? that is to say,

64

103 100 3000 x=2912+ fr. Ans.

RULE OF DISCOUNT.

Discount is a sum to deduct from the amount of a bill or note for a given time, the value of which we would have before its expiration; that is, before it becomes due. It is also a sum that bankers pay upon the value of a bill, payable in a given time, and the interest of which is stipulated at so much per year, or per month, according to the agreement between the borrower and the lender.

Discount is taken in several ways; when, for example, for the sum of $100 lent, a bill is made of $105, payable in a year; the discount is said to be within. But when from the sum lent, the bill of which should be paid in a year, the banker commences by deducting the interest agreed on, we say then that the discount is without. It is evident, that in

this way the banker gains not only the interest of the sum which he lends, but also the interest of the interest.

PROBLEM I.

To find the discount of the sum of $800, at 6 per 100, the discount being within, we shall say

106 : 6 :: 800 : x=$45,28}. Ans.

PROBLEM II.

To find the discount of $800, at 6 per 100, the discount being without.

We shall say, if $100, interest and principal, give 6, how much will $800 give? that is to say,

100 : 6 :: 800 : x=$48. Ans.

When we would discount a bill before its expiration, we proceed thus:

PROBLEM III.

A person has given a bill of $15640, payable at 365 days; he consents to pay the discount at 6 per 100, with the power of diminishing the discount for the time that he may be able to pay before the expiration of the bill; he comes to acquit himself at the end of 240 days. We demand what he should pay.

We shall say, 365 days are to 240 days as $6 are to x.

that is to say, that the interest at the rate of 6 per 100 for a year, is reduced to $39, during 240 days. We shall then

say

106 103 :: 15640 =$15336,82 cts. T. Ans.

RULE OF ALLIGATION OF THE FIRST KIND.

This rule serves to make known the price or the weight of several merchandises or metals mixed together; the price or the weight is called the mean price or weight.

PROBLEM

A wine merchant has mixed together wines of different prices, viz. 15 gallons of wine at 12s., 96 at 15s., and 146 at 11. 6d.

The price of the mixture of these different wines is 3299s. and the number of gallons is 257.

Dividing 3299s. by 257 gallons, we shall find 12s. 107d., which will be the price of a gallon of this mixture.

RULE OF ALLIGATION OF THE SECOND KIND.

PROBLEM I.

A merchant has wine at 21s., at 22s., at 29s., and at 30s, per gallon; he would make a mixture worth 25s. per gallon. How much of each kind must he take?

I write the different prices of the gallon of wine under each other in the order of their value, the highest price being the first. I compare, first, the highest price 30s. with the mean price 25s., and I write the difference 5 opposite to 23, which is the first price inferiour to 25.

I compare, secondly, the second price 29, superiour to 25, with 25, and I write the difference 4 opposite to 21, which is the lowest price.

Afterward, ascending, I compare, thirdly, the lowest price 21 with 25; and I write the difference 4 opposite 29, which is the second price superiour to 25.

I compare, 4thly, the first price 23, which is inferiour to 25, with 25, and I write the difference 2 opposite the highest price; this done, I add the differences 2, 4, 5, 4; their sum gives 15 gallons.

Therefore, to have a mixture the price of which should be 25s., we must take 2 gallons at 30s., 4 at 29s., 5 at 23s., and 4 at 21s.

In effect, 2 gallons at 30s. give 60s., 4 at 29s. give 1168., 5 at 23s. give 115s., and 4 at 21s. give 84s.