350y—240x=310 . . . . . (2)

The two members of the first equation are divisible by 100, and those of the second by 10; they may therefore be replaced by the following:

In order to eliminate r, multiply equation (3) by 8, and then add equation (4); hence

19y=95 •*. y= 5.

Substituting this value of y in equation (3), we have

Then the first rate of interest is 6 per cent., and the second 5 per cent.

PROBLEM 14.

An artisan has three ingots composed of different metals melted together. A pound of the first contains 7 oz. of silver, 3 oz. of copper, and 6 oz. of tin. A pound of the second contains 12 oz. of silver, 3 oz. of copper, and 1 oz. of tin. A pound of the third contains 4 oz. of silver, 7 oz. of copper, and 5 oz. of tin. How much of each of these three ingots must he take in order to form a fourth, each pound of which shall contain 8 oz. of silver, 33 oz. of copper, and 4 oz. of tin?

Let x, y, and z be the number of ounces which he must take in each of the ingots respectively, in order to form a pound of the ingot required.

Since, in the first ingot, there are 7 oz. of silver in a pound of 16 oz., it fol

7

lows that in 1 oz. of the ingot there are oz. of silver, and, consequently, in x

16

oz. of the ingot there must be

7x 16

oz. of silver. In like manner, we shall find

that

represent the number of ounces of silver taken in the second and

12y 4z
16' 16

third ingots in order to form the fourth; but, by the conditions of the problem, the fourth ingot is to contain 8 oz. of silver; we shall thus have

And reasoning precisely in the same manner for the copper and tin, we find

which are the three equations required for the solution of the problem. Clearing them of fractions, they become

In these three equations the coefficients of y are most simple; it will, therefore, be convenient to eliminate this unknown quantity first.

Hence, in order to form a pound of the fourth ingot, he must take 8 ounces of the first, 5 ounces of the second, and 3 ounces of the third.

PROBLEM 15.

There are three workmen, A, B, C. A and B together can perform a certain piece of labor in a days; A and C together in b days; and B and C together in c days. In what time could each, singly, execute it, and in what time could they finish it if all worked together?

Let x= time in which A alone could complete it.

y= time in which B alone could complete it.

z= time in which C alone could complete it.

Since A and B together can execute the whole in a days, the quantity

which they perform in one day is

1

a

; and since A alone could do the whole

1

in r days, the quantity he could perform in one day is

for the same rea

1

; the sum of what

y

son, the quantity which B could perform in one day is they could do singly must be equal to the quantity they can do together; hence

Let t be the time in which they could finish it if all worked together; then, by Prob. 8,

(16) What two numbers are those whose difference is 7 and sum 33 ?

Ans. 13 and 20.

(17) To divide the number 75 into two such parts that three times the greater may exceed 7 times the less by 15.

Ans. 54 and 21.

(18) In a mixture of wine and cider, of the whole plus 25 gallons was wine, and part minus 5 gallons was cider; how many gallons were there of each?

Ans. 85 of wine, and 35 of cider.

(19) A bill of $34 was paid in half dollars and dimes, and the number of pieces of both sorts that were used was just 100; how many were there of each?

Ans. 60 half dollars and 40 dimes.

(20) Two travelers set out at the same time from New York and Albany, whose distance is 150 miles; one of them goes 8 miles a day, and the other 7; in what time will they meet?

Ans. In 10 days

(21) At a certain election 375 persons voted, and the candidate chosen hi a majority of 91; how many voted for each?

Ans. 233 for one, and 142 for the other.

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(22) What number is that from which, if 5 be subtracted, of the remainder will be 40?

Ans. 65.

(23) A post is in the mud, in the water, and 10 feet above the water; what is its whole length?

Ans. 24 feet.

(24) There is a fish whose tail weighs 9 pounds, his head weighs as much as his tail and half his body, and his body weighs as much as his head and his tail; what is the whole weight of the fish?

Ans. 72 pounds.

(25) After paying away and of my money, I had 66 guineas left in my purse; what was in it at first?

Ans. 120 guineas.

(26) A's age is double of B's, and B's is triple of C's, and the sum of all their ages is 140; what is the age of each?

Ans. A's 84, B's 42, and C's 14. (27) Two persons, A and B, lay out equal sums of money in trade; A gains $630, and B loses $435, and A's money is now double of B's; what did each lay out?

Ans. $1500.

(28) A person bought a chaise, horse, and harness, for $450; the horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness; what did he give for each?

Ans. $100 for the horse, $50 for the harness, and $300 for the chaise. (29) Two persons, A and B, have both the same income: A saves of his yearly, but B, by spending $250 per annum more than A, at the end of 4 years finds himself $500 in debt; what is their income?

Ans. $625.

(30) A person has two horses, and a saddle worth $250; now, if the saddle be put on the back of the first horse, it will make his value double that of the second; but if it be put on the back of the second, it will make his value triple that of the first; what is the value of each horse?

Ans. One $150, and the other $200. 12

(31) To divide the number 36 into three such parts that of the first, } of the second, and of the third may be all equal to each other?

Ans. The parts are 8, 12, and 16.

(32) A footman agreed to serve his master for £8 a year and a livery, but was turned away at the end of 7 months, and received only £2 13s. 4d. and his livery; what was its value?

Ans. £4 16s.

(33) A person was desirous of giving 3d. a piece to some beggars, but found that he had not money enough in his pocket by 8d.; he therefore gave them each 2d., and had then 3d. remaining; required the number of beggars?

Ans. 11.

after which,

(34) A person in play lost of his money, and then won 3s.; he lost of what he then had, and then won 2s.; lastly, he lost of what he then had; and this done, found he had but 12s. remaining; what had he at first?

Ans. 20s.

(35) To divide the number 90 into 4 such parts that if the first be increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2, the sum, difference, product, and quotient shall be all equal to each other?

Ans. The parts are 18, 22, 10, and 40 respectively. (36) The hour and minute hand of a clock are exactly together at 12 o'clock; when are they next together?

Ans. 1 hour 55 minutes.

(37) There is an island 73 miles in circumference, and three footmen all start together to travel the same way about it: A goes 5 miles a day, B 8, and C 10; when will they all come together again?

Ans. 73 days.

(38) How much foreign brandy at 8s. per gallon, and domestic spirits at 3s. per gallon, must be mixed together, so that, in selling the compound at 9s. per gallon, the distiller may clear 30 per cent.?

Ans. 51 gallons of brandy, and 14 of spirits.

(39) A man and his wife usually drank out a cask of beer in 12 days; but when the man was from home, it lasted the woman 30 days; how many days would the man alone be in drinking it?

Ans. 20 days.

(40) If A and B together can perform a piece of work in 8 days; A and C together in 9 days; and B and C in 10 days: how many days will it take each person to perform the same work alone?

Ans. A 1434 days, B 17, and C 237

(41) A book is printed in such a manner that each page contains a certain number of lines, and each line a certain number of letters. If each page were required to contain 3 lines more, and each line 4 letters more, the number of letters in a page would be greater by 224 than before; but if each page were required to contain 2 lines less, and each line 3 letters less, the number of letters in a page would be less by 145 than before. Required the number of lines in each page, and the number of letters in each line.

Ans. 29 lines, 32 letters.

(42) Hiero, king of Syracuse, had given a goldsmith 10 pounds of gold with which to make a crown. The work being done, the crown was found to weigh 10 pounds; but the king, suspecting that the workman had alloyed it with silver, consulted Archimedes. The latter, knowing that gold loses in water 52 thousandths of its weight, and silver 99 thousandths, ascertained the weight of the crown, plunged in water, to be 9 pounds 6 ounces. This discovered the fraud. Required the quantity of each metal in the crown.

Ans. 7 pounds 121 ounces of gold, 2 pounds 335 ounces of silver. (43) To divide a number a into two parts which shall have to each other the ratio of m to n.

(44) To divide a number a into three parts which shall be to each other as m:n:p.

(45) A banker has two kinds of change; there must be a pieces of the first to make a crown, and b pieces of the second to make the same: now a person wishes to have c pieces for a crown. How many pieces of each kind must the banker give him?

(46) An innkeeper makes this bargain with a sportsman: every day that the latter brings a certain quantity of game he is to receive a sum a, but every day that he fails to bring it he is to pay a sum b. After a number n of days it may happen that neither owes the other, or that the first owes the second, or that the second owes the first a sum c. Required a formula which