will denote the amount for which the eggs were sold.

.. x = 120, the number of eggs of each sort.

3. A person possessed a capital of $30,000, for which he received a certain interest; but he owed the sum of $20,000, for which he paid a certain annual interest. The interest that he received exceeded that which he paid by $800. Another person possessed $35,000, for which he received interest at the second of the above rates; but he owed $24,000, for which he paid interest at the first of the above rates. The interest that he received annually exceeded that which he paid by $310. Required the two rates of interest.

Let x denote the number of units in the first rate of interest, and y the unit in the second rate. Then each may be regarded as denoting the interest on $100 for 1 year.

To obtain the interest of $30,000 at the first rate, denoted by x, we form the proportion

and for the interest of $20,000, the rate being y,

But by the conditions the difference between these two amounts is equal to $800.

We have, then, for the first equation of the problem,

By expressing algebraically the second condition of the problem, we obtain a second equation,

350y-240 x 310.

Both members of the first equation being divisible by 100, and those of the second by 10, we have

3x-2y=8, 35y-24x=31.

To eliminate x, multiply the first equation by 8, and then add the result to the second. There results

19y=95, whence y = 5.

Substituting for y, in the first equation, this value, and that equation becomes

3x-108, whence x == 6.

Therefore the first rate is 6 per cent; and the second, 5.

VERIFICATION.

and we have

$30,000 at 6 per cent = 30,000 x .06 = $1800,

$20,000 at 5 per cent = 20,000 × .05 = $1000 ;

1800-1000 = 800.

The second condition can be verified in the same manner.

4. What two numbers are those whose difference is 7, and sum 33? Ans. 13 and 20.

5. Divide the number 75 into two such parts that three times the greater may exceed seven times the less by 15. Ans. 54 and 21.

6. 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.

7. A bill of £120 was paid in guineas and moidores, and the number of pieces used of both sorts was just 100. If the guinea be estimated at 21s., and the moidore at 27s., how many pieces were there of each sort?

Ans. 50.

8. Two travelers set out at the same time from London and

York, whose distance apart is 150 miles. 8 miles a day; and the other, 7. In meet?

One of them travels what time will they

Ans. In 10 days.

9. At a certain election 375 persons voted for two candidates, and the candidate chosen had a majority of 91. How many voted for each?

Ans. 233 for one, and 142 for the other. 10. A person has two horses, and a saddle worth £50. Now, if the saddle be put on the back of the first horse, it makes their joint value double that of the second horse; but if it be put on the back of the second, it makes their joint value triple that of the first. What is the value of each horse? One £30, and the other £40.

Ans.

11. The hour and minute hands of a clock are exactly together at 12 o'clock. When will they again be together? Ans. 1h. 5m.

12. A man and his wife usually drank 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.

13. If 32 pounds of sea-water contain 1 pound of salt, how much fresh water must be added to these 32 pounds in order that the quantity of salt contained in 32 pounds of the new mixture shall be reduced to 2 ounces, or of a pound?

Ans. 224 lbs.

14. A person who possessed $100,000 placed the greater part of it out at 5 per cent interest, and the other at 4 per cent. The interest which he received for the whole amounted to $4640. Required the two parts. Ans. $64,000 and $36,000.

15. At the close of an election the successful candidate had a majority of 1500 votes. Had a fourth of the votes of the unsuccessful candidate been also given to him, he would have received three times as many as his competitor, wanting three thousand five hundred. How many votes did each

receive?

Ans. 1st, 6500; 2d, 5000.

chain worth $25.

16. A gentleman bought a gold and a silver watch, and a When he put the chain on the gold watch, it and the chain became worth three and a half times more than the silver watch; but when he put the chain on the silver watch, they became worth one half the gold watch and $15 over. What was the value of each watch?

Ans. Gold watch, $80; silver watch, $30.

17. There is a certain number expressed by two figures, which figures are called digits. The sum of the digits is 11, and if 13 be added to the first digit the sum will be three times the second. What is the number? Ans. 56.

18. From a company of ladies and gentlemen, 15 ladies retire, and there are then left 2 gentlemen to each lady; after which 45 gentlemen depart, when there are left 5 ladies to each gentleman. How many were there of each at first?

Ans. 50 gentlemen, 40 ladies.

19. A person wishes to dispose of his horse by lottery. If he sells the tickets at $2 each, he will lose $30 on his horse; but if he sells them at $3 each, he will receive $30 more than his horse cost him. What is the value of the horse, and the number of tickets? Ans. Horse, $150; No. of tickets, 60.

20. A person purchases a lot of wheat at $1, and a lot of rye at 75 cents, per bushel, the whole costing him $117.50. He then sells of his wheat and of his rye at the same rate, and realizes $27.50. How much did he buy of each?

Ans. 80 bushels of wheat, 50 of rye.

21. There are 52 pieces of money in each of two bags. A takes from one, and B from the other. A takes twice as much as B left, and B takes seven times as much as A left. How much did each take? Ans. A, 48 pieces; B, 28 pieces.

22. Two persons, A and B, purchase a house together, worth $1200. Says A to B, "Give me two thirds of your

money, and I can purchase it alone."

But says B to A,

"If

you will give me three fourths of your money, I shall be able to purchase it alone." How much had each?

Ans. A, $800; B, $600.

23. A grocer finds that if he mixes sherry and brandy in the proportion of 2 to 1, the mixture will be worth 78s. per dozen; but if he mixes them in the proportion of 7 to 2, he can get 79s. a dozen. What is the price of each liquor per dozen ? Ans. Sherry, 81s.; brandy, 72s.

Equations containing Three or More Unknown Quantities. 117. Let us now consider equations involving three or more unknown quantities.

Take the group of simultaneous equations,

To eliminate z by means of the first two equations, multiply the first by 3, and the second by 4. Then, since the coefficients of z have contrary signs, add the two results together. This gives a new equation,

43x-2y=121

(4)

Multiplying the second equation by 2 (a factor of the coefficient of z in the third equation), and adding the result to the third equation, we have

16x+9y=84

(5)

The question is then reduced to finding the values of x and y, which will satisfy the new equations (4) and (5).

Now, if the first be multiplied by 9, the second by 2, and the results added together, we find

419x1257, whence x = 3.