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Let a denote the greater number, and y the less number.
From the conditions of the problem, we have,

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Since a and b may be any numbers whatever, we have these following principles by means of which all similar cases can be solved:

1o. The greater number is equal to the half sum of the two numbers increased by the half difference.

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2°. The less number is equal to the half sum of the two numbers diminished by the half difference.

Ans.

2. If 2 is added to the numerator of a certain fraction, its value will become ; but if 2 is added to the denominator, its value will be. What is the fraction?

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Let a denote the numerator, and y the denominator. From the given conditions, we have the equations of the problem,

x + 2 3

x

y

4

y+2 whence, x=7, and y = 12; hence, the fraction is.

and

1

= ;

3. The hands of a clock are together at 12 o'clock; when are they next together?

Let x denote the number of minute spaces passed over by the minute hand, and y the number of minute spaces passed over by the hour hand.

From the nature of the problem, we have,

x = y + 60,
x = 12y.

x = 651,

y = 5.

Hence, they are together at 1 h. 55 m.

This problem has already been solved by means of a single unknown quantity; many of the following problems can also be solved in the same manner.

4. A person has 22000 dollars at interest, which yields him 1220 dollars annually; a part bears interest at 5 per cent., and the remainder at 6 per cent. How many

dollars in each part?

Let x denote the number of dollars in the first part, and y the number of dollars in the second part.

From the conditions of the problem, we have,

x + y = 22000
x × 150 + y × 180
x = 10000,

= 1220.
y = 12000.

5. A's age is equal to twice A's age was 4 times B.'s age.

B's age; 20 years ago,
What are their ages ?
Ans. A.'s 60; B.'s 30.

6. There are two numbers: the first added to half the second gives 35; the second added to. half the first gives 40. What are the numbers? Ans. 20 and 30.

7. A man has three sons: the sum of the ages of the first and second is 27, that of the first and third is 29, and that of the second and third is 32. What ar the ages of each ? Ans. 12, 15, and 17.

8. Two men are in trade; the stock of the first increased by one third that of the second, is $1700;

the stock of the second increased by one fourth that of the first, is $1800. What is the stock of each ?

Ans.

$1200 and $1500.

9. Find two numbers such that the second shall equal 45, and the the first shall equal 40.

the first plus second plus of

Ans. 50 and 60.

10. The sum of the first and second of three numbers is 13, that of the first and third 16, and that of the second and third 19. What are the numbers ?

Ans. 5, 8, and 11.

11. Bought 100 lbs. of sugar and 80 lbs. of coffee for $28, and afterwards bought at the same rates 200 lbs. of sugar and 60 lbs. of coffee for $36. What did each cost per pound?

Ans. Sugar 12 cents, and coffee 20 cents.

12. There are three numbers; the first increased by twice the second and three times the third, makes 74; the second, increased by twice the third and three times the first, makes 90; the third, increased by twice the first and three times the second, makes 100. What are the numbers ?. Ans. 20, 18, and 6.

13. A butcher bought of one person 12 sheep and 20 lambs for 44 dollars, and of a second person 7 sheep and 13 lambs for 27 dollars, at the same rates. How many dollars did he give apiece?

Ans. $2 for sheep, and $1 for lambs.

14. Divide the number 1152 into three parts, such that 9 times the sum of the first and second shall be

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equal to

times the sum of the second and third;
and if 8 times the first be subtracted from 8 times the
second, the remainder shall be equal to the sum of the
first and third.
Ans. 288, 384, and 480.

15. A farmer mixed rye and oats, forming 100 bushels of the mixture. The rye was worth 96 cents per bushel, the oats 56 cents, and the mixture 72 cents. How many bushels did he use of each ?

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Ans. 40 of rye, and 60 of oats. 16. A person has two sorts of wine, one worth 402 cents a quart, and the other 24 cents. How much of each kind must he use to form a gallon worth 112 cents?" Ans. 1 quart of the first, 3 quarts of the second.

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17. A., and B., trade on a joint stock of 833 dollars,+
and clear 153 dollars. A.'s share of the gain is 45 dol-
lars more than B.'s. What share of the capital did

each possess? +++ = 835 Ans. A., $539; B., $294.54

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5/18. Two laborers, A., and B., received 51 dollars. A.
had been employed 14 days, and B. 15 days; A. re-
ceived for 6 days' labor 1 dollar more than B. got for
4 days' labor. How many dollars did each receive per
day?
Ans. A., 1; B., 2."

19. In 80 pounds of an alloy of copper and tin, there are 7 lbs. of co per to 3 of tin. How much

per must be added to the alloy, that there may be
11 lbs. of copper to 4 of tin? 5
Ans.. 10 lbs.

20. In 3 battalions there are 1905 troops; the number in the first, together with the number in the

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second, is 60 less than the number in the third; the number in the third, together with the number in the first, is 165 less than the number in the second. How many are there in each battalion ?

Ans. 630, 675, and 600.

21. A grocer has three kinds of tea: 12 lbs. of the first, 13 of the second, and 14 of the third are together worth 25 dollars: 10 of the first, 17 of the second, and 11 of the third are together worth 24 dollars; 6 of the first, 12 of the second, and 6 of the third are together worth 15 dollars. What is the value of a pound of each?

Ans. 50 cents, 60 cents, and 80 cents.

22. A. owes $1200, and B. $2500; but neither has money enough to pay his debts. Says A. to B., "lend me of your fortune, and I can pay my debts;" says B. to A., "lend me of your fortune, and I can pay mine." What fortune had each?

Ans. B., had $2400; and A., $900.

23. The united ages of a father and son are 80 years; and if the age of the son be doubled, it will exceed the father's age by 10 years. What is the age of each? Ans. 50, and 30.

24. A. travels uniformly along a certain road, B. starts an hour afterwards in pursuit, and after 4 hours finds by inquiry that he is travelling 14 miles per hour slower than A.; he then doubles his rate of travel, and overtakes A., 6 hours from the time he started in pur

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