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The elder son's share is $7000, as before. The others may be easily found from this.

Again, let z denote the second son's share. The elder son's, being $2000 more, will be x + 2000. The younger son's, being $1000 less, will be x-1000. These added together must make the whole.

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The second son's share is $5000, as before. From this the rest are easily found.

Perform the 5th and 6th in a similar way.

7. At a certain election 943 men voted, and the candidate chosen had a majority of 65. How many voted for each?

8. A person employed 4 workmen ; to the first of whom he gave 2 shillings more than to the second; to the second 3 shillings more than to the third; and to the third 4 more than to the fourth. Their wages amounted to 32 shillings. What did each receive?

9. A cask, which held 146 gallons, was filled with a mixture of brandy, wine, and water. In it there were 15 gallons of wine more than there were of brandy, and as much water as both wine and brandy. What quantity was there of each ?

Observe, that after the question is put into equation, the purpose is to make x stand alone in one member of the equation, equal to a known quantity in the other member, then the value of r is found. In the preceding examples in this Art. x has been found only in the first member, but connected with known quantities by the signs and In the solution of these equations the first thing was to unite all the x's into one term, and all the known quantities into another. Then, if the number which stood on the same side with x, had the sign + before it, that number was subtracted from the other member of the equation; but if it had the sign before it, it was added to the other member. Then the second member was divided by the coefficient of x, and the answer was obtained.

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10. A and B began to trade with equal stocks. year A gained a sum equal to twice his stock and

In the first £27 over;

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B gained a sum equal to his stock and £153 over. amount of both their gains was equal to 5 times the stock of either. What was the stock?

Let a denote the stock. Then A's gain was 2x + 27, and B's was 153. These added together must make 5 times the stock, that is, 5 x.

5x= 2x+27 + x + 153

Uniting the x's in 2d member, and the numbers,

5x3x + 180

Subtracting 3x from both sides,

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11. A young man being asked his age, answered that if the age of his father, which was 44 years, were added to twice his own, the sum would be four times his own age. What was

his age?

12. A man meeting some beggars, gave each of them 4 pence, and had 16 pence left; if he had given them 6 pence apiece, be would have wanted 12 pence more for that purpose. How many beggars were there, and how much money had ne ? 14

Let x represent the number of beggars.

13. A man has six sons, each of whom is 4 years older than his next younger brother; and the eldest is three times as old as the youngest. Required their ages.

14. Three persons, A, B, and C, make a joint contribution, which in the whole amounts to £76, of which A contributes a certain sum, B contributes as much as A and £10 more, and C as much as A and B both. Required their several contributions.

15. A boy, being sent to market to buy a certain quantity of meat, found that if he bought beef, which was 4 pence per pound, he would lay out all the money he was entrusted with ; but if he bought mutton, which was 3 pence per pound,, he would have 2 shillings left. How much meat was he sent for?

16. A man lying at the point of death left all his estate to his three sons, to be divided as follows: to A he gave one half of the whole wanting $500; to B one third; and to C the rest,

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These together will be equal to the whole estate, which was represented by x.

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x = 3600

The whole estate is $3600; the shares are $1300, $1200, and $ 1100, respectively.

17. A father intends by his will, that his three sons shall share his property in the following manner; the eldest is to receive 1000 crowns less than half the whole fortune; the second is to receive 800 crowns less than of the whole; and 13 the third is to receive 600 crowns less than of the whole. Required the amount of the whole fortune, and the share of each.

18. A father leaves four sons, who share his property in the following manner; the first takes 3000 livres less than one half the fortune; the second, 1000 livres less than one third of the whole; the third, exactly one fourth; and the fourth takes 600 livres more than one fifth of the whole. What was the whole fortune, and what did each receive?

19. In a mixture of copper, tin, and lead; 16 lb. less than one half of the whole was copper; 12 lb. less than one third of the whole was tin, and 4 lb. more than one fourth of the whole was lead. What quantity of each was there in the mix

ture?

20. A general having lost a battle, found that he had only 3600 men more than one half of his army left, fit for action; 600 more than one eighth of them being wounded, and the rest, which amounted to one fifth of the whole army, either slain or taken prisoners. Of how many men did his army consist before the battle?

21. Seven eighths of a certain number exceeds four fifths of it by 6. What is that number?

22. A and B talking of their ages, A says to B, one third of my age exceeds its fourth by 5 years. What was his age?

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23. A sum of money is to be divided between two persons, A and B, so that as often as A takes £9, B takes £4. Now it happens that A receives £ 15 more than B. What is the share of each ?

24. In a mixture of wine and cider, 25 gallons more than half the whole was wine, and 5 gallons less than one third of the whole was cider. How many gallons were there of each ?

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IV. 1. A man having some calves and some sheep, and being asked how many he had of each sort, answered, that he had 20 more sheep than calves, and that three times the number of sheep was equal to seven times the number of calves. How many were there of each ?

Let a denote the number of calves.

Then 20 will denote the number of sheep.

7 times the number of calves is 7x; 3 times the number of sheep is 3x+60; for it is evident that to take 3 times x +20, it is necessary to multiply both ferms by 3.

By the conditions these must be equal,

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2. Two men talking of their ages, the first says, your age is 18 years more than mine, and twice your age is equal to three times mine. Required the age of each.

3. Three men, A, B, and C, make a joint contribution, which in the whole amounts to £276. A contributes a certain sum, B twice as much as A and £12 more, and C three times as much as B and £12 more. Required their several contribu

tions.

4. A man bought 7 oxen and 11 cows for $591. For the oxen he gave $15 apiece more than for the cows.

did he give apiece for each ?

Let x denote the price of a cow.

Then the price of an ox will be x + 15.

How much

11 cows at x dollars apiece will come to 11 x dollars. If one ox cost x + 15 dollars, 7 oxen will cost 7 times x + 15, which is 7 x + 105.

The price of the oxen and of the cows added together will make $591, the whole price.

Uniting x's,

11 x + 7 x + 105
18x+105

591

591

Subtracting 105 from both members,

Dividing by 18,

18x486

X

x + 15

27= price of cows. 42 price of oxen.

For

5. A man bought 20 pears and 7 oranges for 95 cents. the oranges he gave two cents apiece more than for the pears. What did he give apiece for each?

6. A man bought twenty oranges and 25 lemons for $1.95. For the oranges he gave 3 cents apiece more than for the lemWhat did he give apiece for each ?

ons.

7. Two persons engage at play, A has 76 guineas, and B 52, before they begin. After a certain number of games lost and won between them, A rises with three times as many guineas as B. How many guineas did A win of B ?

Let x denote the number of guineas that A won of B.

Then A, having gained a guineas, will have 76 + x
B, having lost a guineas, will have only

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52

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A has now three. times as many as B, that is, 3 times 52 — x, which is 156 3 x. It is evident that both 52 and x must be multiplied by 3, because 52 is a number too large by x, therefore 3 times 52 will be too large by 3x.

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