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 52. OC A has now three times as many as B, that is, 3 times 3 x. x, which is 156 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 3 x.

52

first place I subtract

remove it from the out of the second

76 from both members, so as to first member. Then to get 3 member, which is there subtracted, I add 3 x to both members; then the x's are all in the first member, and the known numbers in the other.

N. B. Any term which has the sign +, either expressed or understood, may be removed from one member to the other by giving it the sign; for this is the same as subtracting it from both sides. Thus x+3=10;

a is not so much as 10 by 3, we therefore say x=10—3. Again, 5 x = 18+ 3 x. Now 5 x is more than 18 by 3 x, therefore we may say 5 x 3 x = 18.

Any term which has the sign - before it may be removed from one member to the other by giving it the sign. This is equivalent to adding the number to both sides. Thus 5 x 3 = 17. In this it appears

therefore we say 3.x. Here it ap

that 5x is more than 17 by 3; 5 x = 17+ 3. Again, 5 x = 32 pears that 5 x is not so much as 32 by 3x; therefore we say 5 x + 3 x = 32. This is called transposition. Hence it appears that any term may be transposed from one member to the other, care being taken to change the sign.

In the last example, 76 was transposed from the first member to the second, and the sign changed from + to -; and 3x was transposed from the second member to the first, and the sign changed from to +. This has been done in many of the preceding examples.

When a number, consisting of two or more terms, is to be multiplied, all the terms must be multiplied, and their signs preserved. In the last example, 52 — x, multiplied by 3, gave a product 156-3x.

8. A person bought two casks of wine, one of which held exactly three times as much as the other. From each he drew 4 gallons, and then there were four times. as many gallons remaining in the larger as in the smaller. How many gallons were there in each at first?

Let x denote the number of gallons in the less at first. Then the number of gallons in the greater will be 3 x.

Taking 4 gallons from each, the less will be x- 4

And the greater

3x-4 The greater is now 4 times as large as the less; 4

Ans. Less 12 gallons, greater 36 gallons.

9. A man when he was married was three times as old as his wife; after they had lived together 15 years, he was but twice as old. How old was each when they were married?

Ans. The man's age 45, wife's 15 years. 10. A farmer has two flocks of sheep, each containing the same number. From one of these he sells 39, and from the other 93; and finds just twice as many remaining in the one as in the other. How many did each flock originally contain? Ans. 147 sheep.

11. A courier, who travels 60 miles per day, had been despatched 5 days, when a second was sent to overtake him; in order to which, he must go 75 miles per day; in what time will he overtake the former? Ans. 20 days.

12. A and B engaged in trade, A with £240, and B with £96. A lost twice as much as B; and upon settling their accounts it appeared that A had three times as much remaining as B. How much did each lose? Let a denote B's loss, then 96 x x will denote what

he had remaining.

2x will denote A's loss, and

2402 x what he had remaining, &c.

Ans. A lost £96, and B £48.

13. Two persons began to play with equal sums of money; the first lost 14 shillings, and the other won 14 shillings, and then the second had twice as many shillings as the first. What sum had each at first? Ans. 42 shillings.

14. Says A to B, I have 5 times as much money as you; yes, says B, but if you will give me $17, I shall have 7 times as much as you. How much had each? Ans. A had $20, and B $4,

15. Two men, A and B, commenced trade; A had $500 less than 3 times as much money as B; A lost $1500, and B gained $900, then B had twice as much as A. How much had each at first?

Ans. A $2440, and B $980. 16. From each of 15 coins an artist filed the value of 2 shillings, and then offered them in payment for their original value; but being detected, the whole were found to be worth no more than $145. What was their original value?

Ans. $10. 17. A boy had 41 apples, which he wished to divide between three companions, as follows; to the second he wished to give twice as many as to the first, and three apples more; and to the third he wished to give three times as many as to the second, and 2 apples more. How many must he give to each?

Ans. 3, 9, and 29, respectively. 18. A person buys 12 pieces of cloth for 149 crowns:

2 are white, 3 are black, and 7 are blue. A piece of the black costs 2 crowns more than a piece of the white, and a piece of the blue costs 3 crowns more than a piece of the black. Required the price of each kind. See example 4th of this Art.

Ans. 9, 11, and 14, respectively. 19. A man bought 6 barrels of flour and 4 firkins of butter; he gave $2 more for a firkin of butter, than for a barrel of flour; and the butter and flour both cost the same sum. What did he give for each?

Ans. Flour $4, butter $6. 20. A grocer sold his brandy for 25 cents a gallon more than his wine, and 37 gallons of his wine came to as much as 32 gallons of his brandy. What was

each per gallon.

Ans. Wine $1.60, brandy $1.85. 21. A man bought 7 oxen and 36 cows; he gave $18 apiece more for the oxen than for the cows, and the cows came to three times as much as the oxen wanting $53. What was the price of each?

Ans. Oxen $43, cows $25.

22. A man sold 20 oranges, some at 4 cents apiece, and some at 5 cents apiece, and the whole amounted to 90 cents. How many were there of each sort?

If he had sold 13 at 5 cents apiece, then the number sold at 4 cents apiece would be 2013, or 7.

In the same manner, if he sold ≈ oranges at 5 cents apiece, then he sold 20x oranges at 4 cents apiece. x oranges at 5 cents apiece would come to 5 x cents, and 20 oranges at 4 cents apiece would come to 4 cents, which is 80 4 x cents.

times 20