4. A man being asked the value of his horse and chaise, answered, that the chaise was worth $50 more than the horse, and that one half of the value of the horse was equal to one third of the value of the chaise. Required the value of each.

5. Two persons talking of their ages, the first says, of my age is equal to of yours; and the difference of our ages is 10 years. What are their ages?

6. There are two towns situated at unequal distances from Boston, and on the same road. They are 30 miles apart. of the distance of the second from Boston is equal to of the distance of the first. What is the distance of each from Boston?

7. A man being asked the value of his horse and saddle, answered, that his horse was worth $114 more than his saddle, and that of the value of his horse was 7 times the value of What was the value of each?

his saddle.

8. A hare is 40 rods before a greyhound, but she can run only as fast as the greyhound. How far will each of them run before the greyhound will overtake the hare?

9. A gentleman paid 4 labourers $136; to the first he paid 3 times as much as to the second wanting $4; to the third one half as much as the first, and $6 more; and to the fourth 4 times as much as to the third, and $5 more. How much did he pay to each ?

10. A man bought some cider at $4 per barrel, and some beer at $7. There were 6 barrels more of the cider than of the beer; and of the price of the beer was equal to of the price of the cider. Required the number of barrels of each.

of the first.

11. Two men commenced trade together; the first put in £40 more than the second, and the stock of the first was to that of the second as 14 to 5. What was the stock of each ? 14 to 5 signifies the second is 12. A man's age when he was married was to that of his wife as 3 to 2; and when they had lived together 4 years, his age was to hers as 7 to 5. What were their ages when they were married?

13. A and B began trade with equal sums of money. In the first year A gained £40, and B lost £40; but in the second, A lost one third of what he then had, and B gained a sum less

by £40 than twice the sum A had lost; when it appeared that B had twice as much money as A. What money did each begin with?

Then r

Let be the number of pounds each had at first. +40 will be the sum A had at the end of the first year; and X- 40 the sum B had.

The second year A lost of what he then had, consequently

he saved; his sum will then be

2x + 80

3

B gained twice as much as A lost wanting £40; his will be

x- -40+

2x + 80

40.

3

B had now twice as much as A,

4x + 160
3

Multiplying by 3,

4x+160=3x-120+2x+80-120.

Transposing and uniting,

x= 320.

Transposing again,

320=x,

Ans. £320.

Note. In this example the result had the sign-in both members, but by transposing it has the sign +. It would have been the same thing if the signs had been changed without transposing. The result would have come out right if the first member had been made the second, and the second first, in the first equation.

14. A person playing at cards, cut the pack in such a manner, that of what he cut off were equal to of the remainder. How many did he cut off?

15. Divide $183 between two men, so that of what the first receives, shall be equal to of what the second receives. What will be the share of each?

16. A man sold 20 bushels of grain, rye and wheat; the rye at 5s. and the wheat at 7s. per bushel; of the rye came to as much as of the wheat. How much was there of each?

17. What number is that from which if 5 be subtracted two thirds of the remainder will be 40 ?

18. A man has a lease for 99 years; and being asked how

much of it was already expired, answered, that two thirds of the time past was equal to four fifths of the time to come. Required the time past, and the time to come.

19. It is required to divide the number 50 into two such parts, that three fourths of one part added to five sixths of the other may make 40.

20. Two workmen received equal sums for their work; but if one of them had received 18 dollars more, and the other 3 dollars less, then of the wages of the latter would have been equal to of the wages of the former. How much did each receive?

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21. A certain man, when he married, found that his age was to that of his wife as 7 to 5; if they had been married 8 years sooner, his age would have been to hers as 3 to 2. What were their ages at the time of their marriage?

VI. 1. Divide the number 68 into two such parts, that the difference between the greater and 84, may be equal to three times the excess of 40 above the less.

Let the less.

Then 68. — x = the greater.

68x must be subtracted from 84.

is not so great as 68 by x.

Therefore if I subtract 68 from 84, I shall subtract too much by the quantity x, and I must add ☛ to obtain the true result.

Then we have 84-68+x for the difference between 84 and 68. Ꮖ .

The excess of 40 above the less is 40-x, and 3 times this

26 ess.
26: = 42 = greater.

42,

Note. In this question 68- x was subtracted from 84. Instead of x, now put its value, 68 - 26. Now 68 - 26 that is, the number to be subtracted from 84 is 42, and the answer must be 42. When 68 is subtracted from 84, the result is 16, which is too small by 26, the value of x; to this it is necessary to add 26, and it makes 42, the true result, 84 2642. This shows that we did right in adding x after subtracting 68. This will always be found true. Therefore,

68+

when any of the quantities to be subtracted have the sign -before them, they must be changed to + in subtracting, and those which have+must be changed to

2. A gentleman hired a labourer for 20 days on condition that, for every day he worked, he should receive 7s., but for every day he was idle, he should forfeit 3s. At the end of the time agreed on he received 80 shillings. How many days did he work, and how many days was he idle?

Let a the number of days he worked.

Then 20.

x the number of days he was idle. x days, at 7s. a day, would come to 7 x shillings. 20, at 3s. per day, would be 60-3x shillings. This must be taken out of 7 x.

By the above rule 60 3 x, subtracted from 7 x, leaves 7 x 60+3; for 60 is too much to be subtracted by 3 x.

By the conditions,

3. Two men, A and B, commenced trade; A had twice as much money as B; A gained $50, and B lost $90, then the difference between A's and B's money was equal to three times what B then had. How much did each commence with?

4. Two men, A and B, played together; when they commenced they had $20 between them, after a certain number of games, A had won $6, then the excess of A's money above B's was equal to of B's money. How much had each when they commenced?

5. Divide the number 54 into two such parts that the less subtracted from the greater, shall be equal to the greater subtracted from three times the less. What are the parts?

6. It is required to divide the number 204 into two such parts, that of the less being subtracted from the greater, the remainder will be equal to of the greater subtracted from four times the less..

Let x denote the less number, and solve the question again. Note. Observe, that after multiplying by 5 in the above example, the signs of both terms of the numerator were changed, that of 408 to and that of 2 x to +; this was done because it was not required to subtract so much as 408 by 2x. The change of signs could not be made before multiplying by 5, because the sign - before the fraction showed that the whole fraction was to be subtracted. If the signs of the fraction had been changed at first, it would have been necessary to put the sign + before the fraction. This requires particular attention, because it is of great importance, and there is danger of forgetting it.

7. A man bought a horse and chaise for $341. Now if of the price of the horse be subtracted from twice the price of the chaise, the remainder will be the same as if of the price of the chaise be subtracted from three times the price of the horse. Required the price of each.

8. Two men, A and B, were playing at cards; when they began, A had only as much money as B. A won of B $23; then of B's money, subtracted from A's, would leave one half of what A had at first. How much had each when they began?

9. A man has a horse and chaise. The horse is worth $44 less than the chaise. If of the value of the horse be subtracted from the value of the chaise, the remainder will be the same as if from the value of the horse you subtract of the ex

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