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3. Two couriers start from points distant 200 miles, and travel towards each other. The first travels 9 miles per hour, and the second 8 miles per hour. How long before they will meet, and how far will each have travelled?


x = ... 9x == 81x

Let x denote the number of hours.

T'hen will 9x

denote the distance the first travels, and 81 denote the distance the second travels. From the conditions,


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113, the number of hours required.

102, the number of miles the first travels. 97, the number of miles the second travels.

4. A hare starts 50 leaps before a dog, and makes 4 leaps to the. dog's 3; but 2 of the dog's leaps are equal to 3 of the hare's. How many leaps must the dog make to overtake the hare?

Let 2 denote the number of leaps that the dog makes. 4x Then will denote the number that the hare makes " 3 in the same time; and if we take the length of the hare's leap as the unit of distance, the whole distance to

be passed over by the dog, will be denoted by + 50.


But the dog passes over a distance of units at each leap; hence, in x leaps, he will pass over a distance denoted by 3x. But, from the conditions of the prob lem, these two quantities are equal; hence,


... x = 300. Ans.

+ 50 =

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5. A can do a piece of work in 9 days, and B can do the same work in 10 days. In how many days can they both do it together?

Let x denote the number of days required.

If we denote the work by 1,

in 1 day, and in x days he can do of it in 1 day, and in x



of it.

But, from the conditions, the sum of these two will be equal to the entire work; that is,



= 1,


do of it.

can do of it



days he can do



... x = 414. Ans.



B can


6. If to a certain number its half and its eighth part be added, the sum will be equal to 78. What is the number? Ans. 48.

7. A bought a horse, harness, and wagon, for 250 dollars he gave for the harness one fourth as much as for the horse, and for the wagon as much as for the horse and harness together. What did he give for each?

Ans. $100 for the horse, $25 for the harness, and $125 for the wagon.

8. A drover sold from a flock of sheep one half, and two more; he then sold half that remained, and two more, and then he had 22 left. How many had

be at first?

Ans. 100.

9. If from 3 times a certain number we subtract 8, half the remainder will be equal to the number itself diminished by 2. What is the number? Ans. 4.

10. Ten years ago, a boy's age was

of his

father's; now, it is of it. now?

What is the father's age
Ans. 60 years.

11. The sixth part of a number added to its eighth part gives 56. What is the number?

Ans. 192.

12. Two boys had, together, 35 marbles. One fourth of the number that the first had was equal to one third of the number that the second had. How many had each? Ans. 20 and 15.

13. A man spent half of his money, and afterwards lost one third of what he had left, when he found that he had remaining $30. How much had he at first?

Ans. $90.

14. What number is that, from which if tracted, one half the remainder is equal to 15?

5 be sub

Ans. 35.

15. Divide $116 amongst three persons, so that the second shall have two thirds as much as the first, and the third shall have two fifths as much as the second. Ans. $60, $40, and $16.

16. A wheat field yielded 72 bushels, which was divided between landlord and tenant in such a way, that for every five bushels that the landlord received, the tenant got seven. How many bushels did the tenant receive?

Ans. 42.

17. The half of a number exceeds its third part by 8. What is the number? Ans. 48.

18. A sum of money is divided between A, B, and C so that A has $8, B has as much as A, together with one fifth as much as C, and C has as much as A and B together. How much has C? Ans. $20. If 20 be

19. There are 180 sheep in two flocks.

taken from the second and added to the first flock, the first flock will then contain twice as many as the second How many sheep are there in each flock?

Ans. 100 and 80.

20. A post feet in the air.

in the mud, in the water, and 10 What is its entire length?

Ans. 28 feet.

21. There are two numbers whose difference is 8, and the first is 5 times the second. What are the numbers? Ans. 10 and 2.

22. A merchant gains he finds that he has $8436.

per cent. on his capital, when
What was his capital?
Ans. $7400.

23. A has 3 times as much money as B; but if A were to give to B $100, B would then have 3 times as much as A. How much have they each?

Ans. A $150 and B $50.

24. A laborer was engaged for 30 days, on condition that for every day he labored, he was to receive $2, and for every day he was idle, he was to forfeit $1. At the end of the time he received $21. How many days did he labor?

Ans. 17.

25. A is twice as old as B, but 10 years ago he was three times as old. How old is B now.

Ans. 20 years.

increased by 9, diminished by 7 Ans. 45

26. Find that number which, being the result divided by 2, the quotient the result will be 20.

27. Divide the number 37 into three parts, such that the first shall be 3 less than the second, and the second5 greater than the third. Ans. 12, 15, and 10

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28. A man spends of his it come for board, of the remainder for clothing, and has remaining $70. What is his income? Ans. $630.

29. Divide 1000 into two parts such, that one of them shall be of the other. Ans. 375 and 625,

30. A person after spending 50 dollars more than half of his income, had remaining 125 dollars more than a third of it. How much was his income?

Ans. 1050 dollars.

31. In a naval action of a fleet was taken, of it sunk, and 2 ships burnt; of the remainder were afterwards lost in a storm, when 24 ships were left. How many ships were there in the fleet? Ans. 60.

32. A sum of 990 dollars was divided between 4, B, and C. B received as much as A, and Cas much as A and B together. How many dollars did Ans. A, 300; B, 240; and C, 450.

each receive?

33. A courier A starts 1165 of his own steps ahead of a courier B, and takes 5 steps whilst B takes but 4; now if 3 steps of B are equal to 4 of the courier A, how many steps must B make to overtake A?

Ans. 13980.

34. The hands of a clock are together at 12 o'clock, when are they next together? Ans. At 1 h. 5 m.

35. A grazier spent of his money for horses, for oxen, and of the remainder for sheep, when he had 980 dollars left. How many dollars had he originally? Ans. 2400.



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