23. James was asked how many marbles he had, and replied, I have 55, black and red, and of the black make just as many as of the red. Pray how many have I of each sort?

24. A laborer engaged to work 20 days, and was to receive 9 shillings for every day he worked, and pay 3 shillings a day for his board, every day that he was idle; in settling, he received 84 shillings: how many days did he work?

25. A church, including the steeple, is 188 feet high. If the height of the steeple is equal to the height of the body of the building, what is the height of each ?

LESSON XV.

Analysis of Questions.

1. William chases Henry, who is 42 feet in advance, around a circular walk of 100 feet. Their steps are each 3 feet, but William takes 6 steps while Henry takes but 5: how many steps must William make to overtake Henry?

2. A hare is 25 of his own leaps before a greyhound, which is pursuing him. The greyhound makes 2 leaps while the hare makes 5; but 1 leap of the greyhound is equal to 3 of the hare's: how many leaps will the greyhound make before he overtakes the hare ?*

3. James is in pursuit of John, and 12 of John's steps behind him. James steps 3 times while John

* ANALYSIS. Since the greyhound makes 2 leaps while the bare makes 5, in the time that the greyhound makes 1 leap the hare will make 2 leaps. But 1 leap of the greyhound is equal to 3 leaps of the hare; hence, every time the greyhound jumps he will gain on the hare of a hare's leap: therefore, he must make 50 leaps to overtake him.

steps 4 times; but James' steps are twice as long as John's: how many steps must James make to overtake him?

4. Mary is 24 years old and Jane is 4: how many years must elapse before Mary's age will be just double Jane's ?*

5. A father is 60 years old and his son is 35: how long since the age of the father was double that of the son? What was then the age of each?

6. A mother is 36 years old and her daughter 12: how long before the age of the mother will be double that of the daughter? What will then be the age of each?

7. A mother is 48 years old and her daughter 30 : how long since the age of the daughter was half that of the mother? What was then the age of each?

8. A mother is 48 years old and her daughter 10: how long before the age of the daughter will be onethird that of the mother? What will then be the age of each ?

9. A mother is 54 years old and her daughter 20: how long since the age of the daughter was one-third that of the mother? What was then the age of each?

* ANALYSIS.-At Jane's birth, 4 years ago, Mary was 20 years old, and Jane's age was 0. Twenty years from that date, Mary will be twice as old as Jane: Hence, if Mary's age be diminished by Jane's, and the remainder be multiplied by 2, the product will denote Mary's age when it is double Jane's.

ANALYSIS. At the birth of the daughter, the mother was 38 years old. What number added to 38 years will give a sum equal to 3 times the number added?

If 3 times the number added is equal to 38 plus the numher, then, twice the number added will be equal to 38; and the number must be one half of 38, or 19 years. Hence, if the daughter's age be subtracted from the mother's, and the remainder divided by 2, the quotient will be the daughter's age, when it is one-third the mother's. If the difference be divided by 3, the quotient will be the daughter's age when it is one-fouth the mother's; and so on.

10. A father's age is 45 years, and his son 9: how long before the age of the son will be one fourth that of his father? What will then be the age of each?

11. A father is 54 years old, and his son 30: họw long since the age of the father was 4 times that of he son? What was then the age of each ?

LESSON XVI.

Analysis of Questions by means of Unity.

WE have said, that any number, regarded as a whole, may be called UNITY.

What may any number be called, when it is regarded as a whole?

1. What number added to twice itself will give a sum equal to 12?*

2. What number is that which added to three times itself will give a sum equal to 24?

3. What number is that which added to half itself will give 6 ?

4. What number added to half itself will give 9? 5. What number added to one-fourth of itself will give 20?

6. What number is that which added to twice itself, and the sum to 3 times itself will give 30?

7. What number added to half of itself, and to one fourth of itself will give 28?

8. James being asked his age, said, I am half the age of my father, and the sum of our ages is 60 years: What is the age of each ?

* ANALYSIS.-Call the number sought, unity. Then by the conditions of the question, unity plus twice unity is equal to 12. But unity plus twice unity is equal to 3 times unity, which is equal to 12. Then, if 3 times unity is equal to 12, once unity is equal to 12 divided by 3, which is 4.

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SUGGESTION.Let the pupil see if 4 will fulfil the conditions: 4 + 4 X2 4+8 12. Let every question of the lesson be analyzed and verified in a similar way

9. What number is that to which if its be added, the sum will be 35 ?

10. A man being asked his age, replied, if to my age you add one-third of it and then of it, the sum will be 57: How old was he?

11. John being asked how many marbles he had, aid, that one half of what he had, increased by 1, and diminished by 1, was equal to 14.

12. Divide the number 12 in two such parts, that one shall be three times the other?

13. Divide 24 into 3 such parts, that the second shall be 3 times the first, and the third 4 times the first: What are the numbers?

14. Divide 32 into 3 such parts, that the second shall be 4 times the first, and the third 3 times the first: What are the numbers?

15. Divide 27 into two such parts, that the second shall be of the first?

16. Divide 20 into three such parts, that the second shall be one half of the first, and the third, one third of the second ?

17. James asked Robert how many marbles he had. Robert answered, "If you will give me half as many as I now have, and afterwards give me of what I shall then have, my number will be 54." How many

had he?

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18. A young chap asked an old gentleman his age, who replied, "When you was born, I was of my present age: one third of your age plus one fourth of it, is equal to 7 years. Can you now tell how old I

am ?"

19. A tailor buys a piece of cloth for 6 dollars a yard, and a piece of equal length for $2 a yard, and sells the whole at $4 a yard: does he make or lose?

20. John and James together have 45 marbles: if John's is equal to of James, how many has each ?

21. A market woman bought geese and turkeys; 36 in all; one seventh of one sort was equal to one half of the other: how many of each kind?

22. A pole 72 feet long has one half as much in the mud as in the water, and twice as much in the air as in the mud and water together: how many feet in each ?

23. What number added to, to, and to of itself will give a sum equal to 50?

24. Mr. Wilson bought a hat, a coat, and a cloak he paid for the coat 11 as much as for the hat, and for the cloak, three times as much as for the coat; and for all he paid 42 dollars; how much did he pay for each ?

LESSON XVII.

Analysis of Questions by means of Unity.

1. What number is that to which if 10 be added, the sum will be 16?*

2. James being asked how many marbles he has, replied, if to 1⁄2 of what I have you add 8, the sum wil be equal to 18: how many had he?

3. What number is that whose half exceeds its third by 4?

4. What number is that, to which if of itself and 4 be added, the sum will be equal to 22?

5. What number is that, which being added to on third of itself, and to 3 times itself, will give a sum equal to 26?

6. What number is that, which being diminished by its half and its third, the remainder will be equa to 4?

* ANALYSIS.-Denote the required number by unity. Then by the conditions of the question, unity plus 10 equals 16. But if unity plus 10 equals 16, unity must be equal to 16 minus 10, or 6.

For, if the same number be subtracted from two equal rumiers, the remainders will be equal.