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3. If the pieces were of a yard long, how many would there be? How does the number compare with the number when the pieces are ț of a yard long?

4. If-the pieces were of a yard long, how many would there be? How does the number of pieces compare with the number when the pieces are of a yard long?

5. How many times is contained in 1? ? ? ? ?

6. Since is contained in 1, eight times, how many times will it be contained in 1? What part of 8 times will it be contained in 1?

7. Since gis contained in 1 } of 8 times, or times, how many times will it be contained in 1? How many times in ? ??

8. What is the value of 1:-1? Of 1: 4? 1:42 4:-4?

9. Into how many parts of 3 eighths of a dollar each, can 6 eighths of a dollar be divided ?

10. How many sacks containing % of a barrel each, can be filled from io of a barrel of flour? How many times is o contained in 1? 4 in q? o in z/? &- in 47?

11. How many pine-apples at $1 cach, can be bought for $1? 12. How many times is contained in 2? } in į? fin ? 13. How many times is { contained in 1? In 1? In ļ? 14. How many times is contained in 1? In }? In 3?

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WRITTEN EXERCISES.

1. Divide by PROCESS.

ANALISIS.— } is contained in 1, 5 *=*= 4 X = 21

times; and is contained in 1, one-
third of 5 times, or times.

And since is contained in 1, f 4:$=3! = 3!= 1

times, in 4 it will be contained of

= ? i times. Or, 4 is equal to {}, and is equal to št. 21 thirty-fifths are contained in 20 thirty-fifths zi times.

Or,

DIVISION.

X Rule.— Multiply the dividend by the divisor inverted. ) Or,

Reduce the dividend and divisor to similar fractions and divide the numerator of the dividend by the numerator of the divisor. X When possible use cancellation.

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14. What is the quotient of į of of 5; divided by of

of 3?

PROCESS.

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ANALYSIS.— In the soof of 1 of off

lution of examples like

this, all mixed numbers **$*¥*$*$x*=*=64 should be changed to in

proper fractions, and all fractions that are factors of the divisor, inverted, and the product found as in previous examples.

15. Divide 4 of of 16 by of of 5).
16. Divide of 13 of 5} by 41 times } of 16.
17. Divide } of of 4 by 4 of 4.
18. Divide of of 1 by 5 times of 4.
19. Divide of 4 of 15 by of į of 6.
20. Divide 1 of Ï of 22 by { of 5 of 16.
21. Divide 3 of 3 of 6 by of 6 times 14.
22. Divide 84 times of 7 by of of 5.

23. How many pieces of ribbon 1 of a yard in length, can be made from ļof fr of a yard ?

24. If a man spends $; per day for cigars, in how many days will he spend $174?

25. How many yards of cloth at $3$ per yard can be bought for $317* ?

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26. At $per bushel, how many bushels of potatoes can be bought for $171?

27. If a family uses of a barrel of flour a week, how long will 54 barrels last?

28. If a boy earns $daily, how long will it take him to earn $3?

29. A certain number multiplied by 1 is equal to . What is that number?

30. If a man can saw 11 cords of wood in one day, how lung will he require to saw 17} cords?

31. If a horse eats 12). bushels of oats in 5 weeks, how much does he eat in a day?

32. When wheat is selling at $1} per bushel, how many bushels can be bought for $3168?

FRACTIONAL FORMS.

189. Expressions of unexecuted division are often written in the form of a fraction.

190, A fractional form having an integral denominator and a fractional numerator is called a Complex Fraction.

Thus, and are complex fractions.

Expressions which have a fraction in the denominator can not properly be regarded as Complex Fractions, though they are commonly classified as such.

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PROCESS.

4

ANALYSIS.— is an expression === *=of division, and is the same as 5 + 5,

of Soler

which is equal to in.

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191. To find the relation of one number to another.

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1. What part of 5 cents is 1 cent? 2 cents ? 3 cents ? 4 cents ?

2. What part of 9 acres is 5 acres? 7 acres ? 3 acres ? 4 acres ?

3. What part of 4 apples is 1 apple? į of 1 apple? of 1 apple? ã of 1 apple?

4. What part of $5 is $2? $1?$1? $}? $3? 5. What part of $6 is $1? $1? $3? $ž?

6. Henry had $5 and gave his brother 3: What part of his money did he give his brother?

7. James earned $7, and his brother $2. What part of the whole did each earn?

PRINCIPLE.Only like number's can have relation to each other.

8. What is the relation of 5 to 9?
Analysis.—1 is 1 of 9; and 5 is 5 times į of 9, or,
Hence 5 is of 9.

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What is the relation

9. Of 7 to 21 ? 10. Of 12 to 16? 11. Of 10 to 28?

12. Of 9 to 18?
13. Of 8 to 32?
14. Of 32 to 48?

15. Of 15 to 24?
16. Of 14 to 35?
17. Of 18 to 81?

18. What is the relation of to 2 ?
ANALYSIS.--1 is 1 of 2, and of 1 is of 1 of 2, or 1 of 2.
Hence is io of 2.

What is the relation

19. Of f to 4? 20. Of 4 to 9? 21. Of to 6?

22. Of á to 8?
23. Of

to 15?
24. Of 4 to 25?

25. Of to 6?
26. Of j to 18?
27. Of 5 to 12?

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28. What is the relation of to 4 ?

ANALYSIS.--1 is off, and 1 is 7 times } of 5, or } of z; and since 1 is of , f of 1 is z of off, or 3 of 3.

Hence is of 3.

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192. A number and its relation to another number giveu, to find the other number.

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1. 2 cents are 1 of how many cents? of how many cents ? 4 of how many cents?

2. 3 is of what number? ; of what number? t of what number?

3. 8 is of what number? of what number?. of what number?

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