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each part called ? How do you express one part? Two parts ? Four parts ?. Six parts ? Nine parts ?

12. If a unit be divided into ten equal parts, how do you express one of the parts? How do you express two of them? Ten of them?

13. How many fifths are there in one unit? How many tenths are there? How much greater is a fifth than a tenth ? How many tenths are equal to two-fifths ? How many tenths are equal to four-fifths ? To fivefifths ?

14. If a unit be divided into eleven equal parts, what is each part called ? How do you express one of the parts? Two of them ? Four of them? Six of them ? Seven of them? Eleven of them ?

15. If a unit be divided into twelve equal parts, what is each part called? How do you express it? How do you express five of the parts? Six of them? Twelve of them ?

16. How many sixths are there in one unit? How many twelfths are there? How much greater is a sixth than a twelfth ? How many twelfths are equal to twosixths ? How many to three-sixths? To four-sixths ? To five-sixths ? To six-sixths ?

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17. What is the half of one-half? What is the half of one-third ? What is the half of one-fourth? What is the half of one-fifth ? What is the half of one-sixth ?

18. What is the sum of one-half and one-half? What is the sum of one-third and one-third ? What is the sum of one-third and two-thirds ?

19. What is the sum of one-fourth and two-fourths ? Of one-fourth and three-fourths ? Of two-fourths and two-fourths ?

20. What is the sum of one-fifth and two-fifths ? What is the sum of one-fifth and three-fifths ? Of two-fifths and ?three-fifths ?

21. What is the sum of one-sixth and four-sixths ? What is their difference? What is the sum of threesixths and two-sixths ? What is their difference ?

22. What is the sum of one-seventh and four-sevenths ?

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What is their difference? What is the sum of five-sev. enths and two-sevenths ? What is their difference ?

23. What is the difference between five-eighths and three-eighths ? What is their sum? What is the difference between six-eighths and one-eighth? What their

sum?

How many

24. What is the difference between seven-ninths and two-ninths ? What is their sum ? What is the differ. ence between four-ninths and three-ninths ? What is their sum?

25. What is the difference between one-tenth and sixtenths ? What is their sum ? What is the difference between six-tenths and four-tenths ? What is their sum?

26. What is the difference between one-eleventh and four-elevenths ? What is their sum? What is the dif. ference between three-elevenths and eight-elevenths ? What is their sum ?

27. What is the sum of three-twelfths and six-twelfths ? What is their difference? What is the sum of five-twelfths and seven-twelfths ? What is their difference? What is the sum of eight-twelfths and four-twelfths ? " What is their difference ?

28. How many halves are there in one ? thirds ? Fourths ? Sevenths? Elevenths ? Twelfths ?

29. How many halves are there in two ? How many thirds ? Fourths? Fifths ? Sixths ? Sevenths ? Twelfths ?

30. How many thirds are there in three ? How many fourths ? Fifths ? Sixths ? Tenths ? Twelfths ?

31. How many fourths are there in four ? How many fifths ? How many sixths ? How many sevenths? Eighths ? Ninths ? Twelfths ?

32. How many sixths are there in five? How many sevenths ? How many eighths ? How many elevenths? How many twelfths ?

33. How many sixths in nine? How many twelfths in ten ? How many elevenths in six ? In seven, how many ? In eight?

34. How many twelfths in two ? In four, how many! In five ? How many ninths in five ? In eight ? In ten? In eleven? In twelve ?

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35. How many whole units in two halves? In four halves? In five halves? In seven halves ? In nine halves ?

36. How many whole units are there in three-thirds ? In six-thirds? In nine-thirds? In five-thirds? In eightthirds ? In eleven-thirds ?

37. How many whole units are there in four-fourths ? In six-fourths ? In eight-fourths ? In eleven-fourths ? In sixteen? In twenty-nine? In thirty-six fourths ?

38. How many whole units are there in five-fifths ? In eight-fifths? In nine-fifths ? In twelve-fifths ? In twenty fifths? In twenty-six fifths ? In twenty-eight fifths ?

39. How many whole units are there in six-sixths ? In twelve-sixthş? In fifteen-sixths ? In eighteen-sixths ? In twenty-five sixths? In twenty-six sixths? In thirty. seven sixths ?

40. How many whole units are there in seven-sevenths ? In fourteen-sevenths ? In nineteen-sevenths ? In twenty-nine sevenths ? įn thirty-five sevenths ? In forty-two sevenths ?

41. How many whole units are there in eight-eighths ? In twenty-four eighths ? In sixteen-eighths? In thirty eighths ? In thirty-four eighths ? In forty eighths ?

42. How many whole units are there in nine-ninths ? In eighteen-ninths ? In twenty-four ninths? In thirtyfive ninths ? In forty-five ninths ? 43. How

many

whole units are there in ten-tenths ? In twenty tenths ? In thirty tenths? In forty-five tenths ? In sixty tenths ?

44. How many whole units are there in eleven-elev. enths? In twenty-two elevenths ? In thirty-three elevenths ? In forty-four elevenths? In sixty-elevenths ? In sixty-seven elevenths ?

45. How many whole units are there in twelve-twelfths ? In twenty-four twelfths ? In twenty-six twelfths ? In forty-eight twelfths ? In fifty twelfths ? In sixty twelfths ?

46. What is the sum of one-half and one-fourth? What is the sum of one-half and three-fourths ? Of threefourths and six-fourths ?

79

OF THE DIFFERENT KINDS OF VULGAR FRACTIONS.

110. There are six kinds of vulgar fractions—Proper, Improper, Simple, Compound, Mixed, and Complex.

À PROPER FRACTION is one in which the numerator is less than the denominator. The value of every proper fraction is less than 1. (See ART. 109.) The following are proper fractions:

}, }, }, , , , , , 111. An IMPROPER FRACTION is one in which the nu. merator is equal to, or exceeds the denominator. Such fractions are called improper fractions because they are equal to, or exceed unity. When the numerator is equal to the denominator, the value of the fraction is 1; in every other case the value of an improper fraction is greater than 1. The following are improper fractions :

, , , , , , , 4. 112. A SIMPLE FRACTION is a single expression. A simple fraction may be either proper or improper. The following are simple fractions :

,,,号,号,曼,曼, 113. A COMPOUND FRACTION is a fraction of a fraction, or several fractions connected together with the word of between them. The following are compound fractions :

1 of }, { of ļof , ļof 3, of į of 4. 114. A MIXED FRACTION is made up of a whole number and a fraction. The whole numbers are sometimes called integers.

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8

9

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they?

110. How many kinds of Vulgar Fractions are there? What aro

What is a proper fraction ? Is its value greater or less than 1?

111. What is an improper fraction ? Why is it called improper ? When is its value equal to 1 ?

112. What is a simple fraction? May it be either proper or im. proper ?

113. What is a compound fraction? 114. What is a mixed fraction ?

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The following are mixed fractions :

31, 41, 63, 53, 65, 31. 115. A COMPLEX FRACTION is one having a fractional numerator or denominator. The following are complex fractions :

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5' 191' 691 116. The numerator and denominator of a fraction, taken together, are called the terms of the fraction. Hence, every fraction has two terms.

117. A whole number may be expressed fractionally by writing 1 below it for a denominator. Thus,

3 may be written i and is read, 3 ones.
5

5 ones.
6

6 ones. 8

8 ones. But 3 ones are equal to 3, 5 ones to 5, 6 ones to 6, and 8 ones to 8. Hence, the value of a number is not changed by placing 1 under it for a denominator. 118. If an apple be divided into 6 equal parts, I will express one of the parts,

two of the parts,
three of the parts,

&c., and generally, the denominator shows into how many

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&C.,

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115. What is a complex fraction? Give an example of a proper fraction. Of an improper fraction. Of a simple fraction. Of a com. pound fraction. Of a mixed fraction. Is four-ninths a proper or improper fraction? What kind of a fraction is six-thirds ? What is its value? What kind of a fraction is nine-eighths ? What is its value? What kind of a fraction is one-half of a third? What kind of a fraction is two and one-sixth ? Four and a seventh? Eight and a tenth?

116. Wbat are the terms of a fraction? What are the terms of the fraction three-fourths? Of five-eighths? Of six-sevenths ?

117. How may a whole number be expressed fractionally? Does this alter its value? Give an example.

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