5. Seventy-seven one hundred forty-eighths; thirtyfive one hundred eighty-thirds.

6. One hundred twenty-seven two hundredths; one hundred sixty-three two hundred twelfths.

7. One hundred eight two hundred fortieths; two hundred sixty two hundred seventy-fourths.

8. Two hundred ninety-seven three hundred thirds; three hundred tenths.

9. Three hundred thirteen three hundred eighteenths; three hundred thirty-one three hundred fiftieths.

10. Three hundred seventy-nine four hundred ninths; forty-six four hundred twenty-sevenths.

11. Four hundred thirty-nine four hundred sixtyfirsts; one hundred eleven four hundred ninetieths.

12. Five hundred one five hundred fifths; five hundred fifty-seven hundredths.

93. Fractions Classified.

1. With respect to value, fractions are either proper or improper.

2. A proper fraction is one which expresses a number of relative units fewer than the number required to make the primary unit, as 4, . Its value is less than

one.

3. An improper fraction is one which expresses a number of relative units equal to, or greater than, the number required to make the primary unit, as , . Its value is one or more than one.

4. With respect to form, fractions are simple, complex, or compound.

A simple fraction is one that has a single number for each of its terms, as 4, §.

6. A complex fraction is one that has a fraction in 21/3 4 313 one or both of its terms, as 7 612' 52°

7. A compound fraction is a fraction of a fraction, as of 3.

8. A mixed number is a whole number and a fraction joined together; it is two numbers, as 4, which is read, "four and six sevenths."

94. Principles.

I. If both terms of a fraction are multiplied by the same number, its value is not changed; for, while the number of units is increased, their size is decreased proportionately. (How do you know that the size is decreased?)

ILLUSTRATION: Multiplying both terms of the fraction by 3 gives . There are three times as many units in as in, but the units are only one third as large.

II. If both terms of a fraction are divided by the same number, the value is not changed; for, while the number of parts is diminished, their size is increased proportionately.

ILLUSTRATION: Dividing both terms of the fraction by 2 gives. There are only one half as many units in as in §, but each unit is two times as large. (Why is one fourth twice as large as one eighth ?)

REDUCTION OF FRACTIONS.

95. Reduction in arithmetic is changing the form of a number without changing its value. The form is changed by changing the kind of measuring unit. Of course, then, to preserve the value unchanged, the number of units must be changed proportionately, according to the principles stated on the preceding page.

96. To Change a Fraction to its Lowest Terms.

A fraction is in its lowest terms when the terms contain no common prime factor except one. The process consists in removing all the common prime factors from both numerator and denominator. Is the value of the fraction changed? Why not?

ORAL EXERCISES.

97. Reduce the following fractions to their lowest terms:

18

1. 18, 18, 18, 11, 18, 14, 16, 18, 14, 18.

2. 7, 18, 16, 17, 14, 13, 48, 1, 13, 18.

49,

49 60

5,

64 43 80

75

3. 흉물, 등급, 죽음, 무음, 죽음, 흉음, 송금, 응응, 급, 10. 86 909 70%.

99. To Reduce an Improper Fraction to an Integer or Mixed Number.

Since the numerator expresses the number of relative units and the denominator shows how many of these make the primary unit or integer, there will be as many integers in any improper fraction as the given number of relative units is times the number required to make the primary unit or integer.

ORAL EXERCISES.

100. Reduce the following to integers or mixed numbers:

1. Reduce 28 to a mixed number.

EXPLANATION: 28 expresses twenty-eight relative units, five of which make a primary unit or integer. Then in 28 there are as many integers as there are 5's in 28. There are five 5's in 28, and 3 relative units remaining. Hence, 28=-53.

87

2. 24, 15, 56, 50, 79, 45, 181, 88, 17, 100.

89

83 30 66 39

3. 22, 47, 45, 12, 40, 400, 8, 10, 11, 11.

28

4. 108, 18, 51, 96, 75,

38 80

55, 63, 18, 18, 100.

92

6

65

5. 10, 8, 12, 21, 80, 22, 53, 15, ft, f§.

70

75

60

6. 10, 13, 135, 18, 100, 18, 188, 78, 96, 117.

109 9

7. 48, 186, 120, 140, 14, 18, 125, 120, H, H.

64 20 16

102. To Change an Integer or a Mixed Number to an Improper Fraction.

The proposed denominator shows how many of the required relative units there are in one primary unit. If this number is multiplied by the given number of primary units, the result is the required numerator.

1. Change 5 to eighths. 1; 5=5 times §, or 4o.

2. Change 4 to sevenths. Here we have two numbers of different denominations to add, for 4 is the same as 4+. The numbers can be made alike by changing the 4 to sevenths. The result is 2+3=34.

103. Change:

ORAL EXERCISES.

5 to fourths; 3 to sevenths; 4 to twelfths. 2. 6 to fifths; 4 to ninths; 7 to sixths.

3. 8 to tenths; 10 to fourths; 12 to thirds.

4. 13 to halves; 11 to ninths; 8 to eighths. 14 to fifths; 16 to thirds; 9 to twelfths.

5.

6. 12 to elevenths; 15 to sixths; 16 to fifths.