divides the difference, 20, and one number, 50, it will divide 70: hence, it is a common divisor of 25 and 70; and since there is no other common factor, it is the greatest common divisor.

Hence, to find the greatest common divisor,

Rule.

Divide the greater number by the less, and then divide the preceding divisor by the remainder, and so on, till nothing remains: the last divisor will be the greatest common divisor.

Examples.

1. What is the greatest common divisor of 216 and 408? 2. Find the greatest common divisor of 408 and 740. 3. Find the greatest common divisor of 315 and 810. 4. Find the greatest common divisor of 4410 and 5670. 5. Find the greatest common divisor of 3471 and 1869. 6. Find the greatest common divisor of 1584 and 2772.

NOTE. If it be required to find the greatest common divisor of more than two numbers, first find the greatest common divisor of two of them, then of that common divisor and one of the remaining numbers, and so on for all the numbers: the last common divisor will be the greatest common divisor of all the numbers.

7 What is the greatest common divisor of 492, 744, and 1044?

8 What is the greatest common divisor of 944, 1488, and 2088?

9. What is the greatest common divisor of 216, 408, and 740?

10. What is the greatest common divisor of 945, 1560, and 22683?

99. How do you find the greatest common divisor, when the numbers are large?

4*

COMMON FRACTIONS.

100. A UNIT is a single thing; as, 1 apple, 1 chair, 1 pound of tea; and is denoted by 1.

If a unit be divided into two equal parts, ecch part is called, one-half.

If a unit be divided into three equal parts, each part is called, one-third.

If a unit be divided into four equal parts, each part is called, one-fourth.

If a unit be divided into twelve equal parts, each part is called, one-twelfth; and if it be divided into any number of equal parts, we have a like expression for each part.

101. The UNIT OF A FRACTION is the single thing that is divided into equal parts.

102. A FRACTIONAL UNIT is one of the equal parts of the unit that is divided.

NOTE. In every fraction, let the pupil distinguish carefully between the unit of the fraction and the fractional unit. The first is the whole thing from which the fractions are derived; the second, one of the equal parts into which that thing is divided.

100. What is a unit? By what is it denoted? What is onehalf? One-third? One-fourth? One-twelfth ?

101. What is the unit of a fraction?

102. What is a fractional unit? What is the difference between the unit of a fraction and a fractional unit?

103. Every whole number, except 1, has a fractional unit corresponding to it: thus, the numbers

2, 3, 4, 5, 6, 7, 8, 9, 10, &c.,

have, corresponding to them, the fractional units

1, 1, 1, 1, 1, 1, 1, 1, 1o, &c.

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If we suppose a class of boys each to have an apple, and that the apple of each be divided into equal parts corresponding to his class number, the first boy will have the whole apple, or the unit of the fraction; the second boy will have the whole apple in the two fractional units, one-half; the third, in the three fractional units, one-third; the fourth, in the four fractional units, one-fourth; and each boy of a higher number, will have the whole apple in as many fractional units as are denoted by his number in the class.

The fractional units of the fourth boy may be derived from those of the second, by dividing each half by 2, giving 4 fourths; the units of the 6th boy may be derived from those of the 2d, by dividing each by. 3, or from those of the 3d, by dividing each by 2; and similarly for any of the higher numbers which are multiples of the lower.

104. An INTEGRAL or WHOLE NUMBER is the unit 1, or a collection of units 1.

105. A FRACTION is a fractional unit, or a collection of fractional units.

103. What is the fractional unit corresponding to 2? To 4? To 6? To 12? To 65? If each of a class of boys has an apple divided into parts corresponding to his number, what will be the fractional unit of the 4th boy? How many fractional units will he have? How may they be derived from those of the second boy? What will be the fractional unit of the 12th boy? From those of what other boys may they be derived? How from the 2d? How from the 3d? How from the 4th? How from the 6th? 104. What is an integral, or whole number?

105. What is a fraction?

106. Any collection of fractional units, is thus written:

Hence, we see that every fraction may be divided into two factors; one of which is the fractional unit, and the other, the number denoting how many times the fractional unit is taken.

107. The DENOMINATOR is the number written below the line, and shows into how many equal parts the unit of the fraction is divided.

108. The NUMERATOR is the number written above the line, and shows how many fractional units are taken.

109. The TERMS of a fraction are the numerator and denominator, taken together; hence, every fraction has two terms.

110. The VALUE of a fraction is the number of times which it contains the unit 1.

111. TO ANALYZE a fraction consists in naming its unit, its fractional unit, and the number of fractional units taken : Thus, in the fraction, the unit of the fraction is 1; the fractional unit, ; and the number of fractional units taken is 3.

106. Explain the manner of writing fractional units. Into how many factors may every fraction be divided? What are they? 107. What is the denominator? What does it show?

108. What is the numerator? What does it show?

109. What are the terms of a fraction? How many terms has every fraction?

110. What is the value of a fraction?

111. What is the analysis of a fraction?

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

3 may be written and is read, 3 ones.

5 ones.

6 ones.

8 ones.

113. Properties of Fractions.

1. All the parts of the unit 1, however divided, make up the unit itself; hence, any fractional unit, multiplied by the number of parts, is equal to 1.

2. If the numerator is less than the number of parts, the value of the fraction is less than 1.

3. If the numerator is greater than the number of parts, some of the fractional units must have come from a second unit; and hence, the value of the fraction will be greater than 1.

Examples in writing and reading Fractions.

1. Analyze the following fractions:

2. Write 12 of the 17 equal parts of 1.

3. If the unit of the fraction is 1, and the fractional unit one-twentieth, express 6 fractional units; express, also, 12 and 18.

4. If the fractional unit is one 36th, express 32 fractional units; also, 35, 38, 54, 6, 8.

5. If the fractional unit is one-fortieth, express 9 fractional units; also, 16, 25, 69, 75.

6. Write forty-nine, one hundred and fifteenths.

7. Write three hundred and sixty-one, forty-sevenths.

112. How may a whole number be expressed fractionally? 113. When is a fraction equal to 1? When less than 1? Whe greater than 1?