Examples.

1. What is the greatest common divisor of 3328 and 4592? 2. What is the greatest common divisor of 2205 and 4501? 3. What is the greatest number that will divide 16082 and 25740?

4. What is the greatest number that will divide 620, 1116 and 1488 ?

5. What is the greatest common divisor of 5270, 5952, 5394, and 3038 ?

6. What is the greatest common divisor of 4617, 7695, 6642, and 8424?

7. A farmer has 315 bushels of corn, and 810 bushels of wheat; he wishes to draw the corn and wheat to market separately in the fewest number of equal loads: how many bushels must he draw at a load?

8. The Illinois Central Railroad Company have 15750 acres of land in one location, and 21725 acres in another. They wish to divide the whole into lots of equal extent, containing the greatest number of acres that will give an exact division: how many acres will there be in each lot?

9. A man has a corner lot of land, 1044 feet long and 744 feet wide. The adjacent sides are bounded by the highway, and he wishes to build a board fence with the fewest panels of equal length what must be the length of the panels?

10. A farmer has 231 bushels of barley, 369 bushels of oats, and 393 bushels of wheat, all of which he wishes to put into the smallest number of bags of equal size, without mixing: how many bushels must each bag contain?

11. Three persons, A, B, and C, agree to purchase a lot of 63 cows at the same price per head, provided each man can thus invest his whole money. A has $286; B, $462; and C, $638: how many cows could each man purchase?

COMMON FRACTIONS

123. AN INTEGRAL, or whole number, is the unit 1, or a collection of such units.

NOTE.-All integral numbers are formed by the continual additions of 1: as, 1+1=2, 2 + 1 = 3, &c.

124. A UNIT is a single thing; as, an apple, a chair, a hat, &c.; and is denoted by 1.

If a unit be divided into two equal parts, each 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.

The, is an entire half; the, an entire third; the 4, an entire fourth; and the same for each of the other equal parts; hence, each equai part is an entire thing, and is called a fractional unit.

123. What is an integral, or whole number? numbers formed?

How are integral

124. What is a unit? By what is it denoted? What is each part called when the unit 1 is divided into two equal parts? When it is divided into three? Into four? Into five? Into twelve?

125. THE UNIT OF A FRACTION is the single thing that is divided into equal parts.

126. THE FRACTIONAL UNIT is one of the equal parts of the unit that is divided.

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

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 fraction is derived; the second, one of the equal parts into which that thing is divided.

128. 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,
1⁄2, 1, 1, 1, 1, 1, 1, 1, 1, &c.

129. Expressing Fractions.

Each fractional unit may, like the unit 1, become the base of a collection: thus, suppose it were required to express 2 of each of the fractional units, we should then write

If it were required to express 3 of each of the fractional units, we should write

which is read 3 halves = × 3.

Hence, if we suppose a second unit to be divided into the same number of equal parts, such parts may be expressed in the same collection with the parts of the first: thus,

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

Thus,

3 may be written and is read, 3 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: Hence, we see,

1. That Fractions are expressed by two numbers, one written above the other, with a line between them.

2. 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.

130. THE DENOMINATOR is the number written below the line. It shows into how many equal parts the unit of the fraction is divided.

125. What is the unit of a fraction ?-126. What is the fractiona unit?--127. What is a fraction ?-128. What fractional unit correspond to the whole number 5? What to 6? What to 14?

129. May a fractional unit become the base of a collection? How are fractions expressed? Into how many factors may every fractiou be divided? What are they?

130. What is the denominator of a fraction?

131. THE NUMERATOR is the number written above the line It shows how many fractional units are taken.

132. THE TERMS of a fraction are the numerator and de nominator taken together: hence, every fraction has two terms.

133. THE VALUE of a fraction is the quotient of the numerator divided by the denominator.

134. THE ANALYSIS of a fraction is the naming of its unitits fractional unit-and the number of fractional units taken.

135.

Analysis of Fractions.

How is the fraction to be interpreted?

1. The unit of the fraction is 1.

2. The unit of the fraction is divided into 8 equal parts; hence, the fractional unit is one-eighth.

3. Seven fractional units are taken. In the fraction, the base of the collection of fractional units is, but it is not the primary base. For, is one-eighth of the unit 1; hence, the primary base of every fraction is the unit 1.

The expression may also be interpreted as the quotient of 7 divided by 8. In the latter case, the thing divided is the number 7; in the former, it was the number 1. The value in both cases is the same; for, seven times one-eighth of 1, is equal to one of the 8 equals of 7. Hence, a fractional expression has the same form as an unexecuted division.

136. Principles and Properties of Fractions.

1. A fraction is a fractional unit, or a collection of such units. 2. The denominator shows into how many equal parts the unit of the fraction is divided.

131. What is the numerator of a fraction ?-132. What are the terms of a fraction? How many terms has every fraction?-133. What is the value of a fraction ?-134. What is the analysis of a fraction?

135. Analyze the fraction. What is the base of the collection! What is the primary base? What else does express?

136. Explain the principles and properties of Fractions.