10. What is the least common multiple of 270, 189, 297, 243? Ans. 187110.

11. What is the least common multiple of 1, 2, 3, 4, 5, 6, 7, 8,9? Ans. 2520.

12. What is the smallest sum of money for which I could purchase an exact number of books, at 5 dollars, or 3 dollars, or 4 dollars, or 6 dollars each? Ans. 60 dollars.

13. A farmer has 3 teams; the first can draw 12 barrels of flour, the second 15 barrels, and the third 18 barrels ; what is the smallest number of barrels that will make full loads for any of the teams? Ans. 180.

14. What is the smallest sum of money with which I can purchase cows at $30 each, oxen at $55 each, or horses at $105 each? Ans. $2310.

15. A can shear 41 sheep in a day, B 63, and C 54; what is the number of sheep in the smallest flock that would furnish exact days' labor for each of them shearing alone?

Ans. 15498.

16. A servant being ordered to lay out equal sums in the purchase of chickens, ducks, and turkeys, and to expend as little money as possible, agreed to forfeit 5 cents for every fowl purchased more than was necessary to obey orders. In the market he found chickens at 12 cents, ducks at 30 cents, and turkeys at two prices, 75 cents and 90 cents, of which he im、 prudently took the cheaper; how much did he thereby for feit? Ans. 80 cents.

CLASSIFICATION OF NUMBERS.

Numbers may be classified as follows:

106. I. As Even and Odd.

107. II. As Prime and Composite.

What is the first classification of numbers? What is an even number? An odd number? Second classification? A prime number? A composite number?

108. III. As Integral and Fractional.

An Integral Number, or Integer, expresses whole things. Thus, 281; 78 boys; 1000 books.

A Fractional Number, or Fraction, expresses equal parts Thus, half a dollar; three-fourths of an hour; seven-eighths of a mile.

of a thing.

109. IV. As Abstract and Concrete.

110. V. As Simple and Compound.

A Simple Number is either an abstract number, or a concrete number of but one denomination. Thus, 48, 926; 48 dollars, 926 miles.

A Compound Number is a concrete number whose value is expressed in two or more different denominations. Thus, 32 dollars 15 cents; 15 days 4 hours 25 minutes; 7 miles 82 rods 9 feet 6 inches.

111. VI. As Like and Unlike.

Like Numbers are numbers of the same unit value.

If simple numbers, they must be all abstract, as 6, 62, 487; or all of one and the same denomination, as 5 apples, 62 apples, 487 apples; and, if compound numbers, they must be used to express the same kind of quantity, as time, distance, &c. Thus, 4 weeks 3 days 16 hours; 1 week 6 days 9 hours; 5 miles 40 rods; 2 miles 100 rods.

Unlike Numbers are numbers of different unit values. Thus, 75, 140 dollars, and 28 miles; 4 hours 30 minutes, and 5 bushels 1 peck.

What is the third classification? What is an integral number? A fractional number? What is the fourth classification? An abstract number? A concrete number? What is the fifth classification? A simple number? A compound number? Sixth classification? What are like numbers? Unlike numbers ?

FRACTIONS.

DEFINITIONS, NOTATION, AND NUMERATION.

112. If a unit be divided into 2 equal parts, one of the parts is called one half.

If a unit be divided into 3 equal parts, one of the parts is called one third, two of the parts two thirds.

If a unit be divided into 4 equal parts, one of the parts is called one fourth, two of the parts two fourths, three of the parts three fourths.

If a unit be divided into 5 equal parts, one of the parts is called one fifth, two of the parts two fifths, three of the parts three fifths, &c.

The parts are expressed by figures; thus,

Hence we see that the parts into which a unit is divided take their name, and their value, from the number of equal parts into which the unit is divided. Thus, if we divide an orange into 2 equal parts, the parts are called halves; if into 3 equal parts, thirds; if into 4 equal parts, fourths, &c.; and each third is less in value than each half, and each fourth less than each third; and the greater the number of parts, the less their value.

When a unit is divided into any number of equal parts, one or more such parts is a fractional part of the whole number, and is called a fraction. Hence

113. A Fraction is one or more of the equal parts of a unit.

Define a fraction.

DEFINITIONS, NOTATION, AND NUMERATION.

87

114. To write a fraction, two integers are required, one to express the number of parts into which the whole number is divided, and the other to express the number of these parts taken. Thus, if one dollar be divided into 4 equal parts, the parts are called fourths, and three of these parts are called three fourths of a dollar. This three fourths may be written

3 the number of parts taken.

4 the number of parts into which the dollar is divided.

115. The Denominator is the number below the line.

It denominates or names the parts; and

It shows how many parts are equal to a unit.

116. The Numerator is the number above the line. It numerates or numbers the parts; and

It shows how many parts are taken or expressed by the fraction.

117. The Terms of a fraction are the numerator and denominator, taken together.

118. Fractions indicate division, the numerator answering to the dividend, and the denominator to the divisor. Hence, 119. The Value of a fraction is the quotient of the numerator divided by the denominator.

120. To analyze a fraction is to designate and describe its numerator and denominator. Thus, is analyzed as follows:

4 is the denominator, and shows that the unit is divided into 4 equal parts; it is the divisor.

3 is the numerator, and shows that 3 parts are taken; it is the dividend, or integer divided.

3 and 4 are the terms, considered as dividend and divisor. The value of the fraction is the quotient of 34, or 3.

How many numbers are required to write a fraction? Why? Dcfine the denominator. The numerator. What are the terms of a fraction? The value? What is the analysis of a fraction ?

EXAMPLES FOR PRACTICE.

Express the following fractions by figures:

1. Seven eighths.

2. Three twenty-fifths.

3. Nine one hundredths.

4. Sixteen thirtieths.

5. Thirty-one one hundred eighteenths.

6. Seventy-five ninety-sixths.

7. Two hundred fifty-four four hundred forty-thirds. 8. Eight nine hundred twenty-firsts.

9. One thousand two hundred thirty-two seventy-five thousand six hundredths.

10. Nine hundred six two hundred forty-three thousand eighty-seconds.

Read and analyze the following fractions:

121. Fractions are distinguished as Proper and Improper. A Proper Fraction is one whose numerator is less than its denominator; its value is less than the unit, 1. Thus, 12, 16, To, are proper fractions.

An Improper Fraction is one whose numerator equals or exceeds its denominator; its value is never less than the unit, 1. Thus, 7, 3, 45, 35, 10, 182 are improper fractions.

122. A Mixed Number is a number expressed by an integer and a fraction; thus, 41, 1718, 9 arc mixed numbers. 123. Since fractions indicate division, all changes in the terms of a fraction will affect the value of that fraction according to the laws of division; and we have only to modify the language of the General Principles of Division (87) by substituting the words numerator, denominator, and fraction, or value

What is a proper fraction? An improper fraction? A mixed number? What do fractions indicate?