3. Therefore, the factor 36 may be taken out of any two numbers, as from 36 and 72, leaving 1 and

2. Now the other number may be multiplied by these factors.

Thus, 1×2×108=216, the 1. c. m. of 36, 72, and 108.

NOTE. This method is not applicable unless the resulting quotients are prime to one another, but it is often found convenient where the numbers are large.

When the 1. c. m. of several numbers is required, the following RULE is convenient:

Write the numbers in a horizontal line, divide by any prime number that will divide two or more of them without a remainder, and place the quotients and undivided numbers in a line below. Divide this line as before and thus proceed until no two numbers have a common factor greater than 1. The continued product of the divisors and numbers in the last line will be the l. c. m. required.

Example. Find the 1. c. m. of 8, 14, and 21.

Using the rule, find the 1. c. m. of:

9. 100, 240, and 215. 11. 22, 33, 44, and 66. 10. 27, 36, and 45. 12. 18, 27, 36, and 81.

REVIEW

1. Define arithmetic, number, and unit.

2. What is a composite number? A prime number? 3. What is a factor? A multiple?

4. What is the h. c. f. of two numbers? The 1. c. m. of two numbers? Give examples.

5. What is the product of the h. c. f. and the 1. c. m. of 12 and 15?

6. What numbers are exactly divisible by 3? By 4? By 9? By 11?

7. Show that 1188 is divisible by 2, 3, 4, 6, 9, and 11, but not by 5, 7, or 8.

8. Is 11 a factor of 3535? Of 445? Of 96745? Of 7543173? Of 1087362?

9. If A can build 14 rods of fence in a day, B 25 rods, C 8 rods, and D 20 rods, what is the least number of rods that will furnish a number of whole days' work to either one of the four men?

10. A street 399 ft. long and 35 ft. wide is to be paved with square flagstones of equal size, and as large as possible. How long and wide must each flagstone be?

11. Three ships arrive at a certain port, the first every Monday, the second every ten days, and the third every 12 days; they all arrived on Monday, May 1. When will all three arrive together again?

12. A man having a triangular piece of land, the sides of which are 165 ft., 231 ft., and 385 ft., wishes to inclose it with a fence having panels of the greatest possible uniform length; what must be the length of each panel?

CHAPTER IV

FRACTIONS

I. COMMON FRACTIONS

98. A fraction (Latin frangere, to break) is one or more of the equal parts of a unit.

Two units are involved in a fraction, the unit of the fraction and the fractional unit.

99. The unit of the fraction is 1, or the one thing divided into equal parts.

100. The fractional unit is one of the equal parts into which the unit of the fraction has been divided.

In the fraction bu., the fractional unit is bu., and the unit of the fraction is 1 bu.

101. A common fraction is a fraction expressed by two numbers, one above and one below a horizontal line; as, which is read three fifths.

102. The terms of a fraction are the numerator and the denominator.

103. The numerator (Latin numerare, to number) is the number above the line. It shows the number of fractional units taken.

104. The denominator (Latin denominare, to name) is the number below the line. It shows the number of parts into which the unit of the fraction has been divided.

105. A proper fraction is one whose numerator is than its denominator; as, .

106. An improper fraction is one whose numerato equal to, or greater than, its denominator; as, 3.

107. A mixed number is a combination of an integer a a fraction; as, 43.

108. A simple fraction is a single fraction whose ter are integral numbers; as, §.

109. A complex fraction has one or both of its ter 3 33 3

fractional; as,

7' 4'ᄒ'

110. Similar fractions are those that have like denom nators; as, and §.

111. The reciprocal of a number is 1 divided by tha number; thus the reciprocal of 8 is 1.

112. The three interpretations of a fraction are illus trated by the following:

represents

1. 3 of the eight equal parts into which the unit of the fraction has been divided.

2. of 3 integral units.

3. 3 divided by 8.

113. Fundamental principles involved in fractions:

1. Multiplying the numerator of a fraction by any number multiplies the value of the fraction by the same number.

Reason. This increases the number of parts without changing their size.

2. Multiplying the denominator of a fraction by any

number divides the value of the fraction by the same number.

Reason. This decreases the size of the parts without changing the number of parts.

3. Multiplying both terms of a fraction by the same number does not change the value of the fraction. Reason. This increases the number of parts and decreases the size of the parts in the same ratio.

4. Dividing the numerator of a fraction by any number divides the value of the fraction by the same

number.

Reason. This decreases the number of parts without changing their size.

5. Dividing the denominator of a fraction by any number multiplies the value of the fraction by the same number.

Reason. This increases the size of the parts without changing the number of parts.

6. Dividing both terms of a fraction by any number does not change the value of the fraction.

Reason. This decreases the number of parts and increases the size of the parts in the same ratio.

REDUCTION OF FRACTIONS

114. The foregoing principles should be consciously applied by the learner in the various operations involving fractions.

Example.-1. Reduce to its lowest terms.