When several numbers are prime to each other, what must their L. C. M. equal?

201. The above is a good method of finding the L. C. M. of numbers that are easily separated into their prime factors. For larger numbers, the following method is recommended. ILLUSTRATIVE EXAMPLE II. Find the L. C. M. of 36, 112,

RULE II. To find the L. C. M. of two or more numbers:

1. Express the given numbers in a line as dividends. Make any prime number which is a factor of two or more of the given numbers a divisor of those numbers.

2. Express the quotients and undivided numbers beneath as new dividends, and continue dividing as before, till the last quotients and undivided numbers are prime to each other.

3. The product of all the divisors, last quotients, and undivided numbers is the L. C. M. required.

81. What is the width of the narrowest box that will exactly pack ribbons either 2, 3, or 4 inches wide?

Ans. 12 inches.

82. What is the smallest sum of money that may be made up of either 2-cent, 3-cent, 5-cent, 10-cent, or 25cent pieces?

Ans. $1.50.

83. Charles and Henry, wishing to ascertain how many slats there were in a certain fence, commenced counting them at the same place and counted in the same direction; Charles marked every tenth slat and Henry every twelfth. On which slats in order would both their marks be found? Ans. Every 60th slat.

For Dictation Exercises, see Key.

203. TOPICAL REVIEW IN PROPERTIES OF NUMBERS.

The pupil may illustrate the following topics to his class, using common objects when practicable, and giving definitions and rules: 1. Integral Number. (Art. 172.)

2. Factor. (Art. 173, with notes.)

3. Composite and prime numbers, and prime factor. (Art. 174176.)

4. Odd and even numbers. (Art. 178.)

5. Divisibility of numbers by 2, 3, 4, 5, 6, 8, 9, 10, 100, 1000, &c. (Art. 178.)

6. Finding the prime factors of a number. (Art. 181, 182.)

7. Greatest common factor of two or more numbers. (Art. 185188.)

8. Greatest common factor of two or more numbers, second method. (Art. 190.)

9. Cancellation. (Art. 192.)

10. Least common multiple. (Art. 194-199.)

11. Least common multiple, second method. (Art. 201.)

FRACTIONS.

204. If a unit, as an orange, is divided into two equal parts, each of these parts is one half of the unit.

If a unit is divided

into three equal parts, each of the parts is one third of the unit.

One of the equal parts of a unit is a Fraction or fractional unit.

Define a fraction or fractional unit.

205. A collection of fractional units is a fractional number.

206. The unit of which the fraction is a part is the unit of the fraction. Define the unit of the fraction.

NOTE TO THE TEACHER. -In its broadest sense, any part of a unit is a fraction, and any part of a unit when measured is found to be one of the equal parts of a unit or

But as in operating with fractions we consider the fractional units as separated, each fractional unit is a fraction, and two or more fractional units is a number of fractions or a fractional number.

It is believed that this view of fractions will greatly simplify the subject.

207. A knowledge of a fractional number, as two fourths of an apple, implies a knowledge, first, of the thing divided or the unit of the fraction; second, of the number of equal parts into which the unit of the fraction is divided, namely, "four"; and third, a knowledge of the number of parts taken, namely, "two."

Then in expressing a fraction or a fractional number, we must express the unit of the fraction, the number of equal parts into which the unit of the fraction is divided, and the number of fractional units taken.

To express the unit of the fractional number two fourths of an apple, we employ the word apple; to express the number of parts into which the unit has been divided we employ the figure 4 below a line, thus ; to express the number of fractional units taken, we employ the figure 2 above the same line.

The full expression of this fractional number will then be apple, which is read "two fourths of an apple."

208. EXERCISES.

Express the following fractional numbers in figures : —

1. Three fourths of an apple.

2. Two thirds of a peach.

3. One eighth of an orange.
4. Seven twelfths of a yard.

Name the unit of the fraction of each of the above.
Name the fractional unit of each.

209. We have seen that to express a fourth of anything, as an apple, we write a figure 4 below a line, thus

The number of equal parts, four, into which the unit of the fraction is divided, and which is expressed below the line, gives the name fourth to each of the fractional units; it is, therefore, the namer or denominator of each of the fractional units and of the fractional number.

Define denominator of fractional unit or fractional number. What is the denominator of? of?

210. The number of parts taken, and which is expressed above the line, as 2 in 2, is the numerator of the fraction or of the fractional number.

Define numerator of a fraction or fractional number.
What is the numerator of ? of?

211. The numerator and denominator of a fraction or of a fractional number are called its terms.

What are the terms of a fraction or fractional number? What are the terms of apple, and what does each term show? Ans. Three is the denominator and two is the numerator; the denominator three shows that the unit of the fraction is divided into three parts, and the numerator two that two parts are taken.

212. EXERCISES.

Name the terms of each of the following, and tell what each term

Ans. We cannot add them as now expressed.

3. Add apple and apple.

Ans.apple.

By comparing the above examples we see that fractions which can be added must be like parts of the same or similar units.

Such fractions are like fractions.

Define like fractions.

Numbers whose units are like fractions are like frac

tional numbers.

Define like fractional numbers.

214. EXAMPLES.

Add the following:

1.

orange and orange. | 5. $fo, $1, $.

2. melon and melon.

Ans. $28.

3.

week and 4 week.

6. $70, $to, $fo.

4. $, $, and $ 3.

Ans. $18.