Thus: ( 1-half, 1-third, 1-fourth, 1-fifth, 1-tenth, 1-twelfth, }

3 }

are fractional units.

163. A Fraction is a fractional unit taken one or more

164. The Terms of a fraction are the two numbers used

to express it.

165. The Denominator is that term which names the parts expressed by the fraction. It is written below the horizontal line.

166. The Numerator is that term which numbers the parts expressed by the fraction. It is written above the horizontal line.

Thus the terms of the fraction & are 5 and 6; the denominator is 6, and the numerator is 5.

1o. 5 of the parts when 1 unit has been divided into 6 equal parts.

2o. The fractional unit 1-sixth, or , taken 5 times.

3o. The quotient of 5 divided by 6.

Read each of the following fractions, name the terms, the numerator, the denominator, the fractional unit, the number of fractional units, and give the three meanings of each fraction:

号;; }; 1%; 舌; 18; 5 ninths; 7 twelfths.

167. A Proper Fraction is one of which the numera

tor is less than the denominator; as 4,,, etc.

168. An Improper Fraction is one of which the numer

ator equals or exceeds the denominator; as §, 10, 25, etc.

169. A Mixed Number is an expression consisting of a whole number and a fraction; as 23, 5, 12, etc.

170. The Value of a fraction is the quotient of the numerator divided by the denominator.

The value of is 2; of 12 is 4; of 35 is 5.

When the numerator equals the denominator the value of the fraction equals 1; as 4, 7, 13, etc.

The value of a proper fraction is less than 1, as its numerator is less than its denominator.

The value of an improper fraction is equal to or exceeds 1, as its numerator equals or exceeds its denominator.

171. A Compound Fraction is a fraction of a fraction; as of, of of

MENTAL EXERCISES.

172. 1. How many halves () in one (1)? Why? 2. What is the worth of one orange, if 1 half of it is worth 5 cents? 6c.? 10c.?

3. How many thirds (1) in one (1)? Why?

4. What is the worth of one bale of cotton, if 1 third of it is worth $10? $12? * $20?

5. How many fifths (4) in one (1)? Why?

6. What is the length of a pole, if 1 fifth of it is 3 feet long? 7 feet long? 10 feet?

7. How many sevenths (4) in one (1)? Why?

8. What is the weight of a rock if 1 seventh of it weighs 5 pounds? 9 pounds?

9. How many tenths in one (1)? Why?

10. How many marbles in a box, if 1 tenth of them is 6 marbles? 9 marbles? 25 marbles?

Whose

11. What is the number whose half is 1? third is 2? Whose fourth is 5? Whose sixth is 2? Whose ninth is 3? Whose tenth is 7?

12. How much is 3-fourths (4) of 28?

ANALYSIS.-1 fourth of 28 is 7; 3 fourths of 28 is 3 times 7=21, Ans.

of 15?

of 30?

of 20?

of 40?

of 30?

of $50?

of 60c.?

of 36?

% of 45c.?

of 36?

of $60.

of 24?

18. How much is 2 thirds of 12? 14. How much is 3 fourths of 16? 15. How much is 4 fifths of 20? 16. How much is 5 sixths of 18? 17. How much is 7 ninths of 18? 18. How much is 5 twelfths of 24? 19. If one acre of land cost $12, what is the cost of of an acre?of an acre? of an acre?

20. If one bushel of potatoes cost 30 cents, what is the cost of of a bushel? of a bushel? of a bushel? 21, 5 sixths of a number is 10, what is the number?

ANALYSIS.-If 5 sixths is 10, 1 sixth is of 10, or 2; hence, the number is 6 times 2=12, Ans.

22. What is the number of which 5 sixths is 20? is 20? # is 16? is 10? is 21?

23. What will a melon cost if of it cost 21 cents? If of it cost 18 cents? If § of it cost 35 cents?

FUNDAMENTAL PRINCIPLES.

173. CASE I. To multiply and divide fractional units by whole numbers.

Into how many parts is this bar of soap divided? What is 1 part called? Is it a fractional unit?

If each of the three parts is cut into two parts, how many parts will there be in all? What is 1 part called? Is it a fractional unit

Is 1 part of the first bar equal to 2 times 1 part of

1°. To multiply a fractional unit by a number, we may divide the denominator by that number.

2o. To divide a fractional unit by a number, we multiply the denominator by that number.

174. CASE II.-To change the fractional unit.

Into how many equal parts is this bar of soap divided? What is 1 part called? 2 parts? Is a fractional unit?

Here is an equal bar; into how many parts is it divided? What is 1 part called? 2 parts? etc. Is a fractional unit?

Is 1 part of the first bar, or, equal to two parts of the second, or ?

Are 2 parts of the first bar equal to 4 parts of the second?

What does this show?

Hence,

Ans. That, and = }.

1o. Multiplying both terms of a fraction by the same number does not alter the value of the fraction.

2o. Dividing both terms of a fraction by the same number does not alter the value of the fraction.

EXPLANATION.-Can 24 be written thus: 6 fours? Thus: 12 twos? Are 6 fours equal to 12 twos? Why? Ans. The unit two is half of the unit four, but is taken twice as often, since 12 is twice 6; hence, they are equal.

Are 3 fifths, or, equal to 6 tenths, or? Why? Ans. The unit 1 tenth is half of the unit 1 fifth, but is taken twice as often, since 6 is twice 3; hence,

=

Why, then, is the value of a fraction not changed by multiplying both terms by the same number? Ans. Because it decreases the fractional unit in the same ratio that it increases the number of times it is taken.

Why is the value of a fraction not changed by dividing both terms by the same number? Ans. Because it increases the fractional unit in the same ratio that it decreases the number of times it is taken.

REDUCTION OF FRACTIONS.

175. Reduction of a Fraction consists in changing its terms without altering its value.

176. CASE I.-To reduce a fraction to its lowest terms.

A fraction is in the lowest terms when no number greater than 1 will exactly divide its numerator and denominator.