both 16 and 32 without a remainder, and, therefore, 32nds can be changed to a denomination 16 times greater than 32nds, which is halves.

2. Since halves are 16 times greater than 32nds, there can be but one-sixteenth as many halves as thirty-seconds, which is one half.

Therefore, in there is 2.

22. In the same manner reduce the following fractions to their lowest terms: ; ; ; ; ; ; ; ; #;&#; &; zz.

23. Change the following fractions to their lowest terms: ; ; ; 11; 1; 11; TOO.

QUESTIONS.—What is cancellation? (117.) What is a common divisor of two or more numbers? (118.) What is the greatest common divisor of two or more numbers? (119.) What is a multiple of a number? (120.). What is a common multiple of two or more numbers? (121.) What is the least common multiple of two or more numbers? (122.) What is a fraction? (123.) What is a common fraction? (124.) What is a decimal fraction? (125.) What are the terms of a fraction called? (126.)

LESSON VII.

1. How many 7ths in 5?

SOLUTION. Since in 1 unit there are 7 sevenths, in 5 units there are 5 times 7 sevenths, which are 35 sevenths, which with 8 sevenths added make 38 sevenths.

Therefore, in 5 there are 33.

2. How many 5ths in 93? in 84? in 53? in 7 units? in 8 units? in 10 units?

3. How many 8ths in 63? in 12? in 5? in 9 units? 4. How many 9ths in 53? in 61? in 78? in 114? 5. How many 16ths in 3§?

ANALYSIS. (1.) Change 3 units to sixteenths.

Since there are 16 sixteenths in one unit, in 3 units there are 3 times 16 sixteenths, which are 48 sixteenths.

(2.) Change to sixteenths.

Since there are 16 sixteenths in one unit, in 1 eighth of a unit there is one eighth of 16 sixteenths, which are 2 sixteenths, and in 5 eighths of a unit there are 5 times 2 sixteenths, which are 10 sixteenths.

(3.) Find the sum of the sixteenths.

Since in 3 units there are 48 sixteenths, and in

there are

10 sixteenths, in both there are the sum of 48 sixteenths and 10 sixteenths, which are 58 sixteenths.

Therefore, in 33 there are 58.

6. How many 15ths in 43? in 93? in 113? in 121? in 83? in 8 units? in 1 unit? in 43?

7. How many 18ths in 3? in 81? in 113? in 5? in 127? in 23? in 18? in 4 units? in of a unit? 8. How many units in 35 ?

SOLUTION. Since in 8 eighths there is one unit, in 35 eighths there are as many units as 8 eighths are contained times in 35 eighths, which are 4, with a remainder of 3 eighths of a unit. Therefore, in 35 there are 43.

9. How many units in 47? in 82? in 3? in ? in? in 13? in 21? in 35? in 36? 10. How many units in 37? in 46? in ? in 28?

in ?

in 27?

in 38?

11. How many oranges in §§ of an orange? in 2? in 63? in ? in 48? in 37?

12. How many 7ths of an apple in 3 apples? in 5 apples? in 44? in 94? in 114? in 64?

13. How many twelfths of a dollar in $34? in $54? in $11? in $81? in $68? in $73? in $8? in $11? 14. How many halves in of a dollar? in $33? in $11? in $123? in $9? in $44? in $63?

15. How many 3rds in 3 of a dollar? in $? in $? in $81? in $6,4? in $? in $11? in $91?

QUESTIONS.—What does the denominator of a fraction show? (127.) What does the numerator show? (128.) What is a proper fraction? (129.) What is an improper fraction? (130.) What is a mixed number? (131.) What is a simple fraction? (132.) What is a complex fraction? (133.) What is a compound fraction? (134.) of what are fractions indications? (135.) To what does the numerator answer in division ? (136.) The denominator? (136.)

LESSON VIII.

REDUCTION OF FRACTIONS.

145. To reduce a fraction to its lowest terms, or to à simple fraction of the highest denomination that can be expressed by an integral number of equal parts

Change to its lowest terms.*

MODEL OPERATION.

(a.) ANALYSIS.- is not expressed in its lowest terms, because the numerator and denominator are not prime to each other, being divisible by the common factor 3; therefore:

1. Change to a denomination 3 times greater than 54ths, which is 18ths.

Since 18ths are 3 times greater than 54ths, there can be only one third as many 18ths as 54ths, which is 16 eighteenths.

1 is not expressed in its lowest terms because both the numerator and denominator can be divided by the common factor 2; therefore:

-:

2. Change to a denomination 2 times larger than 18ths, which is ninths.

*NOTE FOR THE TEACHER.-The teacher should, perhaps, use the expression lowest terms, in order to prevent the use of a long and awkward phrase.

Since 9ths are 2 times greater than 18ths, there can be but half as many 9ths as 18ths, which is 8 ninths,

As there is no factor common to both terms of the fraction 8, it is, therefore, expressed in its lowest terms.

(b.) RULE.-Divide the numerator and the denominator of the fraction by any number that will divide them both without a remainder. Continue so dividing until no number greater than ONE will so divide them.

66. 384

T152

67. $34

61.

72. 188

73. $18

60. § 23 QUESTIONS.-When both the numerator and denominator are increased in an equal ratio, how is the value of the fraction affected? (137., a.) When both are decreased? (137., b.) When the denominator alone is increased? (137., c.) When the denominator alone is decreased? (137., d.) When the numerator alone is increased? (137., e.) When the numerator alone is decreased? (137., f.) How is the denomination of a fraction determined? (139.) What is the reduction of fractions? (141.) What is reduction ascending? (142.) What is reduction descending? (143.) When is a fraction said to be expressed in its lowest terms? (144.) What is the difference between an odd and an even number? (107.) (108.)

LESSON IX.

146. To reduce an improper fraction to an integral or mixed number.

Change 17 to a mixed number.

SOLUTION.-Since in 17 seventeenths there is one unit, in 178 seventeenths there are as many units as 17 seventeenths are contained times in 178 seventeenths, which are 10.

Therefore, are equal to 10.

75. Reduce 1476 to an integral or mixed number.

76. Change 374, 463, 864, and 341, to integral or mixed

numbers.

77. How many whole apples in 314? in 5786? in 347? 78. How many whole oranges in 4678? in 3967? in 3478? 79. Change 347, 384, 91874, and 386433, to integral or mixed numbers.

3649

411629

80. Change 307, 30702, 30369, and 3042 to integral or 910 9 4183,

mixed numbers.

61

QUESTIONS.-Are composite numbers ever prime to each other? (111.) What is the difference between prime, and composite factors? (112.) (113.) Is unity ever considered as a factor? (113., a.) What is the difference between an integral number and a fractional number? (106.) (123.) What does the parenthesis, or vinculum denote? (116.) What is the difference between the common and the greatest common divisor of two or more numbers? (118.) (119.) Does a multiple of a number contain all the factors of that number? (120.) Does the least common multiple of two or more numbers contain the highest powers of all the different factors of these numbers? (122., a.)

LESSON X.

147. To reduce an integral or mixed number to an improper fraction.

Change 43 to quarters, or reduce 43 to an improper fraction.