16. What are the prime factors of 14 and 35? What is their greatest common divisor; that is, what prime factors are common to both numbers? What are the prime factors of 16 and 12? What is their greatest common divisor? Mention the prime factors of 16 and 18, and say which are common. What is the greatest common divisor of 28 and 42? Of 16 and 36? 18 and 42? 19 and 57 ? 72 and 30? 20 and 45? 51 and 17? 75 and 125? 39 and 26? 36 and 54? 14 and 63 ? 15 and 125? 27 and 84? 30 and 81?

17. What is the greatest common divisor of 12, 27, and 51? Of 9, 45, and 54? 63, 18, and 36? 15, 39, and 27 ? • 4, 26, and 38? 14, 49, and 63? 15, 105, and 75? 24, 78, and 42 ? 85, 34, and 51 ?

34,

18. What is the greatest common divisor of 4, 32, 12? Of 45, 75, 60? 33, 77, 22? 39, 91, 78? 46, 69, 92? 85, 102? 16, 128, 64? 116, 29, 87?

19. Mention two numbers of which 10 is the common mul-tiple? Of which 15? 22? 26? 34? 35? 42? 46? 51 ? 52? 106? 112?

20. What is the least common multiple of 2 and 3? Of 3 and 5? 7 and 3? 7 and 3? 5 and 11?

21. Is 18 a common multiple of 3 and 2? Its least common multiple? Is 260 a common multiple of 5 and 13? Its least common multiple ?

SECTION III. — Fractional Change of Form.

1. WHEN a bushel of wheat is divided into 4 equal parts, what is one of them called? How many of these fourths make half a bushel? Is 2, then? If a bushel was divided into 6 equal parts, how many of them would make half a bushel ? Is 8=2=1 ? [Show these fractions on blackboard.] If divided into 8 parts, how many would make half a bushel? If divided into 10, 12, 16, 18, &c., parts? [Show a number of these fractions, and let the class observe the relation between the two terms of each fraction.] In how many ways could be represented? [Infinite.] Which is easiest understood, the smallest fraction, or one of the larger ones? 2. When an article is divided into 3 equal parts, what is one of them called? If divided into 6 equal parts, what

would one of them be called? How many of these last would make of the article? Is =, then? [Black-board.] Express in as many different forms as you can. Which is easiest understood, the smallest or one of the larger? Into how many different forms can a fraction be changed without altering its value? [See p. 82, 1. 5.]

3. What is the least common multiple of 2 and 3? How, then, may and be changed to equivalent fractions with a least common denominator? How may and be changed to equivalent fractions with a least common denominator? ? and ? and ? and ?

and

4. Change to an equivalent fraction of the same denomination as §. to same denomination as. Make 1, 2 and §, of same denomination, by a change in the two former? What is the least common multiple of 3, 5, and 25? Change, and, then, to equivalent fractions with the least common denominator. [Use black-board.] Change,, and 15, to fractions with least common denominator, commencing by a change in the last fraction. Change 3, §, 1, 2, and, to fractions with least common denominator, commencing by a change in the second and fifth.

5. Change the following sets of fractions to equivalent ones, with the same lowest denominator by division: and 2; and; and 2; 2, 1, and 1⁄2 ; §, 1, and 16, 3, and 18.

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20

3

3

10

3

6. What is the least common denominator of 4, §, 1, 3, 2o4, ? [Black-board.] How shall be changed to an equivalent fraction of that denomination? How shall ? 1 ? ? 2T ?? What is the least common denominator of §, 21, ? How shall be brought to that denomination??? What is the least common denominator of 4,, fo, 2o 5, 4, 14' 28 ? How shall be brought to that denomination ? How shall ? to? 28? 7? ?? ?

2

??

3

3

25

4

28

The above exercises should be studied without the aid

of the slate. In reciting, the teacher should write the given fractions on the black-board, and call on the pupils to work out the answers mentally.

SECTION IV. Addition and Subtraction of Common

Fractions.

Explanation. Numbers of different denominations can neither be added together nor subtracted from each other. Thus, 6 chairs and 3 tables make neither 9 tables nor 9 chairs; and 3 tables cannot be taken from 6 chairs, nor 3 chairs from 6 tables. Their denomination, however, can be made the same, and then they can be either added or subtracted. Thus, by calling both chairs and tables pieces of furniture, they have the same denomination; and, when added, make 9 pieces of furniture; and the 6 or the 3 may be subtracted from the 9.

The same remark holds good with respect to abstract numbers (that is, numbers used without being applied to things), whether they are whole or fractional numbers. Thus 6ty cannot be added to 3 hundred or to 3 units, because they are of different denominations, and would make neither 9 hundred, 9ty, nor 9 units. Neither can 3ty be subtracted from 6 hundred without changing one of the hundred (mentally) to 10ty, and thus we should have 570. This becomes more evident when applied to fractions. can neither be added to, nor subtracted from, g, while both retain their present form. But can be changed to, and 8 to 19, and, as they have now the same denomination (twelfths), they can either be added or subtracted.

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But the case is entirely different in regard to multiplication and division. In multiplication, as has already been observed, although it is in reality nothing more than addition, yet one of the factors is not to be added. It merely points out the number of times that the other factor is to be taken. Thus, the 6 chairs and 3 tables may be multiplied by 3, giving 18 chairs and 9 tables, because 3 times 6 chairs make 18 chairs, and 3 times 3 tables make 9 tables; and so with any other number whatever. Abstract integers may also be multiplied or divided by numbers of different denominations, since it is evident that 6 hundred or 6ty can be taken three times as well as 6 units can; and it is equally evident that either of these digits can be divided by 3 of any denomination whatever. In like manner, and, or any other fraction, though of different denominations, may be multiplied or divided by 2, 3, 1, or any other number, since the one is only taking each of these fractions so

many times, and the other is only finding how many times they contain the divisor.

1. Can numbers of different denominations be added together or subtracted from each other? Give me examples to show why. What should be done, then, when it is necessary they should be added or subtracted? Can factors of different denominations be used in multiplying? Give an example to show why. May the divisor and dividend be of different denominations? Give an example to show why.

2. By how many methods can a fraction be changed without altering its value? [See p. 83, 1. 36.] Which is more intelligible, a fraction with large or with small terms? Which, then, is the preferable mode of changing the form of a fraction ? Can the form of a fraction always be changed by division? When can division be used? Ans. When its terms have a common divisor. Can be changed by division? Can ! ! ? § ? f ? § ? Why? Can ? ? ? Can every fraction be changed in form by multiplication? Why? Be cause, though every two numbers may not have a common divisor, yet any two, &c.

6

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6

3

3. What is the least common multiple of 3 and 15? What, then, is the sum of and g? What is the least common multiple of 4 and 8? What, then, is the sum of and §? What is the sum of and ? Of,, and? Of 4,2%, and ? Of %, 4, and? Of 1, 2, and? [By division and multiplication the least common denominator becomes 20.] Of 2, , and? Of,, and? [Least common denominator 4.] 4. What is the least common denominator of and g? What is the difference of these fractions, then? The difference of and? Of and ? Of and

In 2? In 5? In

What, then, is the How many fifths in 1? What, then, is the differ

5. How many fourths are there in 1? 9? How many fourths in 11? In 5? difference between 6 and 11? Eighths in 1? Sixths? Ninths? ence between 4ğ and 23 ?

SECTION V.-Contracted Addition and Subtraction of Common Fractions, usually called Multiplication and Division.

1. In how many ways can a fraction be multiplied? [See p. 80, 1. 16, and p. 81, 1. 16.] Name them. By which method is the fraction rendered most intelligible? Which, then, is the preferable mode? Can division be used for multiplying a fraction in all cases? Why not? By how many methods can a fraction be divided? Which, then, is the preferable mode? Can a fraction be divided or multiplied by division in all cases? Why not?

17

2. Multiply by 3 by multiplication; by division. Multiply by 7 by multiplication; by division. Can be multiplied by 3 by division? Why not? Ans. Because 3 is not a factor in Can it be multiplied, then, by multiplication? What is the product? Multiply the following factors by both methods, changing the fraction, when not already so, to its lowest denomination, and observing whether or not the result of the two methods is alike: by 3; by 4; I by 3; 옳 5 28 18

by 7.

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Τ

3. Divide by 3 by multiplication. Can it be done by division? Why? Divide by 7 by multiplication. Can it be done by division? Why? Can be divided by 2 by both methods? Why? Divide the following fractions as indicated, changing the quotient, when not already so, to its lowest denomination, and observing whether or not the result of the twe methods is alike: by 3; by 4; 18 by 2; 18 by 5. 4. Multiply by . [Write it on the black-board.]

6

Suggestive Questions. What part of 2 is ? [See Chap. II., Sect. I., 13.] If be multiplied by 2, then, how many times too large will the product be? If it be 3 times too large, how can it be rectified? [This analysis will be sufficiently clear when exhibited on the black-board, if the teacher write

to

* A fraction may be multiplied and divided in all cases by division; but it becomes complicated when the divisor is not a factor of the dividend. Thus & multiplied by four by division becomes, and 7 divided by 4 by division becomes 1. It is, therefore, more convenient, in such cases, multiply and divide by multiplication, which presents the fractions in the more intelligible forms of 2 and 3. When it is said, then, that one number is not divisible by another, all that is meant is that the quotient would be complicated with fractional parts. But, whenever this would not be so, a change of fractional form by division is always preferable.