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?

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; 3 by 4; by 3; by 7.

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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; 1 by 5. 4. Multiply by 3. [Write it on the black-board.] 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

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* 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 divided by 4 by division becomes. It is, therefore, more convenient, in such cases, to multiply and divide by multiplication, which presents the fractions in the more intelligible forms of 12 and 6. 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.

the answers to the suggestive questions by the class, as follows, especially if the multiplication be only indicated, as below, not performed. Thus, X. First step, 12; second step,

If be multiplied by 2, then, in place of 3, and, because it is 3 times too much, divided by 3, what terms of the two fractions will be multiplied together? Will this be the case, whatever may be the numbers, when one fraction is to be multiplied by another ?

5. What principle, then, may be drawn from this exercise? Ans. To multiply one fraction by another, multiply the for a new numerator, and the for a new denominator.

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6. Multiply by ; by ; by ; by ; 7 by ; repeating, step by step, the above analysis.

7. Multiply by . [Write it on black-board.]

Suggestive Questions.—What factor is common to 5 and 10? What factor is common to 3 and 6? What two factors, then, are common to both terms? What, then, will be the product of the two fractions, if the common factors be dropped; that is, if both terms be divided by 15?

8. Mention the products of each of the following pairs of fractions by inspection merely, casting out the factors common to both terms mentally, so as at one step to present each product in its lowest denomination: X X XX; $X11; $X}; X; X

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9. What addition to the principle developed from the 4th exercise may be drawn from the last two exercises?

Ans. When one or more factors can be found in one or in both of the which can also be found in one or in both of they may be cast out of both before the multiplication, and thus leave the fractional product in its lowest denomination.

the

10. Divide by 3.

Suggestive Questions. What part of 2 is? If be divided by 2, then, instead of 3, how many times too small will be the quotient? How, then, shall it be rectified? [Let the division be indicated on the black-board as before in Ex. 4, as follows. First step; second step 3. If g, then, be divided by 2 in place of, and, because the quotient is 3 times too small, it be multiplied by 3, what terms of the two fractions are found to be multiplied together? [Repeat the

above analysis, step by step, in each of the following problems.] What is ? 7 ÷ 1% ? 8÷?? 8÷34? 4÷7? #÷7? ?÷4? ÷? In place of multiplying crosswise, would the same result be attained by reversing the divisor, and completing the process by multiplying the two fractions? [Show the process on the black-board in one or two of the above cases, as follows: ÷, by division; X (the divisor reversed) by multiplication.] As the result, then, is exactly the same, we shall in future pursue the process by reversing the divisor, and multiplying. Repeat the above problems by the reversing process.

11. What principle may be drawn from the above exercises? Ans. To divide one fraction by another, reverse the and proceed as in multiplication.

12. Divide 3 by . The most simple method of resolving such questions is to give the integer (3) a fractional form, as . But, after a little practice, giving the fractional form becomes a superfluous step. Divide 9 by ; 8 by ; 14 by .

13. How many in 4? Divide 44, then, by 2. How many in 63? Divide 6 by ; 3§ by 9. 7 by 21. 5

by 25.

14. Divide 2 by 7 [4]. 1 by 5; 3 by 8; 5 by 8; 2 by 6; 1 by 4.

15. What does the word of signify when connected with fractions? [See Section I., 13 of this chapter.] How much, then, is of ? Express of of of, in its most simple terms, mentally casting out equal factors. [Black-board.] Express, in their lowest terms, of; § of 13; 4 of 27;

of 2 of 2; † of §.

1. A MERCHANT sold 8 barrels of flour, at 63 dollars per barrel. How much did they come to? [The pupil should explain the process in all the questions that follow.]

2. A countryman sold 4 bushels of cranberries to one store-keeper, and 33 bushels to another, at 33 dollars a bushel. How much money did he receive?

3. One man bought 21 bushels of the same cranberries, and another man bought 1 bushels. How much did the first man pay more than the other?

4. What will 9 bushels of rye come to, at 1 of a dollar per bushel?

5. A man bought 5 yards of cloth for 161 dollars. How much was that for one yard? What would 3 yards cost at that rate?

6. If 6 men can build a piece of wall in 33 days, in what time could 1 man build it? In what time could 4 men build it ?

7. If 3 horses will eat 16 tons of hay in a year, how much will 1 horse eat in the same time? How much will 5 horses? 8. If 3 barrels of flour last a family 63 months, how long will 1 barrel last them? How long will 5 barrels ?

9. If 6 yards of cloth cost 13 dollars, what will 1 yard cost? What will 9 cost?

10. If a man can travel 10 miles in 3 hours, how much can he travel in 1 hour? In 5 hours?

11. If 2 bushels of wheat last a family 3 weeks, how much will last them 1 week? 5 weeks?

12. If 5 boxes of raisins cost 113 dollars, what will be the cost of 1 box? Of 7 boxes ?

13. If 3 ounces of silver cost 3 dollars, what will be the cost of 1 ounce ? Of 8 ounces?

14. If 33 pounds of bread be sufficient for 6 men for a day, how much is that for 1 man? For 5 men?

15. If 9 men receive 11 much is that for each man? earn at that rate?

dollars for a day's work, how How much would seven men

16. If the freight of 9 hogsheads of sugar on a railroad be 16 dollars, what is the freight for 1 hogshead? For 7 hogsheads?

[Repeat the above from Ex. 5 to 16, omitting in each the leading question, "How much for 1," &c. They should be done as follows: 5th 3×51, 6th 2014.]

in

17. There is a pole standing so that of it is in the ground, and of it in the water. How much of it is in the air? 18. A pole is standing so that of it is in the ground, the water, and 10 feet in the air. How many feet are in the ground, how many in the water, and what is the length of the pole?

19. In a certain school, of the pupils are learning to read, studying arithmetic, studying geography, and the rest, 6 in number, learning grammar. How many are reading? how many studying arithmetic? how many geography? and how many in all?

20. In a congregation were men, women, boys, and 100 girls; how many persons were in the congregation?

CHAPTER IV.

DENOMINATE FRACTIONS, OR FRACTIONS EXPRESSED IN CONCRETE WORDS, NOT IN FIGURES.

SECTION I.-Change of Form.

1. TEN cents make a dime, and 10 dimes make a dollar. How many cents, then, make a dollar?

2. Ten dimes make a dollar, and 10 dollars make an eagle. How many dimes, then, in an eagle?

3. Ten mills make a cent, 10 cents make a dime, 10 dimes make a dollar, and 10 dollars an eagle. How many mills, then, in an eagle ?

4. Twelve pence make a shilling, and 20 shillings a pound. How many pence, then, in a pound?

5. Four farthings make a penny, 12 pence a shilling. How many farthings, then, in a shilling?

6. Four farthings make a penny, 12 pence a shilling, and 20 shillings a pound. How many farthings, then, in a pound? 7. As 12 pence make a shilling, and 20 shillings a pound, how many pounds and shillings in 540 pence?

8. Sixteen drams make an ounce, and 16 ounces a pound. How many drams in 1 pound? In 3 pounds? In 7 pounds? How many pounds in 512 drams? In 768 drams?

9. Twelve inches make a foot, and 3 feet a yard. How many inches make 5 yards? 15 yards? 25 yards? 14 yards?