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, } = 2. If be multiplied by 2, then, in place of, 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?

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

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6. Multiply by ; by ; & by ; by 2; 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: XX35; 1X; ex is XX; $×14; #×7; 2 × ; 1×1.

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4

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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 , 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, 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

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above analysis, step by step, in each of the following problems.] What is ? 7÷16? {÷?? 8÷44? 4÷7? #÷7? 우응금? ?÷4? 1÷1? In place of multiplying crosswise, would 웃음? 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: ÷:2, by division; §×2 (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.

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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 .

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13. How many in 4? Divide 4, then, by . How many in 6? Divide 6 by ; 3§ by 9. 7 by 21. 5 by 25.

14. Divide 2 by 7 [2]. 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 g? Express of of 22 of 95, in its most sim2 3 ple terms, mentally casting out equal factors. [Black-board.] Express, in their lowest terms, of; § of 12; 4 of 34; 54 of 2o1 of 2; † of §1.

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65

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 3g 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 14 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 much was that for one yard? that rate?

cloth for 161 dollars. How What would 3 yards cost at

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 163 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 6 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 21 bushels of wheat last a family 3 weeks, how much will last them 1 week? 5 weeks?

12. If 5 boxes of raisins cost 11 dollars, what will be the cost of 1 box? Of 7 boxes ?

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

14. If 3 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.]

17. There is a pole standing so that of it is in the ground, and g 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, & in 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?

10. Three feet make 1 yard, and 5 yards a rod. How many feet make a rod? 5 rods? 8 rods?

11. Sixteen and a half feet make a rod, and 320 rods a mile. How many feet in a mile ?

12. Four quarters make a yard, and 5 quarters make an ell English. How many quarters in 5 yards? How many ells English in 20 quarters? How many ells English, then, in 5 yards?

13. Four quarters make a yard, and 6 quarters an ell French. How many quarters in 12 yards? How many ells. French in 48 quarters ?

14. Five quarters make an ell English, and 6 quarters an ell French. How many ells French, then, in 12 ells English? Solve this question first by multiplication and division, and then by division and subtraction.

15. Four quarters make a yard, 5 an ell English, and 3 an ell Flemish. How many ells English in 15 yards? and how many ells Flemish in the same?

16. Twenty shillings make a sovereign, and twenty-one shillings make a guinea. How many guineas in 63 sovereigns? Solve this question first by multiplication and division, and then by division and subtraction.

17. One of the fields of a farm is exactly square, being 40 rods long and 40 wide, making 1600 square rods. Another field is only 25 rods wide, yet it contains the same number of square rods. Is the last named field longer or shorter than the first, and what is its length?

18. Sixty seconds make a minute, and 60 minutes an hour. How many seconds in an hour? In 5 hours? In 25 hours? 19. Twenty-four hours make a day, and 7 days a week. How many hours in a week? hours in a week? In 4 weeks? In 6 weeks? In 14 weeks? In 19 weeks? [14-15-1; 19-20-1.]

20. Two pints make a quart, 8 quarts a peck, 4 pecks a bushel, and 8 bushels a quarter. How many pints in a quar

ter?

21. Eight quarts make a peck, and 4 pecks a bushel. How many quarts in 75 bushels?

22. Two pints of milk make a quart, and 4 quarts a gallon. How much will a pint cost, at 40 cents a gallon?