10. From the sum of the foregoing fractions, subtract the

sum of the following,,,

11. What is of?

12. What is of?

13. What is of?

14. How do you obtain one ninth of a whole number?

15. How do you obtain one ninth of a fraction?

16. If the denominator of a fraction be 7, into how many

parts is the unit divided ?

17. If you multiply this denominator by 9, into how many parts is the unit now divided; and how much less is the value of each part than before?

18. How does multiplying the denominator divide the fraction?

19. If you multiply the numerator also by 9, will the value of the fraction be affected?

Illustrate.

20. In multiplying by a whole number, how many times do you repeat the multiplicand?

21. If your multiplier be 2, how many times is the multi

plicand repeated ?

22. How many times if the multiplier be 3? if 4?

23. How many times if the multiplier be 1?

24. How many times if the multiplier be?

25. How do you obtain one half of a whole number?

26. Is one half of a fraction obtained by the same process?

27. How do you divide a fraction?

28. What is the quotient of divided by 2?

29. What is the product of multiplied by ?

30. What is the difference between the process of dividing a fraction by a whole number, and multiplying the same by a fraction?

31. If your multiplier be, what part of the multiplicand will your product be?

32. How will you obtain it ?

33. If your multiplicand be a fraction, how will you obtain of it?

34. How will you divide it by 3?

35. If your multiplier be, what part of the multiplicand

will your product be?

36. How will you obtain 2 thirds?

37. How do you multiply fractions by fractions ?

38. Why do you multiply the denominator of the multiplier into the denominator of the multiplicand? and why the numerator of the multiplier into the numerator of the multiplicand?

LESSON TWENTY-FOURTH.

1. What kind of a number is 54?

2. How many sevenths in 54?

If in 1 whole there are 7 sevenths, in 5 wholes there would be 5 times 7 sevenths, which is 35 sevenths, and in 5 wholes and 2 sevenths there would be 37 sevenths; therefore, in 5 are there 37 sevenths.

3. How many fourths in 6?

Give the process.

4. What kind of a fraction is 27?

5. How is a mixed number reduced to an improper fraction?

6. How many wholes in ?

7. How is an improper fraction reduced to a mixed number?

8. How many wholes in ??

How do you know?

9. Is the quotient of 27 divided by 8, 3 wholes and of a whole, or 3 wholes and of 8?

10. Is here employed to express a part of one unit, or of 8 units?

11. of 8 units would be how many units ?

12. A man purchased 8 pounds of tea for 12 dollars;

what did he pay a pound?

13. A man purchased tea for of a dollar a pound; what did he pay for 8 pounds?

14. How many times is 4 shillings contained in 1 pound? Pounds must be reduced to shillings.

15. How many times is contained in ? Fourths must be reduced to eighths.

16. How many times is 21 contained in 84?

17. Is 11 fifths contained in 44 fifths the same number of times that 11 would be contained in 44?

18. What effect upon the quotient has dividing the divisor and dividend by the same quantity ?

19. At 24 dollars a barrel, how many barrels of crackers can be bought for 4 dollars?

In the division of fractions, if the denominator of the divisor and the dividend be alike, divide the numerator of the dividend by the numerator of the divisor.

20. At of a dollar a bushel, how many bushels of apples can be bought for 56 dollars ?

21. At of a dollar a bushel, how many bushels of pears may be bought for of a dollar?

The price of an individual article is always the divisor, and the amount of money to be expended is the dividend. The divisor, &, and the dividend, 윽, both, in their numerators, contain the factor 3. Now, to multiply or divide divisor and dividend by the same quantity, does not affect the quotient. The ques jon may then read, How many bushels may be purchased for of a dollar, at of a dollar a bushel ?

It is evident that at 1 fourth of a dollar a bushel, 4 times as many bushels may be purchased as at 1 dollar a bushel, and that for 1 seventh of a dollar, only 1 seventh as many bushels could be purchased as for 1 dollar. To multiply dividend is the same as to divide divisor; therefore, for of a dollar, at of a dollar a bushel, we could purchase 4 times 1 seventh, which is bushels, the answer.

In division of fractions by fractions, when the numerator i divisor and dividend are alike, divide the denominator of Le divisor by the denominator of the dividend.

22. At of a dollar a pound, how many pounds of tea can be purchased for of a dollar?

Give the process.

23. What is the quotient of

24. What is the quotient of

25. At of a dollar a pound, how many pounds of sugar may be purchased for 7 of a dollar?

In this example the divisor is divided by 8, and the dividend is divided by 9. But since to multiply dividend is the same as to divide divisor, and to multiply divisor is the same as to divide dividend, we may invert the divisor, and proceed as in multiplication. inverted, will be; the question will then read, What is of ?

In all questions in division of fractions by fractions, we may invert the divisor, and proceed as in multiplication.

26. Divide by 4, and illustrate.

30. How do you divide one fraction by another, when their denominators are alike, and their numerators unlike?

31. How when their numerators are alike, and denominators unlike ?

32. How when their numerators are unlike, and their denominators unlike ?

If the terms of the dividend are divisible by the corresponding terms of the divisor, divide the terms of the dividend by the corresponding terms of the divisor. Thus, the quo tient of divided by, is, equal to a unit, or 1.

LESSON TWENTY-FIFTH.

1. Divide by .

2. Multiply by .

3. From subtract §.

4. How do yo divide one fraction by another ?

Illustrate.

5. How do you multiply one fraction by another ? Illustrate.

6. How do you subtract one fraction from another ? Illustrate.

7. What is the difference between and ?

Give the process.

8. What is the product of into?

Give the process.

9. What is the quotient of divided by?

Give the process.

10. A man purchased 9 yards of cloth for 27 dollars;

what did he pay a yard ?

Give the process.

11. A man bought 9 yards of cloth, at 3 dollars a yard; what did the whole cost?

12. A man sold of of of his stock in a railroad; what part of his stock did he sell ?

13. A man sold of of a house and lot, and retained the remainder, which he valued at 100 dollars; what was the value of what he sold?

14. A man purchased off of a threshing machine for 12 dollars; what was the whole machine worth?

15. A man distributed 6 apples among some boys, giving them of an apple each; how many boys were there?

16. A boy gave 12 apples to a number of his companions, giving of an apple to each; how many boys were there? 17. A boy gave to 9 companions of an apple apiece; how many apples did he give away?

18. How many more times will 9 contain, than it will contain 3?

19. How many hours in of a day?

If in 1 day there are 24 hours, in one fourth of a day there would be of 24, which is 6; and if in one fourth there are 6 hours, in three fourths there would be 3 times 6, which is 18 hours.

20. How many minutes in & of an hour ? 21. How many shillings in & of a pound ? 22. How many farthings in of a pound ?