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parts, each part iş one third, if each of is divided into 6 equal pieces, what part of the } is each piece ? 7. What part of the whole orange is each piece ?
8. If a loaf of bread is cut into 4 equal parts, each part is 4. If each is divided into 5 equal pieces, each piece is į of the \, and of the whole loaf, of 4 then is zo
9. If a sheet of paper is cut into 5 pieces, each piece is t. If each } is cut into 3 equal pieces, each piece is of the }, and its of the whole. f of then is ts.
10. If a yard of cloth is cut into 8 equal pieces, and each piece is then cut into 3 equal parts, what part of the whole is each piece ?
11. If a bushel of apples is divided into fourths of a bushel, and each fourth is divided into 6 equal portions, what part of the whole is each portion ?
12. If you divide a pine apple into 3 equal parts, and each of those parts into 6 equal pieces, what part of the whole is each piece ?
13. If you have of a dollar, and wish to give of it to each of 7 children, what part of the whole dollar do you give to each?
14. If you have 5 of a lb. of raisins, and wish to divide it equally between 3 children, what part of a lb. do you give to each?
15. If you have of a yard of muslin, and divide it in. to 8 equal pieces, what part of į is each piece, and what part of the whole yard is each piece ?
16. What part of a unit is į of į?
Ans. If a unit is divided into 6 parts, and each of these parts into 8, the unit would be divided into 48 parts, and each part is als of the whole.
Let the following sums be stated in the same manner. 17. What part of a unit is off? 18. What part of a unit is of 4 ?, 19. What part of a unit is 4 of } ? 20. What part of a unit is off? 21. What part of a unit is į of ? 22. What part of a unit is į of į ? 23. What part of a unit is of
? 24. What part of a unit is į of f? 25. What part of a unit is of it!
26. What part of a unit is of ?
? 30. What is į of
? 31. What is of 32. What is šof ; ? 33. What is of į? What is 4 of ? 34. What is į of į? What is of į? 35. What is of į? What is of 1? 36. What is of? What is of? 37. What is of j? What is to off! 38. What is of į? What is į of ? 39. What is 4 of į? What is it of? 40. What is į of j? What is of ? ? 41. What is į of ji? What is I of 육 ? 42. What is 1 of ? What is of it? 43. What is į of ij? What is 4 of ?
After finding $ of one third we know that of two thirds is twice as much.
1. What is of? What is off?
9. What is i of į? What is of? What is of ? What is į of? What is of ?
10. What is į of ? What is į of ? 11. What is 4 of 1? What is off! 12. What is į of in? What is 1 of 1
? 13. What is of? What is of jy 14. What is į of A ? What is of ? 15. What is of? What is in of } ? 16. What is to of? What is of ? 17. What is į of ? What is of in ? 18. What is of i)? What is of it 19. What is i m? What is of ? 20. What is Ź What is f of ?
21. What is of? What is of 1?
After finding one part of a fraction, we find the other parts by multiplication.
Thus after finding what one fourth of a fraction is, we can find three fourths by multiplying by 3.
Thus į off is įa, therefore of is 3 times as much, or :
1. What is . of ? What is of ? What is of g ? 2. What is į of? What is of? What is of ? 3. What is of? What is of? What is of ? Let the pupil reason thus : What is an ofz? One sixth of one third is ta. One sixth of two thirds is ig. Four sixths of two thirds is 4 times as much, or ia.
4. What is of? What is į of f?. What is of ? What is ofWhat is of? What is fof/? What is of;? What is of ?
6. What is of lo? What is io of iż ? 7. What is il of į? What is lof } ? 8. What is in of į? What is of ? What is 7 of ? What is gof;? What is off ?
In multiplying one fraction by another, we are to take a certain part of one fraction, as often as there are units in the numerator of the other fraction.
Thus, if we are to multiply by 4 we are to take a sixth of four times.
To explain the rule for multiplying, when both factors are fractions, take an example.
What is of ?
One fifth of 4 is, and this is made by multiplying the denominator 6, by the denominator 5.
Three fifths of 4 is three times as much or }}, and this is made by multiplying the numerator 4, by the numera. tor 3.
Therefore multiplying the denominators together obtained one fifth of 5, and multiplying the numerators to. gether, obtained three fifths.
In multiplying 3-5 by 4-6, what effect is produced by multiplying the numerators together? What effect is produced by multiplying the denominators together?
RULE FOR MULTIPLYING WHEN BOTH FACTORS ARE FRACTIONS.
Multiply the denominators together to obtain one part, and the numerators together to obtain the required number of parts.
In performing these sums upon the slate, let the pupil use the signs thus :
Multiply by X=".
EXERCISES FOR THE SLATE, FOR OLDER PUPILS. What is t of 7? What is of? What is of; ? What is off? What is of ? What is softi ?
What is 1 of 4? What is 48 of 137? What is 314 of 2460 ?
Multiply 44 by !? Multiply is by
Division is finding how often one number is contained in another, and thus finding what part of one number is another number,
The number to be divided is called the Dividend.
The first is Simple Division, in which both the dividend and divisor are whole numbers, and ten units of one order, make one unit of the next higher order.
The second is Compound Division, in which other num. bers besides ten, make units of higher orders.
The third is Division of Vulgar Fractions, in which the dividend or divisor (or both) are Vulgar Fractions.
What is the rule for multiplying when both factors are fractions ? What is Division? What are the Divisor and Dividend ? Quotient ? Remain. der? What are the four kinds of Division ?
The fourth is Decimal Division, in which the dividend, or divisor, (or both) are decimal fractions.
SIMPLE DIVISION. How many
9 cents are there in 63 cents ? What part of 63 cents is 9 cents ? How many times is 8 contained in 56 ? If 8 is contained 7 times in 56, what part of 56 is 8?
If 56 is divided by 8, how much smaller is the quotient than the dividend? How many
7 dollars are there in 42 dollars ? What part of 42 dollars is 7 dollars ? How many times is 6 contained in 66 ? If 6 is contained 11 times in 66, what part of 66 is 6?
If 66 is divided by 6, how much smaller is the quotient than the dividend ?
There are many numbers which cannot be divided into equal parts, without making a fraction. For example, if we wish to divide 7 apples into two equal portions, we should have for answer 3 apples and t of an apple.
If we bad 13 apples, and wished to give a third of them to each of 3 friends, we should divide the 13 by 3, and the answer would be 4, and 1 left over. That is, we could give 4 apples to each of the 3 friends, and one would be left to divide among them. This divided by 3, (or into 3 equal parts) would give a third to each one. 13 then, di. vided by 3, gives 4 and fas answer.
If you are to divide 7 apples equally among 3 persons, how many whole apples would you give to each, and what would remain to be divided ?
If you had 14 oranges, and wished to divide them equal. ly among
divide the two that remained ? Ans. Divide each into 6 equal parts, and give one of the parts of each orange to the 6 persons. Each person would then have 2 oranges and %.
If you have two apples, each cut into 12 parts, and take 4 of these parts from each apple, how much do you take ?