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45. How many square feet are there in a gravel-walk 32971 ft. long and 5 ft. wide ?

46. How many casks each holding 2 bushels can be filled from 273 bushels of grain ?

Solution. — Since 1 cask can be filled from 24 bushels, as many casks can be filled from 273 bushels as there are times 2 in 273, to find which we must reduce both to fourths. 23 = 4, 273 = 1092, and ų is contained in 1992 as many times as 11 is contained in 1092, which is 99, ji times. Hence 99 casks can be filled.

Note. — Compare the preceding problem with : “How many casks each containing 2 bu. 3 pk. can be filled from 273 bu. of grain ?"

47. How many coats each containing 24 yards can be made from 45 yards of cloth ?

48. If a man can earn $44 per day, how many days will it take him to earn $17* ?

49. A man who had $147£ exchanged it for quarter-eagles. How many did he obtain ?

50. How many nails are there in a piece of cloth 643] inches long?

81. Multiplication and Division of the Numerator.

(a.) From the nature of fractions, it follows that multiplying the numerator of a fraction must multiply, and dividing the numerator must divide, the number of parts considered, without affecting their size, and must therefore multiply or divide the fraction.

1. What is the effect of multiplying the numerator of by 3 ?

ANSWER. — Multiplying the numerator of the fraction by 3 gives til for a result, which expresses three times as many parts each of the same size as before, and is therefore 3 times as large. Hence, multiplying the numerator of the fraction by 3 multiplies the fraction by 3. What is the effect of multiplying the numerator of — 2. 4 by 2? 4. }} by 6 ?

6. is by 9 ?
3. f by 5 ?
5. by 8?

7. zo by 6?

8. What is the effect of dividing the numerator of 14 by 4?

ANSWER. — Dividing the numerator of the fraction 14 by 4 gives i, for a result, which expresses & as many parts each of the same size as before, and is therefore & as large. Hence, dividing the numerator of 14 by 4 divides the fraction by 4.

Note. — The first of the above solutions is equivalent to : "3 times are 15, just as 3 times 5 apples 15 apples;" and the second is equivalent to : "1 of 1} = ir, just as į of 12 apples = 3 apples.” They are necessary in this form, as a preparation for the exercises which follow. What is the effect of dividing the numerator of

9. 'by 7 ? 11. f7 by 9 ? 13. kby 5 ? 10. 14 by 4? 12. ff by 8? 14. 11 by 6?

82. Multiplication and Division of the Denominator.

/

(a.) From the nature of fractional parts, it follows that

1st. The larger the number of fractional parts into which a unit is divided, or which it takes to equal that unit, the smaller each part will be.

2d. The smaller the number of fractional parts into which a unit is divided, or which it takes to equal that unit, the larger each part will be.

3d. Multiplying the number of fractional parts into which a unit is divided, or which it takes to equal that unit, divides each part.

4th. Dividing the number of fractional parts into which a unit is divided, or which it takes to equal that unit, multiplies each part.

ILLUSTRATIONS. - Halves are larger than thirds, fourths, or fifths, because it takes a less number to equal a unit. Thirds are larger than fourths, fifths, or sixths, because it takes a less number of them to equal a unit.

1. What is the effect of multiplying the denominator of the fraction by 2?

ANSWER. — Multiplying the denominator of the fraction by 2 gives for a result, which expresses the same number of parts each & as large as before. Hence, multiplying the denominator of by 2 divides the fraction

(b.) What is the effect of multiplying the denominator of
2. f by 4?
4. by 7?

6. by 2?
3. Z by 4?
5. it by 3 ?

7. po by 3? 8. What is the effect of dividing the denominator of y by 3?

NSWER. - Dividing the denominator of the fraction / by 3 gives } for a result, which expresses the same number of parts each 3 times as large as before. Hence, dividing the denominator of 7 by 3 multiplies the fraction by 3. What is the effect of dividing the denominator of 9. i by 2? 11. i by 6 ?

13. r by 8?
10. f by 4?
12. fc by 7?

14. Ryby 9?

83. Multiplication and Division of both Terms.

1. What is the effect of multiplying both terms of the fraction

by 4?

ANSWER. — Multiplying both terms of the fraction by 4 gives for & result, which expresses 4 times as many parts each į as large as before. Hence, the value of the fraction is not altered, or = 1 What is the effect of multiplying both terms of 2. Į by 8? 4. f by 5 ?

6. f by 9?
3. & by 2?
5. by 6?

% TT by 3 ? 8. What is the effect of dividing both terms of the fraction 4

by 3?

ANSWER. — Dividing both terms of the fraction is by 3 gives for a result, which expresses f as many parts each 3 times as large as before. Hence, the value of the fraction is not altered, or 15 = $. What is the effect of dividing both terms of 9. by 2?

11. % by 2? 13. tf by 7?
10. f by 3?
12. 1! by 52

14. 1% by 4? 15. What is the effect of multiplying the numerator of a fraction by 5, and the denominator by 6?

ANSWER.- Multiplying the numerator of a fraction by 5 and the denomi. nator by 6 gives for a result of the original fraction, for it gives 5 times as many parts each $ as large as before.

What is the effect of multiplying the numerator of a fraction –

16. By 2 and the denominator by 9 ? 17. By 7 and the denominator by 5? 18. By 13 and the denominator by 4? 19. By 12 and the denominator by 8? 20. By 20 and the denominator by 11?

84. Recapitulation and Inferences. (a.) Multiplying the numerator multiplies the fraction by multiplying the number of parts considered, without affecting their size.

(b.) Dividing the numerator divides the fraction by dividing the number of parts considered, without affecting their size.

(c.) Multiplying the denominator divides the fraction by dividing each part, without affecting the number of parts considered.

(d.) Dividing the denominator multiplies the fraction by multiplying each part, without affecting the number of parts considered.

(e.) A fraction may be multiplied either by multiplying the numerator or by dividing the denominator.

(f.) A fraction may be divided either by dividing the numerator or by multiplying the denominator.

(g.) Multiplying both numerator and denominator of a fraction, both multiplies and divides the fraction.

(h.) Dividing both numerator and denominator of a fraction, both divides and multiplies the fraction.

(i.) Multiplying both numerator and denominator of a fraction by the same number, both multiplies and divides the fraction by that number, and therefore does not alter its value.

(j.) Dividing both numerator and denominator of a fraction by the same number, both divides and multiplies the fraction by that number, and therefore does not alter its value.

85. Reduction to Lowest Terms, and Cancellation.

(a.) A fraction is in its LOWEST TERMS when its numerator and denominator are the smallest entire numbers which will express its value.

(b.) When a fraction is reduced to its lowest terms, its numerator and denominator will be prime to each other.

(c.) A fraction may be reduced to its lowest terms by dividing both numerator and denominator by all their common factors, or, which is the same thing, by their greatest common divisor.

1. Reduce ft to its lowest terms.

1st Solution. — Dividing both terms of ff by their greatest common divisor, 12, gives for a result.

20 Solution.—Dividing both terms of fd by their common factor, 4, gives g for a result, both terms of which contain the factor 3. Dividing them by 3 gives for a result, the terms of which are prime to each other. Hence, 4 reduced to its lowest terms =

(d.) The numbers by which we divide in reducing fractions to lower terms, are said to be CANCELLED.

ILLUSTRATIONS. In the first solution, we cancelled the factor 12. In the second, we cancelled the factor 4 and then the factor 3.

(e.) Reduce the following fractions to their lowest terms:

f, as before.

2. $.

3. pg. 4. 15. 5. 4.

8. 4. 9. 15. 10. 48.

14. %33. 15. 413. 16. *** 17. 18. 18. &4$t. 19. filt:

13. fi

11. 35

103• 6. 18.

91 7. ir. (f.) A fraction is sometimes expressed by the factors of its numerator and denominator.

8 X 15 ILLUSTRATION. - The fraction

which may be read, “8 times 16

9 X 16' divided 9 times 16," or, “ The fraction having 8 times 15 for a numerator and 9 times 16 for a denominator."

(g.) Such fractions should be reduced to their lowest terms before multiplying the factors together.

8 x 15 20. Reduce

to its lowest terms. 9 X 16

Solution. — Cancelling 8 from the 8 of the numerator and from the 16 of the denominator gives 1 in place of the former and 2 in place of the latter.

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