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cand, (see illustrations, pages 36 and 43,) or in other words, we put into the product the number of parts the unit is supposed divided into, as many times as there are units in the whole number, and this product we make a part of the numerator by adding it to the given numerator. By keeping the multiplier for a denominator, it shows how many fractional units in the numerator are required to make a whole unit. If an unit be divided into 5ths, there would be 5 fifths. If we multiply a whole number by 5, we place 5 in the product for each unit in the whole number, and by so doing, the product expresses the value of the whole number divided into 5ths. . . ... . There are 50 fifths in 10, as may be seen in the work of the last question, and by adding the 1 fifth given, the whole value of the mixed number, 10}, is 51 fifths, or *. . . . * , - * 2. What improper fraction is equal to 373 2 - - * * * * Ans. ***. 3. Change 123}} to the form of a fraction. - Ans. "## *. 4. Change 1702.rks to an improper fraction. Ans. * **** 1.

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8. How many shares in that library, which is worth 720 dollars, if each share be worth of a dollar 7 Ans. 2160.

9. If 8 dollars are to be distributed among a certain number of scholars, as premium money, and each scholar is to receive # of a dollar, how many

scholars will receive a premium ? Ans. 56. 10. How many testaments, at # of a dollar a piece may be bought for 31 dollars. Ans. 155.

To change an improper fraction to a mixed number.

RULE,
Divide the numerator by the denominator; the

quotient will be the whole number, and the remainder set over the denominator, will form the fraction. . . .”

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Illustration.—In the last example, the 172 are 8ths, and it requires 8 eighths to make one unit; therefore, as many times as we can take 8 from the 172, there will be so many units. 4, the remainder, being 8ths, and half as many as it requires to make an unit, will constitute 4 of another unit.

The rule is true as it respects any other improper fraction, for the denominator will always show how many fractional units must be taken from the numerator, to equal a whole unit, or an unit in a whole number.

From this illustration and the preceding one, the propriety of dividing the numerator of a fraction by the denominator, to find the value of the fraction, is manifest.

2. Change "o" to a mixed number. - Ans. 231#. 3. What number is equal to :: * * Ans. 405. 4. How many units in 87030 sixths Ans. 14505.

6. How many dollars will 135 yards of #and come to at # of a dollar per yard? Ans. 15. 7. If a man drink a gallon of cider in 4 days, how many days will he be drinking 78 quarts? Ans. 78.

8. If a man travel a mile in 4 of an hour, how

many hours will he be travelling 113 miles 2 ** - Ans. 223.

- o - . .
To reduce a compound fraction to a simple one.
* RULE,

Multiply all the numerators together for a new numerator, and all the denominators together for a new denominator; the new numerator and denominator will be a fraction expressing the same value as the compound one.

JNote 1.--If any part of a compound fraction be a mixed number, it must be changed to an improper fraction, and the improper fraction used instead of the mixed number. If a whole number be connected with the fraction, it must be changed to the form of a fraction by placing 1 for a denominator. The value of the whole number will not be altered, for dividing the numerator by one, the quotient will be the same as the given number.

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Illustration.—A part of a thing is less than the whole; a part of that part is still less. If we multiply 1 by 1 the product is 1. # equal 1, and ; equal 1; if we multiply 3 by 3, the product is #, or 1, for dividing the numerator by the denominator, the quotient is 1, which is equal to the multiplicand. If we multiply by less than 1, that is, by a proper fraction, the product must be less than the multiplicand, for we can take only such a part of the multiplicand as the multiplier is of an unit. Once # is ; ; but let us multiply , by , the product must be less than when we multiplied by 1 ; it can be only half as large, }; and 3 we obtain by multiplying one numerator by the other, and one denominator by... the other denominator. In the first question, # of ; of a dollar is required. Let the 4 quarters of a dollar be each divided into two equal parts, the whole dollar will be divideo into 8 equal parts; but when we divide ; into two equal

parts, one of these parts is half the quarter; it is also } of the whole dollar. If both of the numerators be l’s, the numerator of the product will be 1, but multiplying the denominator of 4 by 2, the other denominator, the denominator in the answer 4, is twice as large as in , and the part of the dollar expressed by the numerator of , is only half as large as the part expressed by the numerator of 4, for multiplying by 2, the denominator becomes twice as large. Or in other words, when we say # of a dollar, the dollar is supposed to be divided into 8 equal parts, and we have one of them; but when we say of a dollar, the dollar is supposed divided into half as many parts, consequently one of those parts must be twice as large. A fraction properly expresses a quotient. ...,'" sigmifies a quotient of 2, for 10+5=2. But the quotient denoted by a proper fraction cannot be expressed by a single number. If we were required to divide 1 dollar equally between 3 men, we should be required to divide 1 by 3, and 3 would denote the quotient, for we cannot actually divide the dividend 1, which is the numerator, by the divisor 3: which is the denominator. Let it be required to find what part of a shilling # of ; is. # implies that 3 is to be divided by 4. Now if we multiply § by , as directed in the rule, we obtain #, the same numerator, as in the last part of the fraction, but a denominator 3 times as large. It is easy to perceive that the quotient 3 divided by 12, can be only a third part as large as the quotient of 3 divided by 45– of a shilling is 9 pence, and 3 of 9 pence is 3 ence, or P, of a shilling, which is equal to # of #. What is # of ; of a shilling 2 The answer must be twice as large as the answer to # of ; of a shilling. By working this example, we obtain the same denominator, or divisor, as before ; but multiplying the numerator of 3 by 2 the numerator, or dividend, is twice as large, for "a of a shilling implies twice as many pence as HoThe truth of the rule rests on this general principle, that the greater the denominators are when compared with the numerators, the less part of all unit, or of another fraction, they express; and if the denominators be larger than the numerators, their product must in the same degree be larger than that of the numerators, consequently the answer must be a less part of an unit than the last part of the compound fraction. But if the numerators are the largest, each part of the compound fraction denotes more than once the next part, or an unit, if it be the last part, and their product will be in the same degree larger than that of the denominators. 2. What is ; of , of ; of a dollar? Here of ; is #, and # of ; is or the answer. 3. What is ; of 4 effor? Ans. #. 4. What part of an unit is # of 4 2 Ans. 31. 5. If ; of a bushel of pears be equally divided between 3 boys, what part of a whole bushel will each boy have 7

Explanation.—Each boy is to receive 4 pears, which is ; of ; of a bushel, and of ; is to of a bushel.

6. If I have ; bushel of wheat, and give # of it to a poor neighbour, what part of a bushel do I give him 2 Ans. #, or #:

7. A man owning for of a share in a certain canal stock, sold #1 of his part; what part of a share did he sell ? Ans. Tor.

8. If 18 men share equally # of a captured ship, what part of the ship does each man possess?

- - Ans. rā, or or. Explanation.—One man’s part is or of 3.

9. If I, with 25 others, build # of a mile of road, and each build an equal proportion, what part of a mile do I build 7 Ans. 13+.

10. At # of a dollar per yard, what part of a dollar will # of a yard cost? Ans. #.

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