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XIX. We observed a remarkable circumstance in the last article, viz. that = and 22. This will be found very important in what follows.
A man having a cask of wine, sold of it at one time, and of it at another, how much had he left?
and cannot be added together, because the parts are of different values. Their sum must be more than 3, and less than or 1. If we have dollars and crowns to add together, we reduce them both to pence. Let us see if these fractions cannot be reduced both to the same denomination. Now, &c. And, &c. It appears, therefore, that they may both be changed to sixths. 14 }=} and, which added together make. He had sold and had left.
A man sold of a barrel of flour at one time, and 4 at another, how much did he sell in the whole?
Fifths and sevenths are different parts, but if a thing be first divided into 5 equal parts, and then those parts each into 7 equal parts, the parts will be thirty-fifths. Also if the thing be divided first into 7 equal parts, and then those parts each into 5 equal parts, the parts will be thirty-fifths. Therefore, the parts will be alike. But in dividing them thus, will make 3, and 4 will make, and the two added together make 4, that is, 16. Ans. 1 barrel.
When the denominators of two or more fractions are alike, they are said to have a common denominator. And the process by which they are made alike, is called reducing them to a common denominator.
In order to reduce pounds to shillings, we multiply by 20, and to reduce guineas to shillings, we multiply by 28. In like manner to reduce two or more fractions to a common denominator, it is necessary to find what denomination they may be reduced to, and what number the parts of each must be multiplied by, to reduce them to that denomination.
If the denominator of a fraction be multiplied by 2, it is the same as if each of the parts were divided into 2 equal parts, therefore it will take 2 parts of the latter kind to make I of the former. If the denominator be multiplied by 3, it is the same as if the parts were divided each into 3 equal parts, and it will take 3 parts of the latter kind, to make 1 of the former. Indeed, whatever number the denominator be multiplied by, it is the same as if the parts were each divided into so many equal parts, and it will take so many parts of
the latter kind to make 1 of the former. Therefore, to find what the parts must be multiplied by, it is necessary to find what the denominator must be multiplied by to produce the denominator required.
The common denominator then, (which must be found first) must be a number of which the denominators of all the fractions to be reduced, are factors. We shall always find such a number, by multiplying the denominators together. Hence if there are only two fractions, the denominators being multiplied together for the common denominator, the parts of one fraction must be multiplied by the denominator of the other. If there be more than two fractions, since by multiplying all the denominators together, the denominator of each will be multiplied by all the others, the parts in each fraction, that is, the numerators must be multiplied by the denominators of the other fractions.
In the above example to reduce and to a common denominator, 7 times 5 are 35; 7 is the number by which the first denominator 5 must be multiplied to produce 35, and consequently the number by which the numerator 3 must be multiplied. 5 is the number, by which 7, the second denominator, must be multiplied to produce 35, and consequently the number by which the numerator 4 must be multiplied.
N. B. It appears from the above reasoning, that if both the numerator and denominator of any fraction be multiplied by the same number, the value of the fraction will remain the same. It will follow also from this, that if both numerator and denominator can be divided by the same number, without a remainder, the value of the fraction will not be altered. In fact, if the numerator be divided by any number, as 3 for example, it is taking of the number of parts; then if the denominator be divided by 3, these parts will be made 3 times as large as before, consequently the value will be the same as at first. This enables us frequently, when a fraction is expressed with large numbers, to reduce it, and express it with much smaller numbers, which often saves a great deal of labour in the operations.
Take for example 1. Dividing the numerator by 5, we take of the parts, then dividing the denominator by 5, 13 the parts are made 5 times as large, and the fraction becomes the same value as. This is called reducing fractions to lower terms. Hence
To reduce a fraction to lower terms, divide both the numɩ
rator and denominator by any number that will divide them both without a remainder.
Note. This gives rise to a question, how to find the divisors of numbers. These may frequently be found by trial. The question will be examined hereafter.
A man bought 4 pieces of cloth, the first contained 23 yards; the second 28; the third 37; and the fourth 171. How many yards in the whole?
The fractional parts of these numbers cannot be added together until they are reduced to a common denominator. But before reducing them to a common denominator, I observe that some of them may be reduced to lower terms, which will render it much easier to find the common denominator. In the numerator and denominator may both be divided by 2, and it becomes 2. may be reduced to, and to. I find also that halves may be reduced to fourths, therefore I have only to find the common denominator of the three first fractions, and the fourth can be reduced to the same.
Multiplying the denominators together 3 X 4 × 5 = 60. The common denominator is 60. Now 3 is multiplied by 4 and by 5 to make 60, therefore, the numerator of must be multiplied by 4 and by 5, or, which is the same thing, by 20, which makes 40, 40. In 2, the four is multiplied by 3 and 5 to make 60, therefore these are the numbers by which the numerator 3 must be multiplied. 45. In the fraction, the 5 is multiplied by 3 and 4 to make 60, therefore these are the numbers by which the numerator 1 must be multiplied. =}• 1=38. These results may be verified, by taking, 2, and of 60. It will be seen that of 60 is 20, the product of 4 and 5; of 60 is 15, the product of 3 and 5; and of 60 is 12, the product of 3 and 4 Now the numbers may be added as follows:
232342345 288283 2840 373373713 17 =
Ans. 107 yards.
I add together the fractions, which make 12727. I write the fraction, and add the 2 whole ones with the others.
A man having 233 barrels of flour, sold 85 barrels; how many barrels had he left?
The fractions and must be reduced to a common denominator, before the one can be subtracted from the other. and. Therefore 233=2311 895815
is larger than 14 and cannot be subtracted from it. To avoid this difficulty, I must be taken from 23 and reduced to 21ths, thus,
Ans. 14 yards.
taken from leaves 29. Then 8 from 22 leaves 14. Ans. 14 yards.
From the above examples it appears that in order to add or subtract fractions, when they have a common denominator, we must add or subtract their numerators; and if they have not a common denominator, they must first be reduced to a common denominator.
We find also the following rule to reduce them to a common denominator: multiply all the denominators together, for a common denominator, and then multiply each numerator by all the denominators except its own.
XX. This seems a proper place to introduce some contractions in division.
If 24 barrels of flour cost 192 dollars, what is that a barrel?
This example may be performed by short division. First find the price of 6 barrels, and then of 1 barrel; 6 barrels will cost of the price of 24 barrels.
Price of 6 bar.
Price of 1 bar.
If 56 pieces of cloth cost $7580.72, what is it a piece? First find the price of 7, or of 8 pieces, and then of 1 piece. 7 pieces will cost of the price of 56 pieces.
Price of 7 pieces
Price of 1 piece
Divide $24674 equally among 63 men. each have?
Share of 7 men
Share of 1 man
First find the share of 7 or 9 men, and then of 1 man. The share of 7 men will be of the whole. The share of 9 men will be of the whole.
Share of 9 men
Share of 1 man
In the first case I divide by 9, and then by 7. In dividing by 7 there is a remainder of 45, which is ; this divided by 7 gives. In the second case, I divide by 7 and then by 9. In dividing by 9 there is a remainder of 5%, which is 41; this divided by 9 gives 21 as before.
Share of 18 men $753,45% (18
How much will
Divide 75345 dollars equally among 1800 men, how much will each have?
First find the share of 18 men, which will be part of the whole. part is found by cutting off the two right hand figures and making them the numerator of a fraction, thus, 753,45%.
1545; this divided by 18 is 14