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Let us now increase the number of fractions by adding three others to these two that we have already. Here let me say, that to reduce four or five fractions to a common denominator, can only be required when we want to add or subtract them; and this being the case, you will observe how the products of the numerators are placed in the following example. Reduce 1, 4, 5, and 5, to a common denominator.
We commence with the numerator
1 X 3 X 5 X 8 X 9 = 1080
1 X 2 X 5 X 8 X 9 720 of the left fraction,
4 X 2 X 3 X 8 X 9 = 1728 and you perceive the
7 X 2 X 3 X 5 X 9 = 1890 denominators of all 5 X 2 X 3 X 5 X 8 = 1200 the others are written
2 X 3 X 5 X 8 X 9 = 2160 in the same line, with the sign of multiplication between them; the whole of the figures are then multiplied together, and the product is 1080
Now this, as we know, is the sum expressive of }, and we accordingly find that there is the same difference between the numerator and denominator, that there was before we altered the fraction; that is, that the denominator is twice the numerator. In the second line we proceed in like manner; and indeed, if you look down the lines you will see that the left figures are the numerators, taken in succession as the fraction stood in the proposition ; and that the remaining figures are the denominators of the other fractions; and that the product of each line becomes the numerator of the fraction whose numerator is on the left of the line. To shew it more fully, we will write the fraction after being reduced to a common denominator under the fraction which each represents. } }
You will find, as I pointed out before, that there is the same difference between the numerator and denominator that there was before they were reduced; and not only that, but that you can reduce them to the smaller fractions they represent respectively, by the rule laid down for reducing fractions to their lowest denomination. have now but to point out how the common denominator is produced. You see in the bottom line of the series that the whole of the denominators are written down, and when multiplied together, that the product is 2160, which is the common denominator.
Example. Reduce 15, 31, and 3}, to a common denominator. In sums like this ex
15 x 32 x 35 = 16800 ample, it does not do for 31 x 16 x 35 = 17360 the learner to depend on
17 x 16 x 32 = 8704 his sagacity; he will there- 16 x 32 x 35 = 17920 fore take two terms, either from the right or left of a line of figures, and multiply them together; he will multiply their product by the next figure or number in the line, and so continue until he has multiplied by the whole of the terms, when the last product will be the numerator of that fraction whose numerator appeared in the line, and he will adopt the same method for finding the common denominator.
After having worked the above examples carefully on your slate, as you read them over, do for practice the following sums. (1.) Reduce , , and }, to a common denominator.
Ans. 144, 189, 194. (2.) Reduce }, 17, and 1], to a common denominator.
8 64 880 960 960 960
(3.) Reduce , 5, 10, and , to a common denominator.
Ans. H114, 119, 119, 1118.
4400 4800 4800 4800 4800
(4.) Reduce o 4, s, and 1), to a common denominator.
. 43 20
4300, (5.) Reduce 1:3, 109, and 312, to a common denominator.
Ans. 6 9 9 3 5 2 0 (6.) Reduce 4, 5, , and 44, to a common denominator.
Ans. 1636, 1996, 147, 1920*
3 963 5 5 2 5 7 3 8 5 6 5 803 92 803 803 92 80 803 92 80
ADDITION OF FRACTIONS. ADDITION, or joining fractions together, is of the utmost importance; and to prepare the pupil for the performance of this rule, we have worked the last six or eight sums in Reduction. And after having so reduced them, we have but to add the numerators together; and, should the amount exceed that of the common denominator, we divide it by that figure or number, and produce either a whole or a mixed number, or otherwise write the common denominator under the sum of the numerators, leaving it an improper fraction.
Take the first sum in the last rule.
Add together the following fractions ; * , and f. The sum being prepared exactly as you did when it last 3 x 6 x 8 = 144 occurred, the numerators are
5 X 4 X 8 = 160 added together for a new nu
7 x 4 x 6 = 168 merator; and the denomi
472 nator being written under it, we have an improper fraction, 4 x 6 x 8 = 192 or one whose numerator exceeds its denominator ; and if the sum was only fractions, leaving it so would do very well; but as they occur frequently with whole numbers, or, as we call
* Bring the mixed number 4 to an improper fraction , and proceed as before directed.
them when so, mixed numbers, it then becomes necessary to reduce the improper fraction to a mixed number, so as that the whole number so produced may be added in with the whole numbers in the proposition. We therefore, as laid down in the reduction of improper fractions into mixed numbers, divide the numerator by the denominator. 192) 472 (2008 Thus we have 2 88 as the sum of
384 the above fractions when added ; and
88 I trust that with another example the learner will be able to do not
192 only the few sums I set down for practice, but also to apply this rule to any proposition where its application may be necessary. Add together the following fractions, }, }, \, and 71.
2 X 8 X 4 X 4 = 256
31 X 5 X 8 X 4 = 4960
The mixed number is reduced to an improper fraction. The numerators, when found, are added, and the common denominator is divided into the total. The answer is, 9426; or, when the fraction is reduced to its lowest terms, it is 90.
After working the above example, do for practice the following sums. (1.) Add together, and
Ans. 13. (2.) Add together, 4, and 12.
Ans. 24. (3.) Add together, 14, and 11: Ans. 22865
(4.) Add together , 1, 1, and 14. (5.) Add together %, 4, 5, 6, and. (6.) Add together 43, 5, 7, and 111.
SUBTRACTION OF FRACTIONS.
To subtract one fraction from another, you reduce them, as in Addition, to a common denominator ; subtract the numerators, the lesser from the greater, and under the difference write the denominator.
From 11 take 13. The numerator 578, 19 X 45 = 855) or that of : 25, is taken 17 X 34 =
578 from 855, or that of 3.;
277 Remainder. the remainder, 277, be
34 x 45 =1530 Com. Den. comes the numerator, the common denominator being written under it. Here you have an opportunity of observing that it is the difference between the numerator and denominator that determines the value of the fraction. In the above example, if we considered the mere figures, we should be apt to imagine the lesser fraction to be the greater ; but we find the difference between the numerator and denominator of the fraction }is only fifteen, while the difference between the numerator and denominator of the fraction is is 28; consequently, this difference being the greater, the fraction } is less than the fraction
Example 2.—What is the difference between 2 and 3%?
In this example it is unnecessary to reduce the mixed numbers to an improper fraction.
2 5 The fractions only being reduced to a com- 11 mon denominator are placed on the right of the whole numbers, the numerators are subtracted one