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of 6 per cent. Now, allowing that I sold the same so as to realize a gain of 20 per cent., how much money did I receive ?
The ordinary mode of solving this sum requires the four following operations; viz. 105: 100 :: 742: the net weight of the wool, which is 706} lb. Again, 7063 X3-8=795, the number of dollars which the wool would have cost, if no deduction had been made for ready money. But a deduction of 6 per cent. was actually made; therefore, 106: 100:: 795: the money paid, viz. $750. Now, on this last sum, I realized a gain of 20 per cent.; hence, 100: 120:: 750 : the money received, viz. $900, Ans. By canceling, these four statements are reduced to one, thus:
742. 100. 9. 100. 120
105. 8. 106. 100 that is, the 742 lb. is to be multiplied by the four succeeding ratios, and the number obtained will equal the number of dollars required. The same canceled : 7 20 3
15 142. 100. 9. 100. 120
and 20X3X15 = $900, Ans. 105. 8. 106. 100'
3 3dly. A great advantage of the canceling system over all others, arises from the expedition it affords in arithmetical solutions. Instead of multiplying and dividing by all the numbers which the nature of the sum proposed would naturally require, the multipliers and divisors are made to cancel each other; that is, equal factors are rejected from both. Hence, they are all made to exert their appropriate influence in procuring the answer, while the labor of multiplying and dividing is avoided. The statement of each sum for canceling is a fractional answer of the same, and it is obvious, that the value of fractions is not affected by rejecting equal factors from their numerators and denominators.
The processes of reduction which occur very frequen in common Arithmetics, are mostly avoided by this system. Suppose it be required to find how many pounds sterling 5 hogsheads of wine would cost at 10 d. per pint. By the canceling system, it is necessary only to write down the numbers required to effect the reduction, and the queslion is then solved by canceling those numbers as far as practicable. Thus:
5. 63. 4. 2. 10
12. 20° The numbers above the line are obviously those, which, when multiplied together, will give the answer in pence. The numbers below the line, are those required to reduce pence to pounds sterling. The above sim canceled :
12. 20; then, 21x5=105 £. Ans.
3. That the above method of solving arithmetical problems is easily
comprehended and applied by the scholar, has been fully tested by the author. The experience of nine or ten years entirely devoted to the business of instruction, leaves him no room to doubi on this point. Being, however, fully aware that his Arithmetic might fall into the hands of some, who would not at once comprehend and apply the principle of canceling, he has introduced the ordinary rules of solution, in connection with those of canceling, and has endeavored to render both modes plain and familiar, by frequent and clear illustrations.
The constant aim of the teacher should be to prepare his pupils for the active duties of life; and, in the department of Arithmetic, this is accomplished only when the scholar has acquired correctness and ecpedition in effecting his solutions.
To make good arithmeticians, it is first necessary to acquire a correct and extensive comprehension of the simple or fundamental rules of Arithmetic. When this is done, their application will be obvious. The danger, therefore, is not, that the scholar will spend too much time on what is usually regarded as the more simple part of Arithmetic, but that he will leave it too soon.
In the use of this treatise, the author would recommend, that, when the pupil shall have passed the simple rules, and commenced those operations to which canceling may be applied, he be required to solve each problem both by the ordinary rule, and by the rule for canceling. More practice will thus be secured, and, consequently, greater expedition acquired.
In the illustrations, which are given in connection with the different rules, it has been the design of the author fully to acquaint the scholar with the nature of the subject presented, without carrying his explanations so far as to take the work which properly belongs to the scholar, out of his hands. No important acquisition can be made, without corresponding effort. This fact seems to have been overlooked, in the preparation of some Arithmetics now in nse, and special effort made to render every thing as easy as possible for the scholar; that is, to enable him to effect the solutions with very little mental labor. The intellectual powers are, however, developed and strengthened only by being brought into vigorous exercise. “In Arithmetic, the young beginner should find just enough assistance to encourage and stimulate him to effort. That is not the best system, which enables the learner to advance from rule to rule with the least amount of study; but that, which, while it helps him over some difficulties, leaves him examples enough to task his powers to the utmost.” (Dr. Humphrey's Thoughts on Education.) With these introductory remarks, the following work is commended to the candor of an enlightened public.
THE AUTHOR. Norwich Academy, May, 1840.
Notwithstanding care has been constantly exercised to avoid errors, the following will be found to have escaped detection : Page 42, line 25, read 6, instead of 5 hundred. 65, in the 4th sum, and in the first statement of that sum, read
21, instead of 4. " 119, line 34, read , instead of . “ 136, line 14, read integers, for integris.
171, line 7, read $420, instead of $1.20.
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