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(35.) Divide 371bu. 1pk. of wheat equally among 270 men; what will each receive? Ans. 1bu. 1pk. 4qt. (36.) A man has to travel 75 miles; he walks the first day 20m. 3fur.; the second, 18m. 5fur. 20rd.; the third, 23m. 7fur. 30rd.; how much of his journey remains every evening? First evening, 54m. 5fur.

? Ans.

Second evening, 35m. 7fur. 20rd.
Third evening, 11m. 7fur. 30rd.

§ 127. The operations which have thus far been presented and required, must have so familiarized the pupil with the combination and decomposition of numbers; have so acquainted him with the two great principles by which all operations in numbers are effected, diminution and increase; and have induced such a degree of mental discipline and such an invigoration of the powers of analysis as to prepare him for higher combinations and less simple deductions.

If this be not so, a review from this point is earnestly recommended. And, in making it, let the pupil realize distinctly that but two principles are involved in all the operations required of him; that of increase, in Addition and Multiplication; that of diminution, in Subtraction and Division.

We first have these connected with simple numbers; then with simple fractions; and lastly, with denominate numbers, that is, representatives of specific quantity. No other principles are applied in all the variety of Cases, and Rules, and pages, which have thus far been presented, or which are ever devoted to the subject of Arithmetic.

This idea realized as it may and should be, will render the mind clear as to the past, and prepare it for higher reflections soon to be presented.

The principles of increase and diminution then, are emphatically the principles-the mode of application alone affords variety.

With these remarks, we pass on to higher reflections on quantity as expressed by numbers; and by analysis, to make an application of the principles, in one mode, sufficient for all practical questions properly connected with this subject.

PART IV.

RATIO.

§ 128. 1. RATIO is the quotient which arises from the division of one number or quantity by another of the same kind; and expresses the relation of one to the other.

2. It is obtained by comparison of the two numbers or quantities; that with which the comparison is made forming the numerator of a fraction, of which that compared forms the denominator; or, in terms of division, that with which the comparison is made, forming the dividend, that compared, the divisor. Thus, the ratio of 5 to 15-often noted 5: 15-is obtained by comparing 5 with 15, by making 15 the numerator or dividend, and 5 the denominator or divisor; 3, the ratio.

3. The two numbers used in a comparison, when named together, are called terms; when spoken of separately, the former of the two, that compared, is called the antecedent, the other the consequent.

4. Simple or abstract numbers, or numbers expressing quantity simply without reference to any other, are compared without difficulty; it being usually obvious which is properly antecedent, and which consequent. Thus, the answer to the question, what is the relation of 6 to 8? is evidently=13; 6 being the number compared, 8 that with which it is compared. Or, that to the question, what is the relation of a piece of cloth, in which there are 18 yards, to another, in which there are 36 yards? The relation or ratio of the former to the latter being obviously =2. Or, that to the question, what is the ratio of $8 to $56? is evidently 57.

18

5. But the comparison of other numbers or quantities requires more reflection. Its use, in such cases, being not to

obtain the ratio simply as a ratio, but by it as a means to attain a further result a final answer to a proposed question-a clear and distinct analysis of the question is necessary, that its true order may be determined.

6. In questions of this kind, two numbers or quantities which are of the same kind or character, are taken for the comparison (§ 128. 1). Their order of arrangement is determined by that number in the question, which is like the final answer desired. The comparison is made with

reference to that number.

7. If the answer should be greater than the number in the question of the same denomination with it, it is plain that the larger of the two numbers compared would be the dividend, to furnish the larger quotient; if the desired answer should be less than the number referred to, the smaller of the two numbers compared would be the dividend, to furnish the smaller quotient (§ 48. 4.).

8. Comparison in all cases expresses the ratio between the terms compared; and united to the number embraced in the question, the character of which forms the subject of the demand, and which is like the desired answer, indicates the answer in the form of a compound fraction.

Thus, if a piece of cloth containing 13 yards was sold for $130, what would be the price of a piece of the same cloth containing 18 yards?

The two numbers which are of the same kind in this question, therefore the two for comparison (§ 128. 1), are 13 and 18; the other number in the question, that like the answer sought, is $130. Now, to determine how the comparison should be made between the 13 and 18, whether the smaller or the greater should be the dividend, we regard the condition of the question and the nature of the demand. These show, in this case, that the final answer is to be cost in dollars; and that that answer is to be greater than the cost, or dollars embraced in the question. For, it is plain that 18 yards must cost more than 13. 18 therefore is the number with which the comparison is made, and 13 that compared. The comparison is therefore, the larger dividend giving the larger ratio. united with the cost in the question, gives the compound fraction

of $130, indicating the final answer required; that is, that 18 yards will cost of $130. Or, 1, the ratio of the two quantities, is the ratio of their respective cost, and

multiplied by the cost given in the question, must give the required cost of the 18 yards.

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Thus,

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10

$180 Ans.

§129. Many of the questions already presented are proper for this mode of solution; and it is proposed to substitute it for that of most of those remaining properly belonging to Arithmetic. It is here introduced in the place usually assigned to Proportions, which, with their principles, are left to the sciences to which they belong.

This is the mode of application of the two great principles of Arithmetic which we have before referred to (§ 127). Its advantages will be apparent on trial.

§ 130. 1. CASE. To solve questions by ratio.

RULE. I. Draw a perpendicular line; and write the number embraced in the question which is like the answer sought at its right.

II. If the desired answer be greater than this number, write the greater of the two remaining numbers under it, and the other at its left, on the left of the line.

III. Perform the operations thus indicated in the shortest mode practicable; by canceling, when it can be done; and, when it cannot, by such abbreviations as may be applicable.

2. Illustrations.

(1.) If 15 men can do a piece of work in 18 days, how long will it take 27 men to do the same?

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In this illustration, we see at once from the question that the answer is to be days; therefore, that 18, the number

in the question expressive of that kind of quantity, is to be first written. This is done, and, over it, the sign or mark of days is placed, to prevent mistake as to the denomination of the answer.

15 and 27, terms which are alike, each expressing the same kind of quantity-men-remain for the comparison. An analysis of the whole question is then made; from which it is apparent, that 27 men will do the work in less time than 15. Therefore, the final answer must be less than 18 days, the time in which the 15 men do the work. We require then the smaller ratio. This is obtained by writing the smaller of the two numbers for the dividend, and making of the other the divisor. The whole, as now written, indicates the answer to be 1 of 18.

We then cancel as far as possible, after which there remain on the right of the line 5 and 2, which, multiplied together, give the true result, 10 days.

(2.) How many yards of matting, 2ft. 6in. wide, will cover a floor that is 27ft. long and 20ft. wide?

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This illustration is a question in duodecimals properly, but is more conveniently solved in this mode, as is the great mass of questions of that class; so also are those of Practice.

Yards in length is the matter to be obtained; consequently yd. 1. is written at the top of the line, on the right, to designate the character of the answer.

But two numbers are embraced in the question, and of them the true order of comparison is apparent.

The two dimensions of length and breadth, multiplied together, make up the surface which is to be covered (§ 114. 5); and this is obviously the dividend. The multiplication is indicated by writing one of those dimensions under the other.

2ft. 6in. is the divisor: this is changed to the fraction of

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