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Examples.

Let 36417,341 be divided by Let 7,384107 be divided by

100 = 364,17341

1. 84968,43 ÷ 10

3. 759,846

100

5. 83,14786 ÷ 10000

1000,007384107

2. 2758,496 ÷ 100

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To contract the work, so as to retain but as many decimal places in the quotient, as may be thought necessary :—

Take as many of the left-hand figures of the divisor as will be equal to the number of figures both integers and decimals, which are to be in the quotient, and find how many times they are contained in the first figures of the dividend as usual.

Let each remainder be a new dividend, and for every such dividend, leave out one figure more on the right-hand of the divisor, remembering to carry for the increase, as in the second contraction in multiplication.

When there are not as many figures in the divisor as are required, begin the operation with all the figures, and continue as usual, till the number of figures in the divisor are equal to those remaining to be found in the quotient, after which, use the contraction.

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RULE OF PROPORTION.

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PROPORTION is the equality of Ratios, and the terms of equal ratios are called Proportionals.

Ratio, when applied to the subject of arithmetic, is the mutual relation of two numbers of the same kind to one another, with respect to quantity; or is the comparative magnitude of two numbers of the same kind.

Numbers are said to have ratio to each other when they are such, that the less can be multiplied, so as to exceed the greater. The two numbers compared are called the terms of the ratio, the first the antecedent, and the latter the consequent.

Ratios are equal when similar parts of their antecedents are respectively contained the same number of times in their consequents, thus, the ratio of 6 to 8 is equal to the ratio of 9 to 12; because 2 the third part of 6, and 3 the third-part of 9, are contained, the one in 8, and the other in 12, each 4 times. And if each of these be expressed as a fraction, making the antecedents the numerators, and consequents the denominators, the expressions thus formed will be equal, that is g= and; in either cases, 6 is said to have to 8, the same ratio which 9 has to 12, or it is simply said, that 6 is to 9 as 8 is to 12.

Ratio is expressed by the aid of 2 dots interposed between the terms of the ratio, and the identity or equality of two ratios by 4 dots interposed between the ratios, thus 6:8::9:12 denotes, that the ratio of 6 to 8 is the same or equal to the ratio of 9 to 12.

Proportion consists of three terms at least..

Three numbers are said to be proportional, when the ratio of the first to the second, is the same as the ratio of the second to the fourth; as 2, 4, 8, where 2, both the same ratio, for 4 is as much greater in comparison of 2, as 8 is greater than 4.

Four numbers are said to be proportional, when the ratio of the first to the second is the same as the ratio of the third to the fourth, as of 2:4:5:10, where =2, both the same ratio.

Four numbers are also said to be proportional, when by comparing them together by two and two, they either give equal products or equal quotients; suppose the former numbers 2:45 10; in comparing them together by multiplication, we have

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therefore, 2 4 5 10 are called proportional numbers. Now let the same numbers be compared by division, and we have

2. ......and.

+ = 21
= 5

= 2 equal. 42 equal.

=

= 1 unequal.

By this comparison the numbers are also said to be proportional.

In this kind of comparing four numbers, there is no need to try for more equal products or quotients than one set of each sort; for either case will determine the proportionality independent of the other. But it must be observed, that in any 4 proportional numbers, there will be but one set of equal products, and 2 sets of equal quotients, the smaller numbers being divisors.

When two numbers of the same kind, of different magnitudes are compared, it is instantly seen, that one of them is greater than the other; thus, if we compare 4 and 1, the former is evidently 4 times greater than the latter, and if each of these be multiplied by any common multiplier, as 2, the products 8, and 2 will bear the same comparative magnitudes to each other, as the original numbers, for in proportion as 4 exceeds 1, so will 8 exceed 2, and the numbers being placed, as under, will form what is called an analogy or proportion.

4:18 2.

The like will hold good, with any numbers or multipliers, and as it is evident, that this position is correct, with respect to the products, arising from a common multiplier; it is equally the case with respect to the quotients, arising from the division of two numbers by a common divisor; thus if we make choice of 4 and 10, and multiply and divide each of them by 2, by the multiplication there will result 8 and 20, and by the division 2 and 5, by which we may form two sepa rate analogies, from the first

and from the second

4 10:8 20

2: 5:4: 10

those numbers may be also interchanged, and still hold the same proportion, as for example,

25:8: 20

therefore, it is evident, that the quotients of the two first terms in any analogy by a common divisor, will be in the same proportion as the original numbers.

If any two ratios be equal, it is plain that their reciprocals must be equal, that is, the consequent of the first ratio is to its antecedent, as the cousequent of the second ratio is to its antecedent. Thus, since 2 : 5 :: 8: 20, we may infer that 5:2 ::20:8; for if 5 be as much greater in comparison of 2, as 20 is in comparison of 8, it follows, that 2 is as much less, in comparison of 5, as 8 is less in comparison of 20.

Again, from any analogy we may infer that the first antecedent is to the second antecedent, as the first consequent is to the second consequent. Thus, since 2:5::8:20, we may infer, that 2:8:: 5:20. For the given ratios could not be equal, unless 5 were as much greater in comparison of 2, as 20 is in comparison of 8; placing each ratio fraction-wise. we have, and again, whence the equality of the ratios is evident.*

In any analogy, the product of the first and fourth terms or extremes, is equal to the product of the second and third terms or means; thus, in the analogy 3: 46: 8, we have 8× 3=6×4. And if three numbers be in continual proportion, the product of the extremes is equal to the square or second power of the means; thus, the numbers 4, 6 and 9 are in continual proportion, and here 9×4=6×6 or 6a.†

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Any four proportional numbers, may be varied as under, and in these several changes, each new set of 4 numbers will still form an analogy.

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This Property may be demonstrated as follows: if the antecedents were equal to the consequents, as in the proportions 8:8; 12: 12, and 2:23:3, it is evident that the product of the extremes would be equal to that of the means; but a proportion may be always brought to that state by multiplying or dividing one of the means and one of the extremes by a proper multiplier or divisor. Thus, suppose the analogy was 3:5 18: 30, here the antecedents and consequents may be made equal, by multiplying the antecedents by 13 for 3×14=5 and 18 x 13=30, the analogy now becomes 5: 5 :: 30, where it is evident the products of the extremes and means are now equal, and it is equally evident they must have been so before the operation of multiplication took place; and the same reasoning will apply to any other numbers.

Hence, if we have given the first three terms of an analogy, we may find the fourth, by taking the product of the second and third terms, and dividing that product by the first. Thus, suppose we have given 3, 4 and 6, and are to find a fourth proportional; that is, such a number that the ratio of 3 to 4, shall be the same as the ratio of 6 to a fourth number to be found; here multiplying 6 by 4, and dividing their product 24 by 3, the quotient 8 is the fourth proportional sought.*

Simple proportion is the equality of the ratios of two quantities, to that of two other quantities.

Compound proportion is the equality of the ratio of two quantities, to another ratio, the antecedent, and consequent of which are respectively the products of the antecedents and con sequents of two or more ratios.

A question is said to belong to the Rule of Proportion, when three terms, whether simple or compound, are given to find a fourth proportional, which is the answer to the question, and in order to resolve such questions, the three terms must be first placed in order, which is called stating the question.Questions of this kind are stated and resolved by the following RULE:

When three numbers are given to find a fourth, which shall bear the same proportion to the third as the second doth to the first.

Bring the numbers to their lowest denominations, fractions to the same name, and mixed numbers to improper fractions; then state the question, that is, place the numbers in such order, that the first and second terms may be of one name or kind, and the third the same as the number required; then multiply the second and third numbers together, and divide the product by the first, which will give the answer in the same name as the third is or was brought into, which bring to a higher denomination if required.

If there happens to be a remainder after the division, it will be of the same name as the third number, and may be called

*That the above rule is true, may be thus proved. Let 4, 6 and 10 be 3 numbers given to find a fourth proportional: here we are given the first extreme and the two means to find the other extreme; now, since every product divided by one of its factors must produce the other factor; and as the product of the extremes has been proved to be equal to that of the means, the fourth term may be found by dividing the product of the given means, by the first term or given extreme, thus in the given example.

4:6 10

6 x 10

4

15 the fourth number required, which is the rule and the ame reasoning may be applied to any other numbers whatever.

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