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CHANGING VULGAR TO DECIMAL FRACTIONS.

by the numerator of the divisor, we shall have 18 = = 2. Or it may be thus illustrated. As the dividend is of the quotient, and being of, we shall obtain of the quotient by taking of the dividend, that is, dividing it by 2, which may be done by multiplying the denominator 10 by 2, the numerator of the divisor, then by multiplying of the quotient by 5, we obtain the whole quotient, 18 = = 2. Hence the following

RULE.

To divide any sum by fractions, prepare the factors as in multiplication of fractions; invert the divisor, and proceed as in multiplication.

EXAMPLES.

1. Divide by 3. 4× 1, Ans.

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2. Divide 12 by 53. 1233 and 57 = 2; then 93 × 8=267-247, Ans.

63 X 32

3. Divide 8 by . 8- and X3103, Ans.

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SECTION XVII.

CHANGING VULGAR TO DECIMAL FRACTIONS.

To

The only difference between Vulgar and Decimal Fractions, consists in the manner of dividing a unit, and the way of expressing them. The former divides it into any number of parts, and the latter into ten, and if a further division is made, each of these are divided into ten, and so on. change vulgar to decimal, then, we have only to divide the unit into ten, a hundred, or a thousand parts, and observe what proportion of these parts is expressed by the vulgar fraction.

Reduce to decimals. and of 10 is .5.

Reduce 2 to decimals.

We divide a unit into 10 parts

Divide 1 into 100 parts; of

100 is .4, and consequently, is twice 4.8.

Hence the following general

RULES.

If the fraction to be reduced is a proper fraction, multiply the numerator by 10, 100, or 1000, &c. and divide it by the denominator.

2. If the fractions are mixed or compound, reduce them to simple ones and proceed as before.

It frequently is true, that the numerator thus multiplied, cannot be divided by the denominator without some remainder. In such cases no Decimal can be obtained which will express the whole value of the Vulgar Fraction. But 5 or 6 decimal figures will ordinarily be sufficient.

EXAMPLE. Reduce to a decimal.

1 x 10 x 10 = 100.

3) 100

.3333+

By continuing to multiply by 10, or what is the same thing, suppose ciphers to be added, and carrying on the division by 3, the decimals might be obtained forever.

SECTION XVIII.

PROPORTION, OR RULE OF THREE.

If a line is a foot in length, and another 2 feet, it is easy to understand, that two others may be proportioned to each other in a similar manner.

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If 5 dollars will pay 8 men for labouring a day, 15 dollars will pay 3 times as many men, 24, because 15 dollars is 3 times 5 dollars. The nature of proportion, &c. is well explained by Mr. Lacroix, to which the attention of the learner is directed.

"The ratio, or relation of two numbers, is the quotient arising from dividing one by the other.

We introduce some examples to illustrate the theory of ratios and proportions.

1. If 13 yards of cloth cost 130 dollars, what will be the price of 18 yards of the same cloth?

If we know the price of one yard of the cloth, it is plain that we can repeat this price 18 times and thus obtain the price of 18 yards. And since 13 yards cost 130 dollars, one yard must have cost the 13th part of 130 dollars; or, performing the division, we find the result 10 dollars, which multiplied by 18, gives 180 dollars as the price of 18 yards.

2. A courier, who travels always at the same rate, having gone 5 leagues in 3 hours, how many will he go in 11 hours? He goes in one hour 3 of 5 leagues, or of one league, and of course in 11 hours he will go 11 times as far, i. e. § of a league multiplied by 11 = 5, that is, 18 leagues and I

mile.

3. In how many hours will the courier of the preceding question go 22 leagues?

If we knew the time he would occupy in going one league, we should have only to repeat this number 22 times, and the result would be the number of hours required. And since it requires 3 hours to go 5 leagues, it will require only of the time, i. e. 3 of an hour, to go one league; this number multiplied by 22 gives 66 13 hours and, that is, 13 hours and

12 minutes.

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In the preceding statements, the known numbers and those required depend on each other in a manner that it will be well to examine.

To do this we may resume the first question, in which it is required to find the price of 18 yards, of which 13 cost 130 dollars.

It is plain that the price of this piece would be double, if the number of yards were double; that the price would be triple if the number of yards were triple, and so on; also that for half or two thirds of the piece, we should have to pay but or of the whole price.

Hence if there be two pieces of the same cloth, the price of the second ought to contain that of the first as many times as the length of the second contains the length of the first, and this circumstance is stated in saying, that the prices are in proportion to the lengths.

The relation, or ratio of the lengths, then, is that number, whether whole or fractional, which denotes how many times one of the lengths contains the other. If the first piece had 4 yards and the second 8, the ratio of the former to the latter would be 2. In the given example the first piece had 13 yards and the second 18; the ratio of the former to the latter is then 18, or 1.

As the prices have the same ratio to each other that the lengths have, 180 divided by 130 must give 18, which is the

case.

The four numbers 13, 18, 130, 180, written in this order, are such that the second contains the first as many times as

the fourth contains the third, and thus they form what is called a proportion.

A relation is not changed by multiplying or dividing each of its terms by the same number.*

To denote that there is a proportion between the numbers 13, 18, 130, and 180, they are usually written thus, 13: 18 130: 180, which is read 13 is to 18 as 130 is to 180; that is, 13 is the same part of 18 that 130 is of 180, or the ratio of 13 to 18 is the same as that of 130 to 180.

The first term of a relation is called an antecedent, and the second a consequent. In a proportion there are two antecedents and two consequents, viz. the antecedent of the first ratio, and that of the second; the consequent of the first ratio, and that of the second. In the proportion 13: 18 :: 130: 180, 13 and 130 are the antecedents, and 18 and 180 the consequents.

To ascertain that there is a proportion between the four numbers 13, 18, 130 and 180, we must see if the fractions 18 and 138 are equal,—and to do this, we reduce the latter to its lowest terms; and if they be equal, as is supposed by the nature of proportion, it follows that by reducing them to the same denominators the numerators will become equal, and consequently that 18 multiplied by 130 will give the same product as 180 by 13. This is actually the case, and proves that if four numbers are in proportion, the product of the first and last, or of the two extremes, is equal to the product of the second and third, or the two means.

We see that if four numbers were not in proportion they would not possess the above property; for the fraction which expresses the first ratio not being equivalent to that which expresses the second, the numerator of the one will not be equal to that of the other, when they are reduced to a common denominator.

The order of the terms of a proportion may be changed, provided they be so placed that the product of the extremes be equal to that of the means. In the proportion 13: 18 :: 130: 180, the following arrangements may be made.

* The learner will do well to remember this.

All the operations in this rule are founded on this principle. The learner should fix it in mind.

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13: 18:130: 180 13: 130 :: 18: 180 180: 130 :: 18 : 13 180: 18:130: 13 18: 13: 180: 130 18: 180: 13: 130 130: 13 :: 180: 18 130: 180: 13 : 18

In each one of these the product of the extremes is formed of the same factors, and the product of the means of the same factors. The second arrangement is one of those which most frequently

occurs.

Since the product of the means is equal to that of the extremes, one product may be taken for the other; and, as in dividing the product of the extremes by one extreme, we must necessarily find the other as the quotient, so in dividing by. one extreme the product of the means, we shall find the other extreme. For the same reason, if we divide the product of the extremes by one of the means, we shall find the other

mean.

We can then find any one term of a proportion, when we know the other three, for the term sought must be one of the extremes or one of the means.

The operation by which, when any three terms of a proportion are given, we find the fourth, is called the Rule of Three. Writers have distinguished it into several kinds; but this is unnecessary, when the nature of proportion and the enunciation of the question are well understood.

Among four numbers which constitute a proportion, there are two of the same kind, ahd two others of the same kind but different from the first two.

If it were required to find how many days it would take 27 men to perform a piece of work, which 15 men, working at the same rate, would do in 18 days; we see that the number of days should be less, in proportion as the number of men is greater, and vice versa. There is still a proportion in this case, but the order of the terms is inverted. The first number of days would contain the second as many times as the second number of workmen contains the first. This order of the terms being the reverse of that assigned them by the enunciation of the question, we say that the number of workmen is in the inverse ratio of the number of days.

GENERAL RULE.

Make the number which is of the same kind with the answer the third term, and the two remaining ones the first and second, putting the greater or less first, according as the third

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