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EXERCISES IN VULGAR FRACTIONS. (1.) Add together , is and Ig of 99 and subtract the result from 54. Ans. 371.

(2.) Subtract 14 of 62 from of 104. Answer. 237 (3.) Reduce 64 of

to a single fraction, and divide it by 3.. Answer. 134% (4.) What number multiplied by will give 44 as the product. Answer 8.

This question is evidently solved by dividing 41 by Lo. A good method of gaining expertness in the management of fractions, is to reason out some simple questions of proportion in the following way: If lb. cost of a shilling, what will & lb. cost at that rate ?

Ib. shilling.
If 4 cost

4 will costs

8 , 35

i » 35 or 1 * s. In the successive steps of the above process, we first consider, that as lb. ost .s. 7 times lb. or 4lbs. will cost 7 times that fraction of a shilling, or $$s. We then take the of 258. (multiplying the denominator as the numerator is not divisible by 4) for the cost of llb. ; and to obtain the cost of 3 lb., or the of 8lbs. we multiply : 5 by 8; (in this case we are enabled to divide the denominator); and lastly, we find the į of the price of 8lbs. (where we are again obliged to multiply the denominator), which gives 1. s. as the answer to the question.

3. DECIMAL FRACTIONS. Notation and Numeration of Decimals.-A fraction, the denominator of which is 10, 100, 1000, or any number consisting of 1 followed by one or more ciphers, is called a decimal fraction, or more shortly, a decimal. The numerator of such a fraction may be any number whatever, but the denominator can only be 10, or one of the series of numbers produced by continued multiplication of 10 into itself. Thus, 5, 61, 732, are decimals. Vulgar fractions, on the other hand, admit of an infinite variety of denominations or divisions of the unit, from halves, thirds, fortieths, &c., upwards.

From the uniformity which thus distinguishes the denomination of decimals, we are enabled by a simple expedient to write down any given decimal, and to ascertain its value on inspection of the numerator, without expressing the denominator, as is necessary in the case of a vulgar fraction. A dot or point placed before the figures, and the use of ciphers to denote vacant places, indicate clearly and readily the character of the denominator. Every unit is supposed to be divided into a certain number of equal decimal parts, as tenths, tenths of tenths or hundredths, tenths of hundredths or thousandths, and so on. This mode of division and sub-division is merely an extension in a decreasing order to the right of the units' figure, of the scale of notation of whole numbers or integers. For example, in the integer 7156, the second figure on the left is ten times as great in that position

as if it occupied the place of units ; the third figure to the left is one hundred times as great, and the fourth, one thousand times as great, as if each stood in the place of units. Let that place be marked by a point written immediately after it, and let a few other figures be annexed, thus :

7156 · 3284 Then proceeding from the point towards the right, we may read the values of the successive figures as follows

* 10 100 10000 or, in words, three tenths, two hundredths, eight thousandths, four ten thousandths, just as we might read the several figures to the left of the point, knowing them to constitute a whole number, as six units, five tens, one hundred, seven thousand. But in the practice of integers it is customary to read such an expression as this, seven thousand, one hundred and fifty six, mentioning first in order the greatest denomination present and advancing from the left towards the right; so in the numeration of decimals it is both usual and convenient to commence in the same way and to express the collective value of the figures present as in the case of a whole number, adding the proper denomination of the decimal, which is always that of the last figure of the series. The entire fraction in question would accordingly be read, three thousand, two hundred and eighty four, ten thousandths. To show that this is the denomination of the sum of the separate fractions composing the decimal, it is only necessary to reduce them to a common denominator, when they will appear, (without any change of value) as

1000036, 13806 1807 1ooo And by adding together the numerators, we obtain for the total, 8.2.8.4. A similar process of reduction and addition may be applied to determine the collective value of the denominations of any other set of decimals; and it will be found generally, that the denominator of the figure furthest from the decimal point is the least common multiple, and, therefore, the proper denominator of the sum of the several decimal figures. Examples. (1.) .55= "+156=96 +=16, fifty five, hundredths. (2.) .603=% +100=90+ Too=76036, six hundred and three

thousandths (3) -4107=6 tot 100o=*%+1685 +1060o=100

four thousand, one hundred and seven, ten thousandths. The subjoined table exhibits the correspondence between the values of figures standing at equal distances, on the right and on the left respectively, of the place of units.

000

Hundreds of thousands.
Tens of thousands.
Thousands.

Ten thousandths or tenths

of thousandths.

In Millions.

Hundreds.

( Hundred thousandths or

hundredths of thou.

sandths.
Millionths.
l
or

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( Ten Millionths or tenths

of millionths.

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Decimals.

This table may easily be carried to any desired extent in either direction by obseving, that if the place of units be made the starting point, the first figure on the left of it represents tens (of units), and on the right, tenths (of units) : the second figure on the left, hundreds, and on the right, hundredths, and so on, the denomination of each figure on the right being at once inferred from the denomination of the figure which occupies the same relative position on the left of the units' place.

Use of ciphers.—In decimals such as, •07, 007, 0007, &c., the placing of ciphers immediately after the decimal point, serves to indicate that there are no tenths, hundredths, thousandths, &c., as the case may be, in the fractions represented. Ciphers are here used to denote vacant places at the commencement of a decimal, just as the same marks would be employed to fill up an interval occurring between significant figures,

The nine digits or figures, 1, 2, 3, &c., are called significant figures, that is, figures having numerical value, to distinguish them from ciphers, when the latter are employed merely for the purpose of keeping the figures in their proper places. Thus: •1 is the decimal expression of 1, first significant figure in place of

10 [tenths.

1

1

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&c.

1000C

100, first significant figure in place of

1 [hundredths. .001

1000, first significant figure in place of

10101 [thousandths. •10101

100000 54.0001

&o. Ciphers placed on the right of a given decimal, that is, annexed to the decimal, have no effect on the value of the fraction.

Examples: 3=30=300= 3000, &c.: or, X=30=300=2000&c., for, to add a cipher is to multiply by 10, to add two ciphers is to multiply by 100, &c., and as a cipher is added to the denominator for every one added to the numerator, no change is thereby made in the magnitude of the fraction.

As the converse of this principle it follows, that the taking away of any number of consecutive ciphers from the end or right hand of a decimal, leaves its value untouched.

Example : •300000=•30000=•3000=300=30=3.

To express a given decimal in the form of a vulgar fraction.-RULE.–For the numerator of the fraction, write down in their order the figures of the decimal, omitting all such ciphers as occur before the first significant figure; and for the denominator, write down 1, and annex to it as many ciphers as there are places, vacant or otherwise, in the decimal,

34

Examples : (1) •0034 stands for 10000 or thirty-fuur ten thousandths

782 (2.) *000782 , 7000000 or seven hundred and eighty-two millionths.

160065 (3.) .0160065 , 1000000 or one hundred and sixty thousand and

[sixty five ten millionths.

If it be desired, reduce the resulting fractions, where practicable, to lower terms.

Numeration of decimals.-Nothing is more common, even amongst tolerably expert arithmeticians, than an inability to express correctly in figures, according to the decimal notation, a fraction stated in words. The best way of attaining quickness and accuracy in this respect, is to learn to associate each figure with the number of places it is distant from the decimal point; to think of tenths, as holding the first place after the point ; of hundredths, as holding the second place, &c., &c.* There would then be no difficulty in making the proper arrangement of figures and ciphers, the moment the fraction had been repeated aloud, or presented on paper. For instance, the expression, twenty-three ten thousandths would at once suggest the writing down of the figures 23, and placing two ciphers between them and the decimal point, as the denomination ten thousandths stands in the fourth place of decimals. Examples : Write in figures as decimals, (1.) Forty-four, hundredths

Answer .44 (2.) Eight hundred and three, thousandths

.803 (3.) Five hundred, hundred millionths

00000500 In the last example, the two final ciphers may be dispensed with, and the expression written, .000005 (five millionths) as ciphers placed on the right of a decimal do not alter its value.

The reverse process, that of translating decimals from figures into words at length-is very easily accomplished ; all that is necessary is to write the value of the figures of the decimal, as if it were a whole number, and to supply the proper denomination by noting the place of the last figure ; thus, .0154 is written, one hundred and fifty-four ten-thousandths, because the figure furthest from the decimal point falls in the fourth place, that of ten-thousandths.f

To reduce decimals to a common denominator.-RULE.-Annex ciphers to the right, where necessary, so that all may have the same number of decimal places. Ex. 4, .06, .005, and .0138 reduced to a common denominator are

4000, 0600, .0050, 0138

That is, 18*8%, 1088 TEST T8 to: As previously explained, the addition of ciphers to the right of a decimal has no effect upon its value, and merely assists us to compare it the more readily with another decimal having a greater number of places.

One of the chief advantages possessed by decimals over vulgar fractions, consists in the facility with which the former are reduced to a common denominator. The labour of equalising the denominators of vulgar fractions, eren by the shortest methods, is in most cases, very considerable, but that process, however tedious, must be gone through before such fractions can be added together or subtracted one from another. Here then, the system of a uniform decimal denomination for

* See on this subject, the "Handbook for Officers of Excise," Chapter ii.

t in reading decimals aloud, on the occasion of comparing transfers of entries in books, &c., it will tend to obviate misapprehension, if the plan be adopted of using the word "point" or "and," whenever the units' figure of a preceding whole number happens to be 0. Thus, 650.7 gallons, should be pronounced 650 point 7, or 650 and 7, to distinguish it from the whole number 657, with which it might otherwise be confounded by the hearer.

all fractions is of great service in computing, since it virtually does away with the most troublesome part of the operations on fractions,--that of equalising the denominators. Suppose it asked, which of the fractions, -36, 425, is the larger ? we at once see, that if we annex a cipher to the first decimal, making it .360, the second fraction must be the larger, since it has the larger numerator referred to a common denominator. When decimals are reduced by this simple process to a common denominator, their sum or the difference of any two of them, is found, as in the case of vulgar fractions, by taking the sum or the difference of the numerators. But if the decimal form is to be preserved, the result, although it may admit of reduction, cannot be brought into any lower terms than are thus obtained. The largest of the given denominators is always the least common multiple, decimally expressed, of the other denominators.

Multiplication and division of decimals are also much more easily performed than the corresponding operations on vulgar fractions, and the superiority of the decimal system is rendered complete by the fact, that every vulgar fraction can be readily converted into a decimal which is either its exact equivalent, or is capable of being brought as near to it in value as we please.

To convert a vulgar fraction into a decimal.-RULE.-Annex ciphers at pleasure to the numerator, or conceive them to be so annexed ; and divide by the denominator until there is no remainder, or until as many quotient figures have been obtained as may be thought necessary. Then, point off the same number of decimal places in the quotient as there have been ciphers employed in carrying the division that far, and to make up this number, if necessary, add ciphers to the left of the quotient. Examples (1.) Reduce to a decimal.

2)1.0

•5 Answer. (2.) Reduce 4, 4s, and respectively, to decimals.

4)1.00( 254)3.00( 75 8)5.000(-625 16)7.0000(-4375 Answers. •25, •75, •625,

•4375. In all these cases the division terminates of itself, and the resulting decimals are exactly equal to the vulgar fractions from which they are derived. (3.) Reduced to a decimal.

243)1.0000000000 (41152
280

Answer .0041152
370
1270

550

64

&o. Here, 7 ciphers have been used up to the point at which the division was stopped, and in accordance with the rule we should make 7 decimal places in the quotient : two ciphers must therefore be prefixed to the quotient figures, giving as the answer, .0041152.

It is usual in the formation of an interminate decimal, such as this—that is where the process of division always leaves a remainder-to specify beforehand the

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