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term in a double rule-of-three stating, is not the third term common to each of the single rule-of-three statings, into which the former may be decomposed; although in arranging the terms of each pair of component ratios, you are guided as to whether the greater or the less 'should be placed first, just as if this third term were that of the corresponding single rule-of-three stating ; you may easily convince yourself that you can never be misled into a wrong disposal of the first two terms by thus consulting the third term. In the second single rule-of-three stating, the true third term is that which actually appears multiplied by a fraction, so that the true fourth term would be the fourth term corresponding to the third wbich actually appears multiplied by the same fraction; and it is plain, that according as one quantity is greater or less than another, so will any fraction of the former be greater or less than that fraction of the latter ; consequently, in inquiring whether the third or fourth term be the greater, the fractional multipliers may be disregarded.
If 12 horses plough 11 acres in 5 days, how many horses would plough 33 acres in 18 days? Here, as the answer is to
hor. give the number of horses, we 11 : 33
:: 12 : put horses for the common third
5 term; and, disregarding days,
hor. just as we should do if the 1 : 3 number of days were the same
:: 2 :
3 : 5 we consider
or merely the acres ploughed; 1 :
hor. hor. 1
:: 2 : 10. Ans. and as more horses are required 1 : 5 for 33 acres than for 11, the first ratio is 11 : 33. Again ; returning to the question, we now disregard acres, and consider only days; and as fewer horses are required for 18 days than for 5 days, the second ratio is 18 : 5; hence the work stands as in the margin. The common factor 11 is struck out of 11 : 33, and the common factor 6 out of the first and third terms, 18 and 12 ; and, lastly, the common factor 3 is struck out of the consequent in the first ratio, and the antecedent in the second : with these simplifications, the work is reduced merely to the multiplication of 2 by 5.
You will, of course, observe that the denomination, common to the first and second terms of each of the given ratios, is wholly disregarded in the work, the abstract numbers alone being used.
2. If 15 men, working 12 hours a day, reap 60 acres in 16 days, in what time would 20 women, working 10 hours a day, reap 98 acres ; 7 men being able to do as much work as 8 women in the same time?
in both cases,
:: 16 : 2623
As the answer here is to be days, we put the given number of days for our third term; then regarding the number of workers 20 : 15
days. days. only, just as if all the other con- 10 : 12 ditions were the same, our first 60 : 98 ratio is 20 :15, because 20 workers 7: 8 is to 15 workers as 16 days to a 15 x 12 x 98 x 8 x 16 less number of days; next con- 20 x 10 x 60 x 7 sidering only the number of hours, 3x 7 x 16 as if these alone varied, the second
5 X 5 ratio is 10 : 12, as there must be
Ans. a greater number of 10-hour days than of 12-hour days, and our answer is to be in 10-hour days. Again, taking the acres only into account, the next ratio is 60 : 98; and, lastly, the ratio of a man's time of doing any amount of work to a woman's is 7 : 8. Hence the complete stating is as in the margin. The factor 15, as also the factor 4 in the 12, may be struck out from the numerator, and at the same time the factor 60 from the denominator. Again the factor 7 may be struck out from 98 in the numerator, and the same factor suppressed in the denominator; we shall thus have the re
3 x 14 x 8 x 16 duced form,
Lastly, striking out the factor
20 x 10 4 from the 8 and the 20, and the factor 2 from the 14 and
3x 7 x 16 the 10, we have finally the fraction
5 x 5 usually occupies less space to work our way to the final result in this manner, than to reach it by successive simplifications of the original stating, as in Ex. 1. That example treated in this
33 x 5 x 12 way, would give at first the fraction
11 x 18 by striking out common factors, becomes simply 5 x 2 or 10.
Exercises. 1. If 14 horses eat 56 bushels of oats in 16 days, how many
horses will 120 bushels keep for 24 days ? 2. If a person walking 12 hours a day travel 250 miles in
9 days, in how many days of 10 hours each could he
walk 400 miles, at the same rate ? 3. If 12s. be paid for the carriage of 2 cwt. 3 qr. 192 miles,
how much should be paid for the carriage of 8 cwt. 1 qr. 128 miles ?
4. If 3000 copies of a book of 11 sheets require 66 reams
of paper, how much paper will be required for 5000
copies of a book of 124 sheets ? 5. If 24 men can finish a piece of work in 36 days of 12
hours each, in what time can 30 men do it when the
working days are only 8 hours long? 6. If 939 soldiers consume 351 quarters of wheat in 168
days, how long will 1404 quarters last 11268 soldiers ? 7. If the sixpenny loaf weigh 32 oz. 8 dwt. when wheat is
608. per quarter, what should the eightpenny loaf weigh
when wheat is 54s. per quarter ? 8. If a family of 13 persons spend £64 in butcher's meat,
in 8 months, when meat is 6d. per lb., how much money, at the same rate, should a family of 12 persons spend in
9 months, when meat is 6 d. per lb.? 9. If the rent of a farm of 13 ac. 1 roo. 11į per. be £50
88. 9d., what should be the rent of another in the neighbourhood, containing 8 ac. 3 roo. 22 per., if 6
acres of the latter be worth 7 of the former ? 10. If £7 108. be the wages of 15 men, who work 10 hours
a day for 6 days, what ought to be the wages of 12 men who work 9 hours a day for 18} days ?
(76.) DECIMALS. I HAVE already explained, at the commencement of this rudimentary treatise, that our notation for integers, or whole numbers, is the decimal notation, inasmuch as the value of any figure of a number is ten times as much as it would be if that figure were removed one place to the right, so that in writing the figures of a number in the usual way, from left to right, every figure we put down is, in value, only the tenth part of what it would be, if it were one place less in advance. Now, the whole number becomes completed as soon as we reach in this way the place of units ; but there is no reason why the decimal notation should not be extended to the right beyond the place of units, still considering the value of each figure we write down, to be only the tenth part of what it would have been, if written in the immediately preceding place. In this way, the first figure written after units, would be tenths ; the next figure, hundredths ; the next,
thousandths; and so on: and, to prevent confusion, we should only have to put some mark of separation between the units and these fractional parts. This extension of the decimal notation is what we now have to consider. The mark employed to separate the decimal integers from the decimal fractions is simply a dot: thus, 234.625, means 234, with 6 tenths, 2 hundredths, and 5 thousandths ; it is therefore the same as 234+%+1+100 Each figure to the right of the decimal point is thus a fraction of known denominator, although that denominator does not appear; and such fractions are properly called decimal fractions, on account of the regular ten-fold increase of the denominators : for brevity, however, they are usually called simply decimals.
This extension of the decimal notation is so natural and obvious, that you can have no difficulty in understanding it; and as soon as an example of it is placed before you, you can as readily pronounce upon the value of a figure to the right of the decimal point, as you can pronounce upon the value of a figure to the left: the place next to units, on the left, is tens ; the place next to units, on the right, is tenths ; the place next to tens, on the left, is hundreds ; the place next to tenths, on the right, is hundredths ; and so on; thus, in the mixed number, 1234.5678, you could as readily tell the value of the 6 as of the 2, each of which is in a third place, the 4 being in the first, or units place : the value of the 2 is 200, or 2 hundreds ; the value of the 6 is 180 or 6 hundredths : in like manner, the value of the 3 is 30, or 3 tens, and of the 5, io, or 5 tenths ; the value of the 1 is 1000; and of the 7, 10o; and, lastly, the value of the 8 is 10800* It is very convenient to be able to express, in this way, decimal fractions without the incumbrance of denominators; and the more so, since, as you will presently see, all fractions may be converted into decimal fractions. From what has just been said, you see, that in order to express a decimal as a common fraction (sometimes called a vulgar fraction), you have only to write the figures of the decimal for numerator, and for denominator, to put 1, followed by as many zeros as will mark the place of the last decimal figure from the decimal point: thus, in the instance before given, it was seen, that •625 = 6+16+10oo, which is obviously 1938, the three zeros corresponding to the third place of the last decimal figure, 5. In like manner, .2438 = % +10o+ 10% +10000
24.3.8; •0342 = 1+100+10o0o =
1000 + 1080 = 108800; and so on; the number of zeros being always equal to the number of decimal places. There is one thing here that will no doubt occur to you; it is this ; that although zeros immediately after the decimal point, that is, before the figures, materially affect the value of those figures; yet, that zeros after them, have no effect at all, and are quite useless : thus, 234:625 is the same as 234.625000, &c.; and .0036 is the same as .0036000, &c.; zeros intermediate between decimal figures, have, of course, the effect of pushing the figures which follow them, lower down in the scale : thus, '62 is too, but .602 is % + Toon = 100% and •6002 is % +10000 = 1000000
(77.) The removal of the decimal point one place to the right, is equivalent to multiplying the decimal by 10; the removal of it two places to the right, is equivalent to multiplying the decimal by 100; and so on: thus, .2438 x 10 = 2.438, where each figure is 10 times what it was before ; •2438 x 100 = 24:38 ; •2438 x 1000 = 243•8, &c. : and the removal of the point in the other direction, is equivalent to dividing the decimal by 10, 100, &c. : thus, .2438:10 = •02438; •2438 • 100 = .002438 ; 2438 • 100 = 2:438, &c. &c. All this is plain from the very notation of decimals.
(78.) I shall now give you an example or two of converting decimals into fractions: 1. 17.5 = 17 * = 171 2. 21.25 = 21206 211. 3. 146.75 = 146100 146: you thus see, that I, 1, expressed in decimals, are :5, 25, •75. 4. 6.14 x 10 = 61.4 61* = 613 5. 3.135 x 100 = 313.5 = 313. 6. 2.76-100 = .0276 = 18760 =
It is further obvious, from the principles of the decimal notation, that when that notation is exchanged for the fractional, as in these examples, by writing the decimals without the point, and putting underneath, for denominator, unity, followed by as many zeros as there are decimal places, we may prefix to the numerator whatever whole number may have preceded the decimal: thus, 146-75 = 146-70% = "1875; 17.5 = 16; 21.25 =
326.047 = ; and so on. This is evidently only the same as reducing a mixed number to an improper fraction.
In writing decimals, you must be careful to put the decimal point against the upper part of the figures, not against the lower. When figures are separated by a point even with the lower part of the figures, the multiplication of the figures separated is understood, the point in that position standing in the place
69 2 500
2 1 2 5
3 2 6 0 4 7