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write them, so that units may stand under units, tenths under tenths, fc..
It is plain that the operations on decimal fractions are as easy as those on whole numbers, but fractions of this kind do not often occur. We shall now see that common fractions may be changed to decimals.
A merchant bought 6 pieces of cloth; the first containing 141 yards, the second 371, the third 44, the fourth 173, the fifth 131, and the sixth 42: How many yards in the whole ?
41 171 19
423 To add these fractions together in the common way, they must be reduced to a common denominator. But instead of reducing them to a common denominator in the usual way, we may reduce them to decimals, which is in fact reducing them to a common denominator; but the denominator is of a peculiar kind.
Ž = = cannot be changed to tenths, but it may be changed to hundredths. I=10 = 1 cannot be changed to hundredths, but it may be changed to thousandths.
ż ; may be reduced to hundredths. Tóo, and 8 = 10 Writing the fractions now without their denominators in the form of decimals, they become
4.25 17.75 19.375 42.65
Ans. 136.125 yards or 136,0 =
== 1361 yards. Common fractions cannot always be changed to decimals 90° easily as those in the above example, but since there will be frequent occasion to change them, it is necessary to find a principle, by which it may always be done.
A man divided 5 bushels of wheat equally among 8 persons ; how much did he give them apiece?
He gave them of a bushel apiece, expressed in the form of common fractions; but it is proposed to express it in decimals.
I first suppose each bushel to be divided into 10 equal parts or tenths. The five bushels make. I perceive that I cannot divide í: into exactly 8 parts, therefore I suppose each of these parts to be divided into 10 equal parts; these parts will be hundredths. 5=1:. But 500 cannot be divided by 8 exactly, therefore I suppose these parts to be divided again into 10 parts each. These parts will be thousandths. 5= 1868.5000 may be divided by 8 exactly, i of 1688 is 621
, or .625. Ans. .625 of a bushel each. Instead of trying until I find a number that may be exactly divided, I can perform the work as I make the trials. For instance, I say 5 bushels are equal to 1: of a bushel. $ of 10 is to, and there are left to be divided into 8 parts. I then suppose these 2 tenths to be divided into ten equal parts each. They will make 20 parts, and the parts are hundredths. of are in, and there are no left to be divided into 8 parts. I suppose these 4 hundredths to be divided into 10 parts each. They will make 40 parts, and the parts will be thousandths. 1 of ifo is noire Bringing the parts for 167, and is together, they make 12 or.625 of a bushel each, as before. The operation may be performed as follows:
I write the 5 as a dividend and the 8 as a divisor. Then I multiply 5 by 10, (that is, I annex a zero) in order to reduce the 5 to tenths. Then of 50 is 6, which I write in the quotient and place a point before it, because it is tenths. There is 2 remainder. I multiply the 2 by 10, in order to reduce it to hundredths. } of 20 is 2, and there is 4 remainder. I multiply the 4 by 10, in order to reduce it to
thousandths. of 40 is 5. The answer is .625 bushels each, as before.
In Art. X. it was shown, that when there is a remainder after division, in order to complete the quotient, it must be written over the divisor, and annexed to the quotient. This fraction may be reduced to a decimal, by annexing zeros, and continuing the division. Divide 57 barrels of flour equally among 16 men.
3.5625 barrels each.
In this example the answer, according to Art. X., is 3.1 bushels. But instead of expressing it so, I annex a zero to the remainder 9, which reduces it to tenths, then dividing, 1 obtain 5 tenths to put into the quotient, and I separate it from the 3 by a point. There is now a remainder 10, which I reduce to hundredths, by annexing a zero. And then I divide again, and so on, until there is no remainder.
The first remainder is 9, this is 9 bushels, which is yet to be divided among the 16 persons; when I annex a zero I reduce it to tenths. The second remainder 10 is so many tenths of a bushel, which is yet to be divided among the 16 persons. When I annex a zero to this I reduce it to hundredths. The next remainder is 4 hundredths, which is yet to be divided. By annexing a zero to this it is reduced to thousandths, and so on.
The division in this example stops at ten-thousandths ; the reason is, because 10000 is exactly divisible by 16. If F take ia of 1880. I obtain 1992, or .5625, as above. There are many common fractions which require so many
figures to express their value exactly in decimals, as to render them very inconvenient. There are many also, the value of which cannot be exactly expressed in decimals. In most calculations, however, it will be sufficient to use an approximate value. The degree of approximation necessary, must always be determined by the nature of the case. For example, in making out a single sum of money, it is considered sufficiently exact if it is right within something less than I cent, that is, within less than 1oo of a dollar. But if several sums are to be put together, or if a sum is to be multiplied, mills or thousandths of a dollar must be taken into the account, and sometimes tenths of mills or ten-thousandths. In general, in questions of business, three or four decimal places will be sufficiently exact.
And even where very great exactness is required, it is not very often necessary to use more than six or seven decimal places.
A merchant bought 4 pieces of cloth ; the first contained 287 yards ; the second 344 ; the third 3015; and the fourth 427 yards. How many yards in the whole ?
In reducing these fractions to decimals, they will be sufficiently exact if we stop at hundredths, since too of a yard is only about į of an inch.
30 (5 200 (7 100 (15 700 (9
is exactly .6. If we were to continue the divison of , it would be .28571, &c.; in fact it would never terminate ; but .28 is within about one į of tão of a yard, therefore sufficiently exact. it is not so much as to, therefore the first figure is in the hundredths' place. The true value is .0666, &c., but because 1000 is more than į of iso, I call it .07 instead of .06. z is equal to .7777, &c. This would never terminate. Its value is nearer .78 than .77, therefore I use .78.
When the decimal used is smaller than the true one, it is well to make the mark + after it, to show that something more should be added, as k =.28 +. When the fraction is too large the mark — should be made to show that something should be subtracted, as t's
The numbers to be added will now stand thus :
Ans. 135.75 yards, or 135,766 = 1357. From the above observations we obtain the following general rule for changing a common fraction to a decimal: Anner a zero to the numerator, and divide it by the denominator, and then if there be a remainder, annex another zero, and divide again, and so on, until there is no remainder, or until a fraction is obtained, which is sufficiently exact for the purpose required.
Note. When one zero is annexed, the quotient will be tenths, when two zeros are annexed, the quotient will be hundredths, and so on. Therefore, if when one zero is annexed, the dividend is not so large as the divisor, a zero must be put in the quotient with a point before it, and in the same manner after two or more zeros are annexed, if it is not yet divisible, as many zeros must be placed in the
Two men talking of their ages, one said he was 37,484? years old, and the other said he was 64}ö years old. What was the difference of their ages ?
If it is required to find an answer within 1 minute, it will 'be necessary to continue the decimals to seven places, for 1 minute is 32 5567 of a year. If tne answer is required only within hours, five places are sufficient; if only within days, four places are sufficient.
64311 = 64.8520000
Ans. 27.5917687 years. It is evident that units must be subtracted from units, enths from tenths, &c. If the decimal places in the two numbers are not alike, they may be made alike by annexing zeros. After the numbers are prepared, subtraction is performed precisely as in whole numbers.