dimes to dimes, cents to cents, and mills to mills. They may be written down thus: $7.375 0.430 0.270 8.000 0.875 0.030 Ans. $16.980 10 at a A man bought 3 barrels of flour at one time, 863 barrels at another, 873 barrel at a third, and 15,78% 100% fourth. How many barrels did he buy in the whole? These may be written without the denominators, as follows: 3.3 barrels, 8.63 barrels, .873 barrel, 15.784 barrels. It is evident that units must be added to units, tenths to tenths, &c. For this it may be convenient to write them down so that units may stand under units, tenths under tenths, &c. as follows: 3.3 .873 15.784 Ans. 28.587 barrels. That is, 28,5 barrels. I say 3 (thousandths) and 4 (thousandths) are 7 (thousandths,) which I write in the thousandths' place. Then 3 (hundredths) and 7 (hundredths) are 10 (hundredths) and 8 (hundredths) are 18 (hundredths,) that is, 1 tenth and 8 hundredths. I reserve the 1 tenth and write the 8 hundredths in the hundredths' place. Then 1 tenth (which was reserved) and 3 tenths are 4 tenths, and 6 are 10, and 8 are 18, and 7 are 25 (tenths,) which are 2 whole ones and 5 tenths. I reserve the 2 and write the 5 tenths in the tenths' place. Then 2 (which were reserved) and 3 are 5, and 8 are 13, and 5 are 18, which is 1 ten and 8. I write the 8 and carry the 1 ten to the 1 ten, which makes 2 tens. The answer is 28.587 barrels. It appears that addition of decimals is performed in precisely the same manner as addition of whole numbers. Care must be taken to add units to units, tenths to tenths, &c. To prevent mistakes it will generally be most convenient to write them, so that units may stand under units, tenths under tenths, &c. 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 14 yards, the second 37, the third 41, the fourth_173, the fifth 193, and the sixth 4213. How many yards in the whole? 14 37 41 17 193 4213 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. can 1,1% cannot be changed to tenths, but it may be changed to hundredths. = 20, £= 7%. not be changed to hundredths, but it may be changed to thousandths. 375. 13 may be reduced to hundredths. 20 = 1009 6 5 = 100° and Writing the fractions now without their denominators in the form of decimals, they become 14.5 37.6 4.25 17.75 19.375 Ans. 136.125 yards or 136, 1361 yards. Common fractions cannot always be changed to decimals so 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 per sons; 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 into exactly 8 parts, therefore I suppose each of these parts to be divided into 10 equal parts; these parts will be hundredths. 58. 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. 5000 may be divided by 8 exactly, of 5000 is 625, or .625. Ans. .625 of a bushel each. 5000 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 50 of a bushel. of 50 is, 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 20 are, and there are 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. of 18 is Too Bringing the parts, 10, and 1500 together, they make 6.25 or .625 of a bushel each, as before. The operation may be performed as follows: 50 (8 48 .625 1000 Then I write the 5 as a dividend and the 8 as a divisor. 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. 57 (16 48 3.5625 barrels each. 90 80 100 96 40 32 80 80 In this example the answer, according to Art. X., is 3 bushels. But instead of expressing it so, I annex a zero to the remainder 9, which reduces it to tenths, then dividing, Į 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 I take of 10000 I obtain 5625 or .5625, as above. 10000 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 consi dered sufficiently exact if it is right within something less than 1 cent, that is, within less than TOO 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 283 yards; the second 344; the third 30; and the fourth 42 yards. How many yards in the whole? In reducing these fractions to decimals, they will be sufficiently exact if we stop at hundredths, since of a yard is only about of an inch. is exactly .6. If we were to continue the division of 4, it would be .28571, &c.; in fact it would never terminate; but .28 is within about one of of a yard, therefore sufficiently exact. is not so much as, therefore the first figure is in the hundredths' place. The true value is 0666, &c., but because is more than ofō, I call 14 it .07 instead of .06.7 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 = .28+. When the fraction is too large the mark should be made to show that something should be subtracted, as = .07 The numbers to be added will now stand thus: |