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 4, the fourth 173, the fifth 193, and the sixth 423. How many yards in the whole? 141 37 41 17 193 423 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. 5 6 } = 1%, } = 1 cannot be changed to tenths, but it may be changed to hundredths. = 250, 232/485 1009 75 7% can not be changed to hundredths, but it may be changed to thousandths. may be reduced to hundredths. 375 1000 65 150, and 13 = 100° 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. = 0 0 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 pcrsons; 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. 598. 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 5000 18885000 may be divided by 8 exactly, 10 1000, or .625. of 5000 is 625 1888 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 0 of a bushel. of 50 is, and there are 2 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 200 are and there are left to be divided into & 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 1400 is To Bringing the parts, To, and together, they make 2 or .625 100 of a bushel each, as before. 2 ቦ 1000 The operation may be performed as follows: 1000 50 48 .625 20 16 40 40 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, I 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. 100009 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 1 cent, that is, within less than 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 303; and the fourth 423 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 7 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 of, I call 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. 6 1000 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 15 = =.07 -. The numbers to be added will now stand thus: From the above observations we obtain the following general rule for changing a common fraction to a decimal: Annex a zero to the numerator, and divide it by the denomina ́tor, and then if there be a remainder, annex another zero, and divide again, and so on, until there is no remainder, cr 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 quotient. Two men talking of their ages, one said he was 37 3847 years old, and the other said he was 643213 years old. What was the difference of their ages? 1 250 14783 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 2560 of a year. If the answer is required only within hours, five places are sufficient; if only within days, four places are sufficient. 642320 = 64.8520000 372473=37.2602313+ Ans. 27.5917687 years. It is evident that units must be subtracted from units, tenths 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 per· formed precisely as in whole numbers. 250)221308 ངོ॰1:|:ཀ3 7250 S |