The answer is $710, or reducing the fraction to cents. $7.30 per yard. If 1.875 yard of cloth is sufficient to make a coat; how many coats may be made of 47.5 yards? In this example the divisor is thousandths, and the dividend tenths. If two zeros be annexed to the dividend it will be reduced to thousandths. 1875 thousandths are contained in 47500 thousandths 25 times, or reducing the fraction to decimals, 25.33 + times, consequently, 25 coats, and of another coat may be made from it. 100 From the three last examples we derive the following rule: When the divisor only contains decimals, or when there are more decimal places in the divisor than in the dividend, annex as many zeros to the dividend as the places in the divisor exceed those in the dividend, and then proceed as in whole numbers. The answer will be whole numbers. At $2.25 per gallon, how many gallons of wine may be bought for $15.375? In this example the purpose is to find how many times $2.25 is contained in $15.375. There are more decinal places in the dividend than in the divisor. The first thing that suggests itself, is to reduce the divisor to the same denomination as the dividend, that is, to mills or thousandths. This is done by annexing a zero, thus, $2.250. The ques tion is now, to find how many times 2250 mills are contained in 15375 mills. It is not important whether the poin be taken away or not. Instead of reducing the divisor to mills or thousandths, we may reduce the dividend to cents or hundredths, thus, $15.375 are 1537.5 cents. The question is now, to find how many times 225 cents are contained in 1537.5 cents. This is now the same as the case where there were deci mals in the dividend only, the divisor being a whole num ber. 1537.5 (225 1350 6.83+gals. Ans. as before. 1875 1800 750 675 75 If 3.15 bushels of oats will keep a horse 1 week, how weeks will 37.5764 bushels keep him? many The question is, to find how many times 3.15 is contained in 37.5764. The dividend contains ten thousandths. The divisor is 31500 ten thousandthis. 375764 (831500 31500 11.929+ weeks. Ans. 60761 31500 292640 283500 91400 63000 254300 253500 500 Instead of reducing the divisor to ten-thousandths, we may reduce the dividend to hundredths. 37.5764 are 3757.64 hundredths of a bushe!. The decimal .64 in this, is a frac tion of an hundredth. 3.15 are 315 hundredths. Now the question is, to find how many times 315 hundredths are contained in 3757.64 hundredths. From the two last examples we derive the following rule for division: When the dividend contains more decimal places than the divisor: Reduce them both to the same denomination, and divide as in whole numbers. N. B. There are two ways of reducing them to the same denomination. First, the divisor may be reduced to the same denomination as the dividend, by annexing zeros, and taking away the points from both. Secondly, the dividend may be reduced to the same denomination as the divisor, by taking away the point from the divisor, and removing it in the dividend towards the right as many places as there are in the divisor. The second method is preferable. The same result may be produced by another mode of reasoning. The quotient must be such a number, that being multiplied with the divisor will reproduce the dividend Now a product must have as many decimal places as there are in the multiplier and multiplicand both. Consequently the decimal places in the divisor and quotient together must be equal to those in the dividend. In the last example there were four decimal places in the dividend and two in the divisor; this would give two places in the quotient. Then a zero was annexed in the course of the division, which made three places in the quotient. The rule may be expressed as follows: Divide as in whole numbers, and in the result point off as many places for decimals as those in the dividend exceed those in the divisor. If zeros are annexed to the dividend, count them as so many decimals in the dividend. If there are not so many places in the result as cre required, they must be supplied by writing zeros on the left. Division in decimals, as well as in whole numbers, may be expressed in the form of common fractions. What part of .5 is .3? What part of 3 is .07! To answer this, 3 must be reduced to hundredths. 3 is 30, the answer therefore is • What part of 14.035 is 3.Š ! 3.8 is 3.800, the answer therefore is 3830 14633* In fine, to express the division of one number by another, when either or boh contain decimals, reduce them both to the lovest denomination mentioned in either, and then write the divisor under the dividend, as if they were whole numbers. Circulating Decimals. XXIX. There are some common fractions which cannot be expressed exactly in decimals. If we attempt to change to decimals for example, we find .3333, &c. there is always a remainder 1, and the same figure 3 will always be repeated however far we may continúe it. At each division we approximate ten times nearer to the true value, and yet we can never obtain it..1666, &c. ; this begins to repeat at the second figure. 6 - 545454, &c.; this repeats two figures. In the division the remainders are alternately 6 and 5. 56 333 .168168, &c.; this repeats three figures, and the remainders are alternately 56, 227, and 272. Some do not begin to repeat until after two or three or more places. It is evident that whenever the same remainder recurs a second time, the quotient figures and the same remainders will repeat over again in the same order. In the last example for instance, the number with which we commenced was 56; we annexed a zero and divided; this gave a quotient 1, and a remainder 227; we annexed another zero, and the quotient was 6, and the remainder 272; we annexed another zero, and the quotient was 8. and the remainder 56, the number we commenced with. If we annex a zero to this, it is evident that we shall obtain the same quotient and the same remainder as at first, and that it will continue to repeat the same three figures for ever. It is evident that the number of these remainders, and consequently the number of figures which repeat, must be one less than the number of units in the divisor. If the fraction is, there can be only six different remainders; after this number, one of them must necessarily recur again, and then the figures will be repeated again in the same or der. |