divide the dividend by the numerator of the divisor, it will generally be necessary to multiply the denominator, of the dividend by the numerator of the divisor. From this article and the preceding, we derive the following general rule, to divide by a fraction, whether the dividend be a whole number or not: Multiply the dividend by the denominator of the divisor, and divide the product by the numerator. If the divisor is a mixed number, it must be changed to an improper fraction. Decimal Fractions. XXV. We have seen that the nine digits may be made to express different values, by putting them in different places, and that any number, however large, may be expressed by them. We shall now see how they may be made to express numbers less than unity (that is, fractions), in the same manner as they do those larger than unity. Suppose the unit to be divided into ten equal parts These are called tenths, and ten of them make 1, in the same manner as ten units make 1 ten, and as ten tens make hundred, &c. In the common way, 3 tenths is written, and 47 and 3 tenths is written 47. Now if we assign a place for tenths, as we do for units, tens, &c. it is evident that they may be written without the denominator, and they will be always understood as tenths. It is agreed to write tenths at the right hand of the units, separated from them by a point (.). Hitherto we have been accustomed to consider the right hand figure as expressing units; we still consider units as the starting point, and must therefore make a mark, in order to show which we intend for units. Thus 47 47 signifies 4 tens and 7 units; then if we wish to write, we make a point at the right of 7, and then write 3, thus 47.3. This is read forty seven and three tenths. Again, suppose each tenth to be divided into ten equal parts; the whole unit will then be divided into one hundred equal parts. But they were made by dividing tenths into ten equal parts, therefore ten hundredths will make one tenth. Hundredths then may with propriety be written at the right of tenths, but there is no need of a mark to distinguish these, for the place of units being the starting point, when that is known, all the others may be easily known. is written 7.04. 83.57 is read 83 and and 10, or since we may read it 83, which is a shorter expression. is Again, suppose each hundredth to be divided into ten equal parts; these will be thousandths. And since ten of the thousandths make one hundredth, these may with propriety occupy the place at the right of the hundredths, or the third place from the units. It is easy to see that this division may be carried as far as we please. The figures in each place at the right, signifying parts 1 tenth part as large as those in the one at the left of it. Beginning at the place of units and proceeding towards the left, the value of the places increases in a tenfold proportion, and towards the right t diminishes n a tenfold proportion. 87 Fractions of this kind may be written in this manner, when there are no whole numbers to be written with them. for example may be written 0.4, or simply .4. T may be written 0.03 or .03. may be written .87. The point always shows where the decimals. begin. Since the value of a figure depends entirely upon the place in which it is written, great care must be taken to put every one in its proper place. Fractions written in this way are called decimal fractions, from the Latin word decem, which signifies ten, because they increase and diminish in a tenfold proportion. XIt is important to remark that 3000 7 1000 2 &c. and that 100 70. 3572 10000 10000 50 500 T0000 = expressed in the same manner. 3 100009 5 Toooo, consequently +10 + 1000 + 0.3572. Any other numbers may be From this it appears that any decimal may be reduced to a lower denomination, simply by annexing zeros. Also any number of decimal figures may be read together as whole numbers, giving the name of the lowest denomination to the whole. 2 5 Thus 0.38752 is actually% +18+ TUOO + 10000 +10000, but it may all be read together 3875, thirty eight thousand, seven hundred and fifty two hundred-thousandths. Any whole number may be reduced to tenths, hundredths, &c. by annexing zeros. 27 is 270 tenths, 2700 hundredths, &c. consequently 27.35 may be read two thousand, seven hundred and thirty five hundredths, 2735 In like manner any whole number and decimal may be read together, giving it the name of the lowest denomination. It is evident that a zero at the right of decimals does not alter the value, but a zero at the left diminishes the value tenfold. 100 It is evident that any decimal may be changed to a common fraction, by writing the denominator, which is always understood, under the fraction. Thus .75 may be written 75 , then reducing it to its lowest terms it becomes. The denominator will always be 1, with as many zeros as there are decimal places, that is, one zero for tenths, two for hundredths, &c. The following table exhibits the places with their names, as far as ten-millionths, together with some examples. In Federal money the parts of a dollar are adapted to the decimal division of the unit. The dollar being the unit, dimes are tenths, cents are hundredths, and mills are thousandths. For example, 25 dollars, 8 dimes, 3 cents, 7 mills, are written $25.837, that is, 25,837 dollars. XXVI. A man purchased a cord of wood for 7 dollars, 3 dimes, 7 cents, 5 mills, that is, $7.375; a gallon of molasses for $0.43; 1 lb. of butter for $0.27; a firkin of butter for $8; a gallon of brandy for $0.875; and 4 eggs for $0.03. How much did they all come to? It is easy to see that dollars must be added to dollars, dimes to dimes, cents to cents, and mills to mills. They may be written down thus: 3 10 1000 A man bought 3 barrels of flour at one time, 8,83 barrels at another, 873 barrel at a third, and 15-78 at a 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, ténths under tenths, &c. as follows: 3.3 .873 Ans. 28.587 barrels. That is, 2850 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, I 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 |