...... Ans. 14. ...Ans. 7. ...Ans. 43. ....Ans. 22. ..Ans. 32. ..Ans. 18. ......Ans. Z. ......Ans. 13. ...... Ans. Te. Ans. 3451. ....Ans. 200. ...Ans. 5379%. .......Ans. 2004. .........Ans. q. .....Ans. 13. Ans. 1. ...Ans. 13. ..Ans. 23 35° ..Ans. 12. ...Ans. 1. 21st. 433?. ...Ans. 64. 207. When the dividend is a mixed number greater than the divisor, the division may be effected without the reduction of the former to an improper fraction; for, if the divisor is a whole number less than the dividend, it is contained in the integral part of the dividend a certain number of times, either exactly, or with a remainder less than the divisor. If exactly, the quotient obtained by dividing the integral part of the dividend by the divisor, + the quotient obtained by dividing the fractional part of the dividend by the divisor, must be the quotient arising from the division of the whole dividend by the divisor, for the product of this quotient by the divisor is equal to the dividend (Art. 70 and 199). If a certain number of times with a remainder less than the divisor, the whole dividend is decomposed into two parts, the first a multiple of the divisor, the second a mixed number less than the divisor. The result of the division of the first part by the divisor is a whole number; that of the second part (found by Art. 205) a proper fraction; and the sum of these partial quotients is, as in the other case, the quotient required. When the divisor is a fractional expression, the dividend may be multiplied by the denominator of the divisor, as in Article 197, and this product divided by its numerator, in the manner already explained. As illustrations of this Article, let it be required, 1st. To divide 748695 by 5. The integral part of this dividend is a multiple of the divisor (Art. 122). Decomposing the dividend into the parts 748695 and g, it is found that 748695+5=149739 and 3+5= Collecting the partial quotients, 149739+ or 149739 is the quotient required. When the divisor is less than 10, the calculation may be made as in Article 86, thus, 5) 7486953 2d. Let it be required to divide 4557 by 9; By Article 127 it is found that 4557 is not a multiple of 9, also that the remainder from the division of 4557 by 9 is 3; therefore 4557% must be decomposed into the parts 4554 and 33. 4554÷9=506, and 33÷9=45 +9=2, consequently 45573÷9=506+3=5063. The process of division may be represented as follows: 9) 45573 506+33 rem., 33=8, and 18+9=2, ..506 is the quotient required. 3d. Let it be required to divide 454543 by 18. Since the divisor=1, it is required to multiply the dividend by 6, and to divide this product by 11; 4545+1 11) 27273, product of dividend by 6. 2479+47 rem., 47-55, and 55+11=1, ..24795 is the quotient required. The calculation by this method is sometimes more concise than that by the general rule for the division of fractional quantities. On this account a rule and appropriate examples are subjoined. Rule. To divide a large mixed number by a whole number; 1st. Divide the whole number of the dividend by the divisor: the result is the integral part of the quotient. 2d. Divide the remainder (which, if a mixed number, must be brought to an improper fraction) by the divisor: the result is the fractional part of the quotient. 3d. Annex the fractional to the integral part of the quotient: the mixed number thus formed is the quotient required. To divide a large mixed number by a fraction or by a mixed number (which must be reduced to an improper fraction); Multiply the dividend by the denominator of the divisor, and divide this product by its numerator. 4th. 156273+}?.... 5th. 48269+12?.... 6th. 5241&?.... 8th. 595635?. 20th. 85622845÷1000?.... .................Ans. 48650, .......Ans. 4768}}. .......Ans. 3747. ....Ans. 195341. ....Ans. 82746171. .....Ans. 6294. ..Ans. 5009}. ...Ans. 105003. ......Ans. 56014. Ans. 90502T ......Ans. 5760. ...Ans. 1610. Ans. 4500. ..Ans. 24484317873............. Ans. 7592. ..Ans. 135000%. ....Ans. 31592. ........Ans. 52494. .......Ans. 85613. 209. Remarks on multiplication and division of fractions. In multiplication, when the multiplier is a proper fraction the product is less than the multiplicand; for, the multiplier being less than unity, the product, which is a like part of the multiplicand, is less than once the multiplicand. In division, when the divisor is a proper fraction the quotient is greater than the dividend; for, the divisor being less than unity, its reciprocal is greater than unity, and the quotient is the product of the dividend by the reciprocal of the divisor. 210. The fractional expressions considered in the preceding rules are fractions of unity;, for example, and of 1 being expressions of the same signification (Art. 146). But a fractional part or fractional parts may, in like manner, be taken of any numbers either less or greater than unity, as of 3, of 2, of 53. To distinguish a fraction of unity from a fraction of any number less or greater than unity, it is usual to term the former a simple and the latter a complex or compound fraction. When the sum or difference of any numbers, of which some are complex fractions, is to be found, it is necessary to reduce the complex to equivalent simple fractions, as without such reduction the complex cannot be brought to the same denominator with the simple fractions or with each other. Let a given complex fraction be of, and let it be required to reduce it to a simple fraction. The signification of an expression such as, or of 1, is that 1 is divided into 4 equal parts, and that 3 of these 4th parts are taken to compose the fraction. So in the complex fraction of, it is to be understood that is divided into 7 equal parts, and that 5 of these 7th parts are taken to compose the fraction. Now, if or of 1 is divided into 7 equal parts, one of these 7th parts is equal to +7, or of 1, and 5 such parts are equal to 5 times of 1 or of 1, that is, to the simple fraction 5 Wherefore of 7 8 15 3x5 28 4x7 Similarly of 2 =8 times the 9th part of 2=2x8=16=173; In these instances the reduction of a complex to a simple fraction is made by multiplying together the numerators of the complex fraction for the numerator of the equivalent simple fraction, and the denominators for its denominator. The method being independent of the particular values of the given fractions, it follows that any complex fraction, consisting of a fraction of some number less or greater than unity, is reducible to a simple fraction by this process, which is identical with that for finding the product of two fractional factors. Next, let it be required to reduce the complex fraction of of to a simple fraction. In this example it is required to reduce of of 5 to a fraction of unity. of 3= 2x5 7x9° 2x5 2x5x3 4 Similarly 3 of but 2x5x3 2x5x3 5 3 of = That is, the complex fraction of of is equal to the simple fraction From a consideration of the method followed in the reduction of these complex fractions to simple fractions, it appears that the process of reduction coincides with that for finding the product of all the fractional expressions composing the complex fraction. Whence, to reduce any complex fraction to an equivalent simple fraction, Rule. Multiply together the numerators of the complex fraction for the numerator of the simple fraction, and the denominators for its denominator. 211. Exercises in the reduction of complex to simple fractions: of?..... the former are termed Vulgar or common, and the latter Decimal, fractions. SECTION VII. OF DECIMAL FRACTIONS. 213. If the figures of any number are taken consecutively from right to left, the first figure expresses units of the first order or simple units; the second, units of the second order, or tens; the third, units of the third order, or hundreds, &c. &c; the units expressed by any figure being ten times greater than those expressed by the figure which is immediately on its right. Conversely, the units expressed by any figure which is on the right of another, are tenth parts of the units expressed by that other figure. This principle may be extended so as to include figures written to the right of that digit which expresses the simple units of any number; thus, if 1 is written to the right of the digit expressing the simple units of a number, its relative value is one tenth part of unity, or, simply, one tenth; similarly the values of the figures 2, 3. . . . . 9, written to the right of the digit expressing the simple units of a number are respectively two tenths, three tenths,.. nine tenths. The relative value of 1 written to the right of the figure expressing tenths is one tenth part of one tenth, or one hundredth part of unity (Art. 210); and similarly the values of 2, 3 ...... 9, in the second place from the figure expressing simple units, are two hundredths, three hundredths, . . . . . nine hundredths of unity. ..... In like manner the relative values of the figures 1, 2, 3 . . . . . 9, written to the right of the figure expressing hundredths, are one thousandth, two thousandths, three thousandths..... nine thousandths of unity. 214. If the relative values of the figures equidistant on both sides from the digit expressing the simple units of a number are compared with each other, Of the figures on the left of the place of simple units: The first expresses tens. The second expresses hundreds. Of the figures on the right of the place of simple units: The first expresses tenths. And, generally, whatever multiple of unity is expressed by a figure on the left of the digit which expresses the simple units of a number, the corresponding part of unity is expressed by the same figure, at an equal distance from that digit, on the right. 215. The fractional numbers expressed by figures written to the right of the digit which occupies the place of simple units, are termed decimal fractions, or, simply, decimals. To distinguish the integral from the fractional part of a mixed number written in this manner, a mark, named the decimal point, is placed after |