divide the numerator by it. The latter mode is to be preferred, when the numerator is a multiple of the divisor. Here, the divisor being inverted, and the fractions being multiplied, we have for the answer 2. Divide 51 by 21. X= Here 51, and 2113; we have, therefore, for the required result, 3×3 3 × 2=186=238. 3. Divide 75 by 9. Τ Here, 75, 57, and 9; we have, therefore, for the required result, 757×!=757=837. Or a fractional number may be divided by a whole number; thus, 9 is contained 8 times in 75, with the remainder 3. Then 37, by reduction to an improper fraction, becomes, which, by dividing by 9 becomes, the remaining part of the quotient. This mode of dividing furnishes an explanation of the rule, in page 43, for finding the true remainder in dividing by the factor of a given divisor. 4. Required the value of the complex fraction Here, 5, and 6333; and the simple fraction required is 3=4×35,388.t 30. 205 *The reason of the operation, and of course the above rule, will appear evident from considering, that divided by for instance, is For if we suppose any whole thing divided first into 10 equal parts, and then into 30 equal parts, the latter being 3 times as many as the former, must each of them be 3 times less than each of the former; and therefore 7 of them must be 3 times less than 7 of the former; or, in other words, seven thirtieths is the third part of seven tenths. Now, if the divisor were, it is plain that the quotient obtained by dividing by must be 4 times the quotient obtained by dividing by 3. We must therefore divide by 3 and multiply the result by 4, which is the same that is done by the inversion of the divisor. From this it appears, that in Division of Fractions, there is in reality, both a division and multiplication. It will also readily appear, that if the divisor be a proper fraction, the quotient will be greater than the dividend. † It may be useful on some occasions to know, that if the given fractions be reduced to equivalent ones having a common denominator, the quotient of the resulting numerators will be the required quotient. Thus, 11. What fractional part of 7 is 3 of 5? Ans. 59 Ans. 233. Ans. 8. Ans. 15. of the re 12. A man spent of a legacy in 5 months; mainder in 7 months; and then had 95 dollars left. was the amount of the legacy? What Ans. $380. 13. A man devised of his fortune to his widow; of the remainder to his eldest son; and the rest to his younger, The elder son's share exceeded the younger's by $750, How much had the widow ? Ans. $3375. After the full illustration of the multiplication and division of fractions, which has been given in the preceding pages, it appears unnecessary to give their application in the doctrine of ratio and proportion: for the ratio of one fraction to another is found by division of fractions; and a third proportional to three given quantities, either mixed or fractional, is found by dividing the product of the second, and third terms by the first. For instance, to find a fourth proportional to the three given quantities, 175, 4, and 34; or, which amounts to the same thing, to find a fourth proportional to 259, 29, and 25: we have 259:29:24: 1987; which is found by multiplying 29 by 25, and dividing the result by 25 thus 2x+25=24×4×5=1987, the answer. : 0875 12691 126913 the fractions in the first example are equivalent to 18 and 2; dividing therefore the numerator 10, by the numerator 21, we have, the same quotient as before; and thus may all the exercises in this rule be performed. What is a fraction ? Questions. How is a fraction expressed? What is the number below the line called? what above the line? What is a proper fraction? What is an improper fraction? What does the numerator of a fraction denote ? If you multiply both terms of a fraction by the same number, does the value of the fraction remain the same ? When can a fraction be reduced to lower terms ? When are numbers said to be prime to each other? How do you find the greatest common measure of two given numbers? How is a fraction reduced to its lowest terms? Repeat the rule for finding the least common multiple of two or more given numbers. How do you reduce fractions of different denominators to other equivalent ones, having a common denominator ? How is a mixed number reduced to an improper fraction? How do you find the value of a fraction in the denominations contained in the integer ? How is a given fraction reduced to another of a lower denomination? Repeat the rule of operation for adding fractional quantities. Repeat the rule for subtracting fractional quantities. CHAPTER VI. DECIMAL FRACTIONS. On the Nature of Decimal Fractions.* 114. A decimal fraction, or a decimal, is a fraction whose denominator is 10, or some number produced by the continued multiplication of 10 as factors, such, 100, 1000, &c. Hence all the rules for the management of fractions in general, are applicable to decimal fractions. 115. In the notation of decimals, the denominator is usually omitted; and, to indicate its value, a point is placed to the left of as many figures of the numerator, as there are ciphers in the denominator. Should there not be a sufficient number of figures in the numerator, as many ciphers are prefixed as supply the deficiency. 2 Thus, 180, 1000, are decimals; and is written ·7; 18, 09; 10000, 0003; 475, 4.75, &c. Hence, conversely : 116. The denominator of a decimal, thus expressed, is the number denoted by a unit with as many ciphers annexed, as there are figures in the given number. 이 Thus, 37 is; 004 is Too; 00083, T, &c. 117. From this notation it is evident, that the figure * The following articles on decimal fractions are merely an adaptation of the general principles of fractional quantities, already delivered, to a particular and important class of fractions. It therefore contains no principles that can be properly called new; but rather a short and easy mode of applying those already given. Every operation on decimal fractions, in fact, may be performed by reducing them to common fractions, as will appear hereafter, and then applying to the results the rules In the preceding articles. immediately following the decimal point denotes tenths, the next figure hundredths; the third, thousandths, &c. 1000 1000 7 Thus, since 476 is equivalent to 400+70+6, the decimal fraction 476, or 475 is equivalent to 100+100+1000, or to totoo, by dividing the terms of the first of these frations by 100, and those of the second by 10. In a similar circumstance it would appear, that 709 is equivalent to seven tenths, no hundredths, and nine thousandths. 118. Hence, the values of figures in decimals, as well as in whole numbers, are increased in a tenfold ratio by removing them one place towards the left hand, and diminished in the same ratio by removing them one place to the right. Thus, in the decimal 004, by removing the point one place to the right, and consequently the figures of the decimal one place to the left, we have 04, which denotes, or 70, and is ten times the given fraction, ; but by introducing a cipher after the point, which removes the figures a place to the right, we have 0004, or too, which is evidently only a tenth part of the given fraction, or 40 From these principles it will appear evident, that : 119. A decimal is multiplied by 10, if the separating point be removed one place towards the right hand; by 100, if two places; by 1000, if three places, &c.; vacant places, when such occur, being supplied in both cases by ciphers. 1000 Thus, 7248 X 10-7.248, or 7,24; 6.347 X100= 634.7; 6.3X1000=6300. Also, 78-33-10=7.833; 736-100-00736; 7.3100073, &c. It appears from the same principles, that: 120. The value of a decimal is not changed by annexing a cipher to the end of it, nor by taking one away, as in each case the significant figures retain the same positions in relation to the separating point. Thus, 505 500 each being equivalent to one half.* *From this view of the nature of decimal fractions, it appears, that there is in every respect the closest resemblance between them and whole |