CASE I.* To abbreviate, or reduce fractions to their lowest terms. RULE Divide the terms of the given fraction by any number, which will divide them without a remainder, and the quotients, again, in the same manner; and so on, till it appears that there is no number greater than 1, which will divide them, and the fraction will be in iw lowest terms. Or, Divide both the terms of the fraction by their greatest common. measure, and the quotients will be the terms of the fraction required. That dividing both the terms, that is, both numerator and denominator of the fraction, equally by any number whatever, will give another fraction, equal to the former, is evident: And if thofe divifions be performed as often as can be done, ●r the common divifor be the greatest possible, the terms of the refulting fraction must be the leaft poffible. Note 1. Any number, ending with an even number or cypher, is divisible by 2. 2. Any number, ending with 5 or 0, is divisible by 5. 3. If the right hand place of any number be 0, the whole is divifible by 10. 4. If the two right hand figures of any number be divitible by 4, the whole is divisible by 4. 5. If the three right hand figures of any number be divisible by 8, the whole is divisible by 8. 6. If the fum of the digits, conftituting any number, be divifible by 3 or 9, the whole is divifible by 3 or 9. 7. If a number cannot be divided by fome number lefs than the square root thereof, that number is a prime. 8. All prime numbers, except 2 and 5, have 1, 8, 7, or 9 in the place of units; and all other numbers are compofite. 9. When numbers, with the fign of Addition or Subtraction between them, are to be divided by any numbers, each of the numbers must be divided: Thus, 6+9+12=2+3+1=9. 3 10. But if the numbers have the fign of Multiplication between them; then only ⚫ne of them must be divided: Thus, 4X6X10 X6X10 2X6X2 24 1X5 CASE II. To reduce a mixed number to its equivalent improper fraction. RULE.* Multiply the whole number by the denominator of the fraction, and add the numerator of the fraction to the product; under which subjoin the denominator, and it will form the fraction required. EXAMPLES. 1. Reduce 36% to its equivalent improper fraction. 36 X8+5 Ans. 293 Or, I multiply 36 by 8, and adding the numerator 5 to the product, as I multiply, the sum 293 is the numerator of the fraction sought, and 8 the denominator: So that 293 is the improper fraction, equal to 36%. 36x8+5_293 Answer as before. 8 4 8 2. Reduce 127 to its equivalent improper fraction. Ans. 2143. TT 19 8. Reduce 653 to its equivalent improper fraction. To reduce a whole number to an equivalent fraction, having a given denomi nator. Multiply the whole number by the given denominator: Place the product over the said denominator, and it will form the fraction required. EXAMPLES. 1. Reduce 6 to a fraction, whose denominator shall be 8. 6x8=48, and 48 the Ans.-Proof 48=48÷8=6. 2. Reduce 15 to a fraction, whose denominator shall be 12. Ans. 180 8. Reduce 100 to a fraction, whose denominator shall be 70. To reduce an improper fraction to its equivalent whole, or mixed number. RULE. Divide the numerator by the denominator: the quotient will be the whole number, and thè remainder, if any, will be the numerator to the given denominator. EXAMPLES. All fraction reprefent a division of a numerator by the denominator, and are saken altogether as proper and adequate expreffions of the quotient. Thus the quotient of 3, divided by 4, is ; from whence the rule is manifeft; for if any number is multiplied and divided by the same number, it is evident the quo.ient muft be the fame as the quantity firft given. + Multiplication and Divifion are here equally used, and confequently the refult is the fame as the quantity first propofed. This cafe is, evidently, the reverfe of cafe 2d, and has its reason in the nature of common divifion. EXAMPLES. 1. Reduce 23 to its equivalent whole, or mixed number. 8,293(36 Ans. 24 53 48 5 Or, 293-293-8=36 as before. 2. Reduce 2163 to its equivalent whole, or mixed number. Ans. 1274. 3. Reduce 1241° to its equivalent whole, or mixed number. 19 4. Reduce 45 to its equivalent whole number. CASE V.* Ans. 653 To reduce a compound fraction to an equivalent simple one. RULE. Ans. 9. Multiply all the numerators continually together for a new numerator, and all the denominators, for a new denominator, and they will form the simple fraction required. If part of the compound fraction be a whole or mixed number, it must be reduced to an improper fraction, by case 2d, or 3d. If the denominator of any member of a compound fraction be equal to the numerator of another member thereof, these equal numerators and denominators may be expunged, and the other members continually multiplied, as by the rule, will produce the fractions required in lower terms. EXAMPLES. 1. Reduce of of of to a a simple fraction. 1×2×3×4 2X3x4x-12= the Answer. 2X3X4X5 Or, by expunging the equal numerators and denominators, it will give as before. 2. Reduce of of of to a simple fraction. Or, by expunging the equal numera That a compound fraction may be represented by a fimple one is very evident; fince a part of a part must be equal to fome part of the whole. The truth of the rule for this reduction may be shown as follows. Let the compound fraction to be reduced, be of. 6 18 6 6 Then 1 of 10=10 4, and confequentiy of 40X31 the fame as by the rule. compound fraction confifts of more numbers than two, the two first may to one, and that one and the third will be the fame as a fraction of two nd so on 5. Reduce 4 of of of 12 to a simple fraction. CASE VI. Ans. 1 To reduce fractions of different denominators to equivalent fractions, having a common denominator. RULE I*. Multiply each numerator into all the denominators except its own, for a new numerator, and all the denominators into each other, continually, for a common denominator. EXAMPLES. 1. Reduce, and to equivalent fractions, having a common denominator. 1x5x8-40 the new numerator for 2×4×8= 64 the new numerator for 3. 5×4X5=100 4x5x8-160 the common denominator. Therefore the new equivalent fractions are Answer. 2. Reduce, 3, 4, tor. ditto for . 40 64 1609 160 and 100 1609 the and to fractions having a common denomina Ans. 576 768 960 864 1008 1132 1132, T732, 1132, 1132° 3. Reduce, of 8, 74, and, to a common denominator. 3 936 1040 14508 432 Ans. 1872 1872 1872 1872* 4. Reduce, of 21, 2, and, to a common denominator. 8448 Ans. 21600 6720 7200 115209 113209 11320, 11320* RULE II. To reduce any given fractions to others, which shall have the least common denominator. 1. By Problem 2, Page 61, find the least common multiple of all the denominators of the given fractions, and it will be the common denominator required. 2. Divide the common denominator by the denominator of each fraction, and multiply the quotient by the numerator, and the product will be the numerator of the fraction required. EXAMPLES. 1. Reduce, and to fractions, having the least common denominator possible. 4) By placing the numbers multiplied properly under one another, it will be feen that the numerator and denominator of every fraction are multiplied by the very fame number, and consequently their values are not altered. Thus, in the first example. In the fecond rule, the common denominator is a multiple of all the denominators, and confequently will divide by any of them: Therefore, proper parts may be taken for all the numerators as required: I 4)3 4 8 312 4×3×2 = 24 = least common denominator. 24÷3×1-8 the first numerator; 24÷÷4x3=18 the second numerator; 24-8x7-21 the third numerator. 2 8 18 21 Whence, the required fractions are 24 24 2. Reduce 1, 3, 4, and to fractions having the least common denominator. Ans. CASE VII. 30 40 +5 and 48 To reduce a fraction of one denomination to the fraction of another, but greater, retaining the same value. RULE. Reduce the given fraction to a compound one by comparing it with all the denominations between it and that denomination you would reduce it to; lastly, reduce this compound fraction to a single one, by case 5th, and you will have a fraction of the required denomination, equal in value to the given fraction. EXAMPLES. 1. Reduce of a cent to the fraction of a dollar. 10 By comparing it, it becomes of of, which, reduced by case 5, will be 4x1 x1 2. Reduce of a mill to the fraction of an eagle. 3. Reduce of a mill to the fraction of a dollar. Ans. 1 1280° Ans. guinea. 2688 of a shilling to the fraction of a moidore. Ans. moidore. 7. Reduce Ans. lb. f9. Reduce of a pound to the fraction of a guinea. Ans. guin. 10. Reduce of a pwt to the fraction of a pound Troy. 78 Ans. 1920 lb. 11. Reduce of a lb. Avoirdupois to the fraction of 1 Cwt. 12. Express 5 furlongs in the fraction of a mile. *The reason of this and the next rule is explained in the rule reducing compound |