EXAMPLES. 1. Reduce 23 to its equivalent whole, or mixed number. 8)293(36 Ans. 24 53 48 Or, 233 293-8363 as before. 5 2. Reduce 21 to its equivalent whole, or mixed number. Ans. 127 19 3. Reduce 123 to its equivalent whole, or mixed number. Ans. 653 4. Reduce 45 to its equivalent whole number. CASE V.* Ans. 9. To reduce a compound fraction to an equivalent simple one. KULE. Multiply all the numerators continually together for a new numperator, 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 3 of 3 of 4 to a simple fraction. 1x2×3× 4 24 1 2×3×4×5-120- the Answer. Or, by expunging the equal numerators and denominators, it will give as before. 2. Reduce of of of to a simple fraction. Ans. Or, by expunging the equal nu 660 11 1440 24 merators and denominators, it will be 3x11 6×12 ==11 as before. That a compound fraction may be reprofented by a simple 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 3 of 0 Then of and confequently 3 of the fame as by the rule. If the compound fraction confifts of more numbers than two, the first two may be reduced to one, and that one and the third will be the fame as a frace sion of two numbers, and fo on. 3. Reduce of of 1 to a a simple fraction. Ans. 3. 5. Reduce of of of 12 to a simple fraction. Ans. 1 CASE. VI. 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. 2x4x864 the new numerator for 3. 4X5X8=160 the common denominator. 40 64 1805 Therefore the new equivalent fractions are 4%, and 188. the answer. 2 5 2. Reduce, 3, 3, and 7 to fractions having a common denom inator. Ans. 578 763 3. Reduce,of, 73, and, to a common denominator. 3 864 960 1008 1152 1152 1152 1182' 1152' 1872 1872 1872 432 1872 4. Reduce, of 21, 72, and §, to a common denominator. Ans. 936 1040 To reduce any given fractions to others, which shall have the least common denominator. 1. By Problem 2, Page 73, 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. *By placing the numbers multiplied, properly under one another, it will bè seen that the numerator and denominator of every fraction are multiplied by the very same number, and consequently their values are not altered. Thus, in the first example. In the second rule, the common denominator is a multiple of all the denominators, and consequently will divide by any of them: Therefore, proper part? may be taken for all the numerators as required. EXAMPLES. 1. Reduce, and to fractions having the least common denominator possible. 4)3 4 8 3 1 2 4x3x2=24= least common denominator. 243x1-8 the first numerator; 24÷4X3=18 the second numerator; 24-8×7=21 the third numerator. 18 21 24 24 Whence, the required fractions are, 312. Reduce,,, and to fractions having the least commonl denominator. Ans. 38, 48, 43, and 48. CASE VII. 30 To reduce a fraction of one denomination to an equivalent fraction of a higher denomination. RULE.* Multiply the given denominator by the parts in the several denominations between it and that denomination to which it is to be reduced, for a new denominator, which is to be placed under the given numerator: Or, compare the given fraction with the several denominations between it and that denomination to which it is to be reduced, and then, by case 5th, reduce the compound fraction thus formed, to a single one, and the equivalent fraction of the required denomination will be obtained. Let this fraction be reduced to its lowest terms. As there are *The reason of the rule may be seen in the following manner. 12 pence in fhilling, four-fifths of one penny can be only a twelfth part as much of 12 pence or a fhilling, as it is of one penny. Hence, to reduce four fifths of a penny to the fraction of a fhilling, the given fraction must be diminished 12 times, or one twelfth of it will be the equivalent fraction of a fhilling. A fraction is diminished in value, according to the note to Cafe 1. by multiplying the denominator by the whole number. Thus four fifths of a pen4 I 4 1 4 of of a filling. For the fame rea 12 60 ny= 4 5X12 of a hilling fon, four fixtieths of a fhilling can be only one twentieth as much of a pound, 4 60X 20 66 2912007 300 of a pound. Put these two operations together, and you have four-fifths of a penny, 5X12×20 The fame operation might have been performed thus. In a pound there are 240 pence. Then, four-fifths of a penny 1 as before. And in general the fraction of one denomination must be as -300 much diminished to be an equivalent fraction of a higher denomination, as is indicated by the number of parts of the given denomination to make one of the higher denomination. EXAMPLES. 1. Reduce of a cent to the fraction of a dollar. By comparing it, it becomes of of, which, reduced by case 5, will be 4X1 X1 = 4 D. Ans. Ans. 358 E. of a mill to the fraction of a dollar. Ans. D. of a penny to the fraction of a pound. Ans. of a farthing to the fraction of a pound. Ans. T of a penny to the fraction of a guinea. and 7x10x10= 700 2. Reduce of a mill to the fraction of an eagle. 3. Reduce 1. Reduce 5. Keduce 6. Reduce 7. Reduce Ans. guinea. Ans.moidore. 8. Reduce of an ounce to the fraction of a . Avoirdupois. Ans.. *9. Reduce of a pound to the fraction of a guinea. Ans. 4 guin. 10. Reduce of a pwt. to the fraction of a pound Troy. Ans.. 11. Reduce of a 1. Avoirdupois to the fraction of 1 Cwt. ៖ 12. Express 5 furlongs in the fraction of a mile. CASE VIII. Ans. Cwt. To reduce a fraction of one denomination to an equivalent fraction of a lower denomination. RULE. Multiply the given numerator by the parts in the denominations between it and that denomination you would reduce it to, for a This rule is the reverse of the preceding, and the propriety of it may be seen in a similar manner. The fraction of a higher denomination is obviously less than the equivalent fraction of a lower denomination; for instance, of a pound is & shillings or 5 shillings. Whence the value of the fraction must be increased, to render it an equivalent fraction of a lower denomination, so many times as there are parts of the less denomination in the higher. But, by the Note to Case I, the value of a fraction is increased by multiplying the numerator by a whole number. To reduce £ to the fraction of a shilling, I 400 as there are 20 shillings in a pound, we have; X20= 400 new numerator, which place over the given denominator: Or, only invert the parts contained in the integer, and make of them a compound fraction as before, then, reduce it to a simple one. EXAMPLES. 1. Reduce of a dollar to the fraction of a cent. By comparing the fraction it will be 3 = 1 X 10 X 10 175 X 1 X 1 2. Reduce 3. Reduce 11 4. Reduce 5. Reduce 6. Reduce 7. Reduce 8. Keduce 9. Reduce 10. Reduce 11. Reduce 15000 1 28 100 4 = c. Answer. 175 7 of of 10; then of an eagle to the fraction of a mill. of a guinea to the fraction of a pound. Ans. m. Ans. 11m. Ans. d. 19 Ans. qr. Ans. §d. Ans. 12s. ounce. Ans. 40%. Ans. £. of a Troy to the fraction of a pwt. Ans. Zpwt. of Cwt. to the fraction of a b Avoirdupois. Ans. 15. CASE IX. To find the value of a fraction in the known parts of the integer, as of coin, weight, measure, &c. RULE.* Multiply the numerator by the parts of the next inferior denomi nation, and divide the product by the denominator; and if any thing Thus let£ be the fraction, This rule follows from the preceding. whose value is to be found. By the preceding rule, £=5 of 133s. And on the same principle, s. of of a penny3d. ZL=1833. = 4d. Whence penny, = d. =6d. and, therefore 8. = a farthing,= 7qr.gr. Therefore, 7£ s. = 8s. 6d. The same process is obviously applicable to every similar case. Or, the process 78. 8s.67d. But 4.=7 of ī of 3 4 6 3 Ps. Ed. 3. gr. |