REDUCTION OF FRACTIONS. CASE 1st.-TO REDUCE FRACTIONS TO THEIR LOWEST TERMS; OR, TO FIND THE LOWEST TERMS BY WHICH THE VALUE of A GIVEN FRACTION CAN BE EXPRESSED. RULE.-Divide both numerator and denominator by any number that will divide them both WITHOUT REMAINDER; then divide the quotients obtained, in the same manner, and so continue to do till there is no number greater than 1, that will divide them. The last quotient will be the numerator and denominator required. Ex. 1. Reduce to its lowest terms. Operation, 4÷8=, and 2, its lowest term. 2. Reduce 124 196 to its lowest terms. 3. Reduce 1296 to its lowest terms. 4. Reduce 96 to its lowest terms. 5184 108 56 5. Reduce 42 to its lowest terms. 6. Reduce 960 to its lowest terms. 1240 7. Reduce 76 to its lowest terms. 80 8. Reduce 1465 to its lowest terms. 67185 Ans. . Ans. . Ans.. Ans. J. Ans. 1. 9. Reduce 78943 to its lowest terms. 10. Reduce 6786 67860 to its lowest terms. 11. Reduce 468 to its lowest terms. 1060 12. Reduce 104 to its lowest terms. 56 Ans. . Ans. 13. CASE 2d.-TO REDUCE A WHOLE NUMBER OR A MIXED QUANTITY TO AN IMPROPER FRACTION. RULE.-If the given quantity be a whole number, multiply it by the proposed denominator; the product will be the numerator: but if it be a mixed quantity, multiply the whole number by the denominator of the fraction, and to the product add the given numerator; then under the number thus produced, write the denominator. Ex. 1. Reduce 21 to a fraction whose denominator is 9. Operation, 21 x9=189, the numerator; the fraction therefore is 189, 14 16. Ans. 251. 2. Reduce 8 to an improper fraction. Operation, 8×3=24, and 24+1=25, the numerator; therefore, 25 is the answer. 3. Reduce 16 to an improper fraction. Ans. 67. 4. Reduce 171 to an improper fraction. 5. Reduce 47 to an improper fraction. 6. Reduce 135 to an improper fraction. 7. Reduce 115 to an improper fraction. 8. Reduce 17283 to an improper fraction. Ans. 46677 9. Reduce 9 to an improper fraction. Ans. 19. 10, Reduce 12% to an improper fraction. Ans. 77. 11. Reduce 8 to a fraction whose denominator shall be 9. Ans. 12. 27 12. Reduce 16 to a fraction whose denominator shall be 12. Ans. 192. CASE 3d.-TO REDUCE AN IMPROPER FRACTION TO A WHOLE OR MIXED NUMBER. RULE.-Divide the numerator by the denominator; the quotient will be the whole number. If there be any remainder, place it over the denominator at the right of the whole number. Note. The true quotient includes both the whole number and fraction. In all cases of division, therefore, the remainder (if any) constitutes the numerator of a fraction of which the divisor is the denominator. Ex. 1. Reduce 147 to a mixed number. OPERATION. 17)1 41 (8 5 rem.; therefore, 8 is the answer. 2. Reduce 346 to a mixed quantity. Ans. 15. 23 79 3. Reduce to a mixed quantity. Ans. 131. 4. Reduce 49 to a mixed quantity or whole number. Ans. 7. 5. Reduce 456 to a mixed quantity. 79 6. Reduce 3564 to a mixed number. 346 7. Reduce 847 to a mixed number. 15 8. Reduce 1246 to a mixed number. 22 9. Reduce 108 to a whole number. 9 73 10. Reduce to a mixed number. Ans. 59. Ans. 6. 11. Reduce 128 to a whole or mixed number. Ans. 288. 6 12. Reduce 56789 to its proper number. Ans. 27041 21 CASE 4th.-TO REDUCE COMPOUND FRACTIONS TO SIMPLE ONES. RULE 1st.-Multiply all the numerators together for a new numerator, and all the denominators for a new denominator, and reduce the new fraction to its lowest terms, by Case 1st. Ex. 1. Reduce of of to a simple fraction. Performed, 2×3×5=30, the new numerator; and 3×4×6=72, the new denominator; therefore, 39 is the fraction required, but susceptible of being expressed in lower terms; therefore, 39÷6=1, Ans. Compound fractions may be reduced to simple ones, however, much more expeditiously, by canceling. The labor of reducing to lower terms is thereby avoided. RULE.-Draw a horizontal line and place all the numerators above the line and all the denominators below it. Cancel the numbers as far as practicable, as taught in the Rule for Canceling; then make the product of the numbers remaining above the line the new numerator, and the product of those remaining below, the new denominator. Note 1st. If there be nothing remaining above the line after canceling, 1 will always be the numerator of the new fraction. The same is true of the denominators. Ex. 2. Reduce of of to a simple fraction. 3. Reduce of of 14 to a simple fraction. Statement, 6. 1. 14 Canceled, 6. 1. 14 Ans. . Note 2d. Whenever the product of any two numbers on one side of the line will cancel any number on the opposite side, they may be so canceled; as in the last example, 7 and 2 below the line cancel 14 above it. 4. Reduce of of of to a simple fraction. Statement, 5. 12. 4. 7 6 2 Canceled, 9. 13. §. 8' and 7×2= 14, numerator; and 13 x 3 = 39, 2 denominator; to a simple fraction. of of to a simple of to a simple fraction. Note 3d.-If any term of a compound fraction be a mixed number, it must be reduced to an improper fraction before stating. 1. 6. 14. 4. 3 8. Reduce of of 4 of of to a simple fraction. 43, therefore, statement, ; which canceled will give the answer, 3. 7. 3. 5. 4 9. Reduce of 7 of 2 of 24 to a simple fraction. Ans. 216=6,85. 35 10. Reduce of of of of to a simple fraction. Ans. CASE 5th.-TO CHANGE FRACTIONS FROM ONE DENOMINATION TO ANOTHER, WITHOUT ALTERING THE VALUE. 1st. To reduce fractions of low denominations to those of higher value. RULE.-Divide the fraction, or what is the same thing, multiply the denominator by such numbers as are required to reduce the given quantity from the GIVEN to the REQUIRED DENOMI NATION. The Ex. 1. Reduce of a penny to the fraction of a pound. numbers required to reduce pence to pounds, are 12 and 20; therefore, of a penny is to be divided by these numbers; and since this can be effected in the present case only by multiplying the denominator, the operation will be, 5 5 and 1440' 6 X 12 X 20 this, by Case 1st, is reduced to, The canceling principle may however be successfully applied in the solution of sums of this character. RULE FOR CANCELING.-Place the numerator of the given fraction above a horizontal line, and its denominator below it; then place also BELOW the line, such numbers as are necessary to reduce the denomination given to that required. Cancel, &c. as before. 5 We will solve the above example by this rule also. Statement, The scholar should compare the statement with the rule, to see that he understands its application. The above 6. 12. 20 statement canceled, 6. 12. 20 and 6 × 12×4=288, the denomi 4 nator as before, and nothing remains as a numerator; therefore, as before, of a pound is the answer. (See Note 1st, Case 4th.) 2. Reduce of a farthing to the fraction of a shilling. By the common rule, 4X4×12-192, which, by Case 1st, equals Ans. By the rule for canceling, 3 3 3 The same can 3. Reduce of a penny to the fraction of a pound. Statement, 4. Reduce of a gallon to the fraction of a hogshead. Ans. 5. Reduce of an ounce Troy to the fraction of a pound. Ans. 36 6. Reduce 19 of a minute to the fraction of a day. Ans. 1530 7. Reduce of a pound Avoirdupois to the fraction of a Ans. 1 cwt. |