7. A fraction is divided by another fraction, by inverting the divisor, and multiplying them together as before. A unit is contained in the fraction, three fourths of once; is consequently contained in the same fraction twice as often, viz. of a time; and, three times as often, viz. of a time, which fractions are obviously obtained by multiplying by and inverted. Again, suppose it be required to find how many times is contained in. As before, a unit or 1 is contained in, seven eighths of a time; would be contained in it four times as often, viz. 28 of a time, and would be contained in the same only one third as often as 4, viz. 24 of a time=1, or 14. This result is obtained by inverting the divisor, and multiplying it into the dividend; thus, x=24. The following examples may now be performed: 2. Divide by . 3. Divide by 9 4. Divide by 1 5. Divide by 1 40 by . Ans. 2. 6. Divide by 2. 7. Divide by 4. 45 Ans. 36 44 Ans. 9 60. Ans. 20. 8. Divide by 9. Divide 10. Divide 8. If the numerator and denominator of any fraction be both multiplied or both divided by the same number, the value of the fraction will not be altered. The value Of this principle no explanation is necessary. of the fraction being the quotient arising from dividing the numerator by the denominator, it is obvious that if both the terms be doubled, or repeated any number of times, the value of the quotient will not be affected. 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 4 to its lowest terms. Operation, CASE 2d.-TO REDUCE A WHOLE NUMBER OR A MIXED QUAN TITY 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 × 9-189, the numerator; the fraction therefore is 189. 4 14 2. Reduce 8 to an improper fraction. Operation, 8x3=24, and 24+1=25, the numerator; therefore, 25 is the answer. 3. Reduce 16 to an improper fraction. Ans. 67. 4. Reduce 1713 to an improper fraction. 5. Reduce 47 to an improper fraction. 6. Reduce 135 to an improper fraction. 7. Reduce 135 to an improper fraction. 8. Reduce 17282 to an improper fraction. 9. Reduce 9 to an improper fraction. 10. Reduce 123 to an improper fraction. Ans.. 22 Ans. 251. Ans. 19. 11. Reduce 8 to a fraction whose denominator shall be 9. Ans, 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 141 to a mixed number. 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. 6. Reduce 3564 to a mixed number. 79 346 15 7. Reduce to a mixed number. 8. Reduce 1246 to a mixed number. 22 9. Reduce 10 to a whole number. Ans. 59. 346 Ans. 612 11. Reduce 1728 to a whole or mixed number. Ans. 288. 6 12. Reduce 56789 to its proper number. Ans. 27045 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 5 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=' Ans. 5 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 off of to a simple fraction. 5 14, numerator; and 13 x 3 = 39, denominator; therefore, the new fraction is . and 7×2 5. Reduce Ans. of 13 of 1 of 1 of 1 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, 9. Reduce of 7 of 216 =633• 3. 7. 3. 5. 4 of 24 to a simple fraction. Ans. 10. Reduce of of of of to a simple fraction. 16 147° 11. Reduce 9 12. Reduce 13. Reduce 1 of of 1 to a simple fraction. ΤΟ of of of to a simple fraction. ៖ of 1 of 51 to a simple fraction. of of of to a simple fraction. 15. Reduce of 8 of 53 to a simple fraction. 14. Reduce or 701. CASE 5th.-TO CHANGE FRACTIONS FROM ONE DENOMINATION TO ANOTHER, WITHOUT ALTtering 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. Ex. 1. Reduce of a penny to the fraction of a pound. The 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 |