The simple rule may now be repeated for solving any question which may arise in Multiplication and Division of Fractions by Whole Numbers-Multiplication and Division of Whole Numbers by Fractions-Multiplication and Division of Fractions by Fractions. RULE. Place all those numbers which are to be multiplied together for a numerator, or dividend, on the right of the perpendicular line, and those numbers which are to be multiplied together for a denominator, or divisor, on the left of the line, and proceed to cancel, as before directed. PROMISCUOUS EXAMPLES. Art. 81.-1. A man owning of of 2 of 3 of 7 of a ship, sold of of of his share. What part of the ship did he sell? 4 7 3 8 2 2 15 Thus: 21 32 Or thus: 43 54 $7 73 38 2 Ans. Fractions connected by the word of, are called compound fractions. They are reduced to simple fractions, by multiplying all the numerators together for a new numerator, and all the denomi152=Ans. nators for a new denominator. By cancelling, the process of multiplying and reducing the fraction is performed at once. 32 2. Reduce of 3 of 5 of 3 of 5 of to a simple fraction. Ans. 25 84. 3. A man owning of of of of a factory, sold of of of his share. What part of the factory did he sell? Ans. 4. What simple fraction is equivalent to of 22 of 7, of of of 3 of 9, of of 18, of of 2. Ans. 172. 7. Multiply by 1. 8. Divide 4 by 31. 9. Divide of by 5 of 8 of 9. Ans. 10. Divide of 9 by of 7. 161 123 Ans. 11. Having reduced the terms of divisor and dividend to improper 181 17 fractions, it will be found that 11. Divide by the numerators and denominators themselves become fractions. Thus, the numerator of the dividend, 163=49, and the denominator, 181=73 4 3 cess illustrated in the following manner. tion, and the The numerator is 49 divided by 3. To multiply the denominator is the same as to divide the numerator. To multiply the numerator is the same as to divide the denominator. Therefore Let the scholar reduce the divisor, and illustrate in a similar of 5 of 7 of 18 by of 5 of 7 of 12; mul 13. Divide of tiply by of of ofofofof Ans. 4 14. A man who owns of a farm, sells his half. What part of the farm does he sell? 15. Multiply 12 by of 3, divide by 1 of 1, multiply by of 6, divide by of 14, multiply by of 18, divide by of 27. Ans. 9. bers. Addition of Fractions. Fractions are added on the same principle as whole numAs tens can only be added to units, and pounds to shillings, by first reducing the higher denomination to the lower, so fractions of different denominations, or which have different denominators, can only be added by first reducing them to the same. Art. 82.-Fractions which have a common denominator may be added by the following RULE. Add their numerators, and write their sum over the denominator. Art. 83.-Addition of fractions whose denominators are different, and one is a multiple of each of the others. 1. Add and . In this example sixths is the lowest denomination mentioned; thirds must, therefore, be reduced to sixths. That is, must be reduced to an equivalent fraction, whose denominator is 6. (See Art. 58.) OBS.-That fraction is of the lowest denomination whose denominator is the largest. To ascertain how many of the smaller fractions make one of the larger, we divide the denominator of the smaller by the denominator of the larger; 6 contains 3 twice, or contains twice. That is, 2 sixths make. If, therefore, we multiply the denominator of the fraction by 2, we reduce the fraction to sixths. If we multiply its numerator by 2, the value of the fraction is preserved. Hence the RULE Divide the denominator of that fraction whose denominator is a multiple of each of the other denominators, first by the denominator of one of the other fractions; multiply its numerator into the quotient, and write the product over the denominator thus divided; and so continue to do, until the fractions are all reduced to the same denomination, or to a common denominator. Art. 84.-Addition of fractions when no denominator is a multiple of each of the other denominators. 1. Add 3, 1, . In this example we have no common denomination, or denominator given, to which each of the fractions may be reduced. If we multiply all the denominators together, we shall obtain a common multiple of all the denominators, for every product is divisible by all its factors, 3X4X5=60. We have now to reduce each fraction to equivalent fractions whose denominator shall be 60. This may be done by the foregoing Rule. But since the quotient of 60, divided by any one of the denominators, must be the product of all the others, we may adopt the common RULE. Multiply all the denominators together for a common denominator, and each numerator into all the denominators except its own for a new numerator. Operation 1st. Denominators, 3×4×5=60 com. denominator. Then, 60-3×2=40 60÷4×115 new numerators. Operation 2d. Denominators, 3×4x5=60 com. denominator. 1st numerator, 2×4×5=40 1. Reduce,,, and to fractions having a common denominator. 2. Reduce, 2, 14, and denominator. 336 224 384 504 Ans. 1344, 1344' 1344' 1344 to fractions having a common Ans. 3696 16632' 3. Add together 5, 7, 2, and 3. OBS. 1.-Reduce the fractions to a common denominator, find new numer. ators, and add them together. 4. Add 1, 2, 3, and 7. 5. Add together of and of 13. 6. Add of 96 and 7 of 14 together. OBS. 2.-Compound fractions must be reduced to simple fractions. OBS. 3.-Mixed numbers may be reduced to improper fractions, or the fractional parts may be reduced to a common denominator, and added as in the foregoing examples. If their sum amount to an integer, add it to the whole numbers. 9. Add together 142 and 163. Operation. 8 1412 12, and 3=2; then 12+12=}}=112· We find the common denominator to be 12, and the new numerators to be 9 and 1612 31 Ans. 8, which when added are 1=112. Write the under the Ans. 557. fractions, and carry 1 to the whole numbers. 10. Add together 17, 18, 193. 11. A grocer sold the following parcels of sugar, viz: 16 lbs., 191, 133, 204, 25, 30%, and 11 lbs. pounds did he sell in all ? Subtraction of Fractions. RULE. How many Ans. 13637. Art. 85.-Prepare the fractions as in Addition, and subtract the less numerator from the greater, and under the difference write the denominator. QUESTION.-What is the rule for the subtraction of fractions? |