Operation. 512 243 10|3 As the numerators of the fractions are to be multiplied together for a new numerator, or dividend, they are placed on the right Ans. of the line, and the denominators, which are to be multiplied for a new denominator, or divisor, are placed on the left of the line. The numbers are cancelled and multiplied as in preceding examples. Or thus: 3 2 3 The factor 2 in the numerator, cancels 2, one of the X 24 5 To Ans. factors of 4, in the denominator. 2. What is the product of ×; of 72×3? 3. Multiply by 2, and by 8. 4. A boy having of a dollar, gave did the toys cost him? Ans. 36. of it for toys; what Ans. of a dollar. 5. At of a dollar per yard, what will 6. At tea cost? Ans. of a yard cost? of a pound of of a dollar. Ans. of a pound of of a dollar. 7. At of a dollar a pound, what will coffee cost? 8. At 2 dollars per bushel, what will 63 bushels of wheat cost? Ans. $133. Ans. $445. 9. If a house lot be worth 100 dollars, what is of the lot worth? 10. If a flock of sheep be worth 75 dollars, what is of the flock worth? Division of Fractions by Fractions. Ans. $18. 1. If a bushel of corn cost of a dollar, how many bushels may be bought for of a dollar? It is evident that will contain twice: +32=2, Ans. In this example, both dividend and divisor are divided by 3. It has been shown, that to divide the dividend is the same as to divide the quotient, and to divide the divisor is the same as to multiply the quotient; and also, that to multiply and divide any number by the same quantity does not affect its value. Art. 76.-Therefore, when the denominator of dividend and divisor are alike, divide the numerator of the dividend by the numerator of the divisor, and the quotient will be the answer. 2. If a bushel of corn cost of a dollar, how much may be bought for of a dollar? Were the money to be expended 3 dollars, and the price 3 dollars, the answer would be 1 bushel, 3÷3=1. But suppose the price 3 dollars, and the money to be expended 3 dollars divided by 7, equal. The answer would then be÷3=4 of a bushel. To divide the quotient is the same as to divide the dividend. But the price is not 3 dollars, but 3 dollars divided by 4. To multiply the quotient is the same as to divide the divisor. Therefore, the true answer is, 4 divided by 7 equal . Again, x3, and 3=1×3. To multiply dividend and divisor by the same quantity does not affect the quotient. Art. 77.-Therefore, when the numerators of divisor and dividend are alike, RULE. Divide the denominator of the divisor by the denominator of the dividend. Art. 78.-3. If a bushel of oats cost of a dollar, how many bushels may be bought for of a dollar? Operation:=1 Ans. 4. If a bushel of rye cost of a dollar, how many bushels may be bought for of a dollar? In this example, the terms of the dividend cannot be divided by the corresponding terms of the divisor without a remainder; but to multiply the numerator is the same as to divide the denominator. Hence the RULE. Divide the terms of the dividend by the corresponding terms of the divisor, when it can be done without a remainder; otherwise, invert the divisor, and proceed as in Multiplication. Operation. 917. 518 dividend, is placed on the right of the 4556=113 Ans. line, and the numerator of the divisor, on the left. This is the same as inverting the divisor. }÷}=}}={8=1 Ans. Division of fractions by fractions may be variously illustrated. 1. To multiply the dividend is the same as to divide the divisor, (see Art. 43.) To multiply the denominator divides the fraction. Therefore, x+5=7x=58=1} Ans. 2. Multiplying the numerator of by 8 reduces it to eighths. Multiplying the denominator by 5 divides the fraction, (see Art. 72, Obs.) 1. If a bushel of potatoes cost of a dollar, how many bushels may be bought for of a dollar? Ans. 8. 2. If of a bushel of apples cost will 1 bushel cost? 3. How many bushels of rye, at may be bought for of a dollar? 4. If of a ton of hay cost per ton? 5. If 4 6. If of a dollar, how much of a dollar of a dollar, what does it cost pounds of tea cost 31 dollars, what is it per lb. ? Ans. of a dollar. of a dollar buy 1 pound of tea, how much will 31 dollars buy? Ans. 4 pounds. 7. Divide 17 by 7 and 183 by Ans. 2; 561. Multiplication and Division of Fractions. MULTIPLICATION. Art. 79.-1. A man owning of a house, sold of his share. What part of the house did he sell? 3. If a bushel of salt cost 19 DIVISION. Art. 80.-2. A man sold 35 of a house, which was of his share. What part of the house did he own? 4. If of a bushel of salt cost of a dollar, what will of a bush-1 of a dollar, what does it cost el cost? per bushel? 5. If a peck of coal cost of 6. If of a peck of coal cost a dollar, what will of a peck of a dollar, what will one peck cost? cost? 15 244 8. If of a cord of wood cost 31 dollars, how much is it per cord ? 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 of of 7 of a ship, sold of of of his share. What part of the ship did he sell? 2 3 4 1 $ 2 2 X = X Thus: 21 32 2345 4 32 Or thus: 1 15 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 denominators for a new denominator. By cancelling, the process of multiplying and reducing the fraction is performed at once. 2. Reduce of of of of of to a simple fraction. Ans. 25. 152=Ans. 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, of of of 3 of 9, of of 18, of of 2. 4 Ans. 17. 10. Divide of 9 by of 7. 161 122 11. Divide by 181 171 Ans. 113. Having reduced the terms of divisor and dividend to improper fractions, it will be found that the numerators and denominators themselves become fractions. Thus, the numerator of the dividend, 163=49, and the denominator, 1873 4 3 cess illustrated in the following manner. 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 13. Divide of of 5 of 7 of 18 by of 5 of 7 of 12; multiply by of of 3 of 2; divide by ofofof 6. 14. A man who owns of a farm, sells of his half. What part of the farm does he sell? Ans. 75. of 3 of 7 of Ans. 4 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 3 of 27. Ans. 9. Addition of Fractions. Fractions are added on the same principle as whole numbers. As tens can only be added to units, and pounds to shillings, by first reducing the higher denomination to the lower, |