reduction can be made, the 2 and 8 are to be multiplied together, for a divisor or denominator: (multiplying the denominator divides the fraction.) The answer, then, is 1 divided by 16, or 16 1 Or thus, o Ans. 2 X 8 3. If a pair of oxen plough, in 4 days, 3 part do they plough in one day? of a field, what Ans. 1 4. If 6 men earn 18 of a guinea in 1 day, what part of a guinea does 1 man earn in the same time? 5. If 16 hats cost $64, what does 1 hat cost? · 3 Ans. 28: Ans. $415. 6. What will 1 pipe of molasses cost, if 14 pipes cost $7074? Ans. $5015. 7. What will 1 pound of rice cost, if 50 pounds cost $1503? Ans. $3200. 96 17. If 1 pint of wine cost of 18. If 12 quarts of wine cost a dollar, how much will 12 quarts 13 dollar, what is it per pint ? cost? 19. Multiply by 11. 20. Divide 9ğ by 11. Art. 71.-Multiplication of whole numbers by fractions. 1. If a barrel of flour cost $9, how much will of a barrel cost? Had the cost of 2 barrels been required, the price of 1 barrel being given, we should multiply the price of 1 barrel by the number of barrels. The same is now to be done; that is, the price of one barrel is to be multiplied by the part or parts of a barrel taken. To multiply by 1, is to repeat the units of the multiplicand once. To multiply by 2, is to repeat the units of the multiplicand twice. 2 To multiply by of 1, is to repeat one-half of the units of the multiplicand once. 2 The product of any number multiplied by a fraction is proportionally as much less than the multiplicand as the multiplier is less than the unit, or-1. Therefore, if we multiply 9 by 3 of 1 the product will be of 9, or 6. Hence, it appears, that, to multiply by a fraction, is to repeat such a part of the multiplicand as the fraction is part of a unit. If 9 dollars be the cost of 1 barrel, then the quotient of 9 divided by 3 will be 3 dollars, the cost of of a barrel, and 3×2=6, the cost of 3. Thus it appears, that the only difference between multiplying by a whole number and a fraction, is, that in the last case the multiplier is a number divided; now to divide the multiplicand or the product is the same as to divide the multiplier. Hence, to multiply a whole number by a fraction, RULE. Multiply the whole number by the numerator of the fraction, and divide the product by the denominator; or divide the whole number by the denominator of the fraction, and multiply the quotient by the numerator. OBS.-As multiplying by a fraction is repeating a part only of the multiplicand, we divide by the denominator of the fraction, to obtain that part of the multiplicand to be repeated, and multiply by the numerator to repeat that part. Thus, to multiply by, we divide by 3, and obtain one-third of the multiplicand, which is to be repeated twice, or multiplied by 2, the numerator. The same principle is applicable to the multiplication of fractions by fractions. 2. What will 16 yards of cloth cost, at of a dollar per yard? Were the cost of 1 yard 3 dollars, the cost of 16 yards would be 16 times 3 dollars; but the cost of one yard is 1 of 3 dollars; therefore the cost of 16 yards will be of 16 times 3 dollars. Excluding equal factors from divisor and dividend, or from numerator and denominator, does not affect the result. In this example we exclude the factor 4, and the remaining factors 3 and 4 multiplied together, being factors of the dividend or numerator, give 12, the answer. 3. What will 40 yards of carpeting cost, at per yard? of a dollar Ans. $35. a dollar per Ans. $24. 4. What will 64 bushels of oats cost, at 3 of bushel? 5. What will 24 bushels of corn cost, at of a dollar per bushel? 3 6. Multiply 21 by 2, by 4, by . 7. What is the product of 324 multiplied 8. What will 9 pounds of tea cost, at pound? Ans. $15. 9. What will 56 pounds of butter cost, at pound? 10. What will 124 pounds of sugar cost, at per pound? 11. Multiply 32 by 1, by 2, by 3, by 7. 8 12 of a dollar per Ans. $14.. 16 Ans. 8, 8, 12, 28. Ans. 4. Division of Whole Numbers by Fractions. Art. 72.—1. If of a barrel of flour cost $6, what will be the cost of 1 barrel? QUESTIONS.-15. What is the product of any number multiplied by 1? 16. How many times greater than the multiplicand is the product of a multiplier greater than 1 ? 17. How much less than the multiplicand is the product, when the multiplier is less than 1? 18. What is the product of a whole number multiplied by a fraction? 19, Rule? As this question is the reverse of question 1st, in the preceding section, the reverse of the rule there given will be the rule for solving this question. Had 6 dollars been the cost of 2 barrels, we should divide the price by the number of barrels. The same is now to be done; that is, the price is to be divided by the parts of a barrel taken. The quotient of any number divided by a unit, or 1, is the same as the dividend. The quotient of any number divided by a greater number than 1, is as many times less than the dividend, as the divisor is times greater than a unit, or 1. On the same principle, if the divisor be less than a unit, or 1, the quotient will be greater than the dividend. If the divisor be , the quotient will be, or twice the dividend. If the divisor be, the quotient will be 3, or three halves of the dividend. Therefore, if we divide 6 by 2 of 1, the quotient will be of 6, or 9. It is evident, that 6 will contain of 2 three times as often as it will contain 2. Again: if 6 dollars be the cost of of a barrel, the quotient of 6 divided by 2 will be 3 dollars, the cost of of a barrel, and 3×39 dollars, the cost of 3=1 barrel. barrel. Thus it appears, that the difference between dividing by a whole number, or by a fraction, is, that in the latter case the divisor is a number divided; for, To multiply the dividend or the quotient, is the same as to divide the divisor. Hence the RULE. Divide the dividend by the numerator of the fraction, and multiply the quotient by the denominator; or multiply by the denominator, and divide the product by the numerator. Operation. 216 3×3=9 Ans. 3 Regarding the numerator of the fraction as a whole number, we say, if 2 barrels cost 6 dollars, the quotient of 6 divided by 2 will be the cost of 1 barrel; therefore, place 6, the dividend, on the right, and 2, the divisor, on the left of the line. But the 2 barrels is 2 divided by 3; therefore, place 3, the divisor, or denominator, on the opposite side of the line, as a multiplier. To multiply the dividend, or quotient, is the same as to divide the divisor. We then say, 2 in 6, three times; cross 6 and 2, and multiply 3 into 3. 3X3 $9 Ans. OBS.-Were the foregoing question, How many times will 6 bushels contain 2 pecks? we should multiply the dividend by 4, to bring it into pecks, or fourths of a bushel. 6 bushels equal 24 pecks, or 24 of a bushel, and 2 pecks equal of a bushel. So, to divide 6 units by, we multiply the dividend by 3, the denominator of the fraction, to bring it into thirds. The same is true in division of fractions by fractions. 2. If of a ton cost $15, how much will 1 ton cost? Ans. $18. 3. If 3 of an acre be worth $12, what is 1 acre worth? Ans. $32. 4. If 2 of the number of rows in a corn-field be 30, what is the whole number? 5 Ans. 40. 5. Divide 20 by 1, 1, 1. Ans. 40, 80, 100. 6. If a pound of tea cost of a dollar, how many lbs. may be bought for $60 ? Ans. 100 lbs. 7. In what time. can a man build 7 rods of wall, if he build 21 of a rod in an hour? 52 6 8. At of a dollar for building one rod many rods may be built for $69? 9. At $33 per yard, how many yards $80 ? Ans. 174 hours. of stone wall, how Ans. 80. may be bought for Ans. 21 yards. 10. If 14 bushel of wheat sow an acre of land, how many ...res will 12 bushels sow? 11. How many times is contained in 56 ? 12. How many times is 24 contained in 21? Ans. 9 acres. Ans. 64. Ans. 43. QUESTIONS-20. How much greater than the dividend is the quotient of a whole number divided by a fraction? 21. How do you divide a whole number by a frac tion? |