22. What is the product of 99 by 43? SOLUTION. Multiplying 9 by 4, we obtain 393; and multiplying 9 by, we obtain 64. Adding these products, we find the sum to be 46. Or, reducing 99 and 43 to improper fractions, we multiply one by the other as before. Canceling like factors, the result is 46. RULE. To find the product of mixed numbers, either multiply by the integer and fractional part separately, and then add their products; or, reduce both numbers to improper fractions, and then multiply as in the foregoing rule. Ans. 1083. 23. Multiply 12 by 81. 24. What will be the cost of 63 cords of wood at $24 a cord? 25. What will of 2 tons of hay cost at $113 a ton? Ans. $21. 26. What will 8g cords of wood cost at $23 per cord? Ans. $2247. 27. What must be paid for # of 61 tons of coal at of $7 per ton? 28. A man owning of a farm, sold of his share. What part of the whole farm had he left? Ans. 14. 29. A man bought a horse for $125, and sold him for of what he cost. What was the loss? Ans. $253. 30. A owned of 1235 acres of land, and sold of his share. How many acres did he sell? Ans. 49 acres. 31. If a family consumes 11 barrels of flour a month, how many barrels will five such families consume in 4% months? DIVISION. EXAMPLES. 138. To divide a fraction by an integer. 1. If a horse eats of a ton of hay in 3 months, what part of a ton will last him 1 month? yard cost? OPERATION. SOLUTION. of a dollar, what will 1 Here we cannot exactly divide +3=, Ans. the numerator by 3; but, since a fraction 189 is divided by multiplying the denominator (124, II), we multiply the denominator of the fraction, §, by 3, and have, the required result. RULE. To divide a fraction divide its numerator, or multiply its denominator. We divide the numerator when it is exactly divisible by the divisor; otherwise we multiply the denominator. 8. If 6 pounds of sugar cost of a dollar, what will 1 pound cost? 9. At $7 a barrel, what part of a barrel of flour can be bought for of a dollar? Ans. of a barrel. 10. If a yard of cloth costs $5, what part of a yard can be bought for & of a dollar? Ans. of a yard. 11. If 9 bushels of barley cost $7, what will 1 bushel 12. If 12 barrels of flour cost $764, what will 1 barrel cost? OPERATION. SOLUTION. We first divide as in simple num12)761 bers, and have a remainder of 43. We reduce this remainder to an improper fraction, 24, which we $63, Ans. divide (as in Ex. 1), and annex the result, , to the partial quotient, 6, and we have 63, the required result. 13. How many times will 16 vessel that holds 3 gallons? gallons of cider fill a Ans. 5 times. pounds of meat in a 14. If 9 men consume of 9 day, how much does each man consume? 15. A man paid $992 for 4 cows. that apiece? Ans. of a pound. How much was 139. To divide an integer by a fraction. 1. At & of a dollar a yard, how many yards of cloth can be bought for 12 dollars? FIRST OPERATION. 12 4 3)48 16 yards. SECOND OPERATION. 3)12 4 16 yards. RULE. SOLUTION.Integers cannot be divided by fourths, because they are not of the same denomination. Reducing 12 dollars to fourths by multiplying, we have 48 fourths; and 3 fourths is contained in 48 fourths 16 times, the required number of yards. SOLUTION. If the cloth cost $3 a yard, we could buy 4 yards for $12; but since it costs only as much, we can buy 4 times as much for the same money. Therefore, we can buy 4 × 4 yards 16 yards. To divide a whole number by a fraction, multi ply it by the denominator, and divide by the numerator of the divisor. 2. Divide 18 by 3. 3. Divide 63 by 1 Ans. 48. Ans. 117, 8. If of an acre of land sells for $21, what will an acre sell for at the same rate? Ans. $49. of a dollar a bushel, how many bushels of 9. When potatoes are worth and corn of a dollar a bushel, potatoes are equal in value to 16 bushels of corn? Ans. 22 bushels. 10. If a man can chop 23 cords of wood in a day, in how many days can he chop 22 cords? OPERATION. 23 = 4. 22 4 11)88 SOLUTION. We reduce the mixed number to an improper fraction, and then divide the integer in the same manner as by a proper fraction. 8 days, Ans. 11. Divide 75 by 134. 12. Divide 149 by 241. Ans. 518 Ans. 6245 13. A farmer distributed 15 bushels of corn among several persons, giving them 13 bushels apiece. Among how many persons did he divide it? 14. Divide of 320 by of 91. Ans. 255. 15. I bought of 74 cords of wood for of $32. How much did 1 cord cost? Ans. $3. 16. A father divided 183 acres of land equally among his sons, giving them 454 acres apiece. had he? How many sons 17. A man bought & of 23 yards of cloth for $8. How much would 5 yards of the same cloth cost? Ans. $20. 140. To divide a fraction by a fraction. 1. How many pounds of tea can be bought for 11 of a dollar, at of a dollar a pound? tained in = =18, Ans. times, and is con 3 times as many times as 1, or 3 times, which is times; is contained but as many times as }, and divided by 2 gives, equal to 1. In the operation, we have multiplied the dividend by the denominator of the divisor, and divided the result by the numerator of the divisor, which is in accordance with 138 for dividing a fraction. Hence, by inverting the terms of the divisor, the two fractions will stand in such relation to each other that we can find the product of the two numerators for the numerator of the quotient, and of the two denominators for the denominator, as shown in the operation. RULE. I. Reduce integers and mixed numbers to improper fractions. II. Invert the terms of the divisor, and proceed as in multiplication. 1. The dividend and divisor may be reduced to a common denominator, and the numerator of the dividend be divided by the numerator of the divisor; this will give the same result as the rule. 2. Apply cancellation where practicable. 7. How many times is contained in? Ans. 1 RAC. AR. -8 |