10. If a man receive of a dollar for one day's labor, what will he receive for 21 days' labor? 11. What cost 56lb. of chalk at Ans. 1661. Ans. 3526. Ans. 43. Ans. $73. of a cent per lb. ? Ans. $0.42. of a 12. What cost 396lb. of copperas at 13. What cost 79 bushels of salt at cent per lb. ? Ans. $3.24. of a dollar per bushel ? Ans. $69. ART. 154. To multiply a whole number by a fraction. Ex. 1. Multiply 15 by g. FIRST OPERATION. 5) 15 3 x3=9 quotient by 3, the of it, which is 9. SECOND OPERATION. 15 3 Ans. 9. In the first operation we divide the whole number by the denominator of the fraction, and obtain of it. We then multiply this numerator of the fraction, and thus obtain In the second operation we multiply the whole number by the numerator of the fraction, and divide the product by the denominator, and obtain 9 for the answer, as before. Therefore, Multiplying by a fraction is taking the part of the multiplicand denoted by the multiplier. 455 9 RULE 1. - Divide the whole number by the denominator of the frac tion, when it can be done without a remainder, and multiply the quotient by the numerator. Or, RULE II. Multiply the whole number by the numerator of the fraction, and divide the product by the denominator. QUESTIONS. Art. 154. How do you multiply a whole number by a fraction according to the first operation? How by the second? What inference is drawn from the operations? What is the first rule for multiplying a whole number by a fraction? The second? ART. 155. To multiply a whole and mixed number to gether. Ex. 1. Multiply 17 by 62. OPERATION. 17 62 102 of 17 = 12₫ 1142 Ans. 1143. We first multiply 17 by 6, the whole number of the multiplier, and then by the fractional part,, which is simply taking of it, and add the two products together. Ex. 2. Multiply 7% by 4. Ans. 30%. We first multiply in the multiplicand by 4, the multiplier; thus, 4 times are 12, equal to 23, which is in effect taking of the multiplier, 4. We then multiply the whole number by 4, and add the two products together. RULE.- Write the less number under the greater, and if the FRACTION is in the multiplier, take a part of the multiplicand denoted by the fraction; but if it is in the multiplicand, take a part of the multiplier denoted by the fraction, and in each case add the product thus obtained to the product of the whole numbers. EXAMPLES FOR PRACTICE. 3. Multiply 93 by 5. 4. Multiply 12g by 7. 5. Multiply 9 by 811. 6. Multiply 10 by 74. 8. What cost 7lb. of beef at 5 cents per pound? 9. What cost 23 bbl. of flour at $6 per Ans. $0.37. 10. What cost 8 yd. of cloth at $5 per yard? Ans. $417. 11. What cost 9 barrels of vinegar at $63 per barrel? Ans. $57. QUESTIONS. Art. 155. What is the rule for multiplying a whole and mixed number together? Does it make any difference which is taken for the multiplier ? 12. What cost 12 cords of wood at $6.37 per cord? Ans. $76.50. 13. What cost 11cwt. of sugar at $93 per cwt.? Ans. $103. 14. What cost 43 bushels of rye at $1.75 per bushel? Ans. $7.65§. 15. What cost 7 tons of hay at $ 113 per ton? 16. What cost 9 doz. of adzes at $10§ per 17. What cost 5 tons of timber at $3 18. What cost 15cwt. of rice at $7.621 19. What cost 40 tons of coal at $8.37 Ans. $831. doz.? Ans. $95§. per ton? Ans. $15. per cwt. ? per ton? Ans. $335 ART. 156. To multiply a fraction by a fraction. (Art. To multiply by is to take of the multiplicand 154). Now, to obtain of, we simply multiply the numerators together for a new numerator, and the denominators together for a new denominator (Art. 138). Therefore, Multiplying one fraction by another is the same as reducing compound fractions to simple ones. RULE I. Multiply the numerators together for a new numerator, and the denominators together for a new denominator; then reduce the fraction to its lowest terms. Or, RULE II. Cancel all the common factors in the numerators and denominators, and then multiply the remaining factors together as before. QUESTIONS. Art. 156. What is the first rule for multiplying one fraction by another? How does it appear that this operation multiplies the fraction of the multiplicand? What is the inference drawn from it? What is the second rule? 9. What cost bushel ? Ans. Ans. T Ans. of a bushel of corn at of a dollar per 10. If a man travels of a mile in an hour, how far would he travel in of an hour? 11. If a bushel of corn will buy how much salt might be bought for Ans. of a mile. of a bushel of salt, of a bushel of corn? Ans. 2 of a bushel. 12. If of of a dollar buy one bushel of corn, what will of of a bushel cost? IT Ans. of a dollar. 13. If of of of an acre of land cost one dollar, how much may be bought with of $18? Ans. 19 acres. ART. 157. To multiply a mixed number by a mixed number, it is only necessary to reduce them to improper fractions, and then proceed as in the foregoing rule. 5. Multiply 12 by 115. Ans. 60%. Ans. 45. Ans. 99 Ans. 147. 6. What cost 7 cords of wood at $5 per cord? Ans. $413 7. What cost 78yd. of cloth at $3 per yard? 8. What cost 6 gallons of molasses at 23 lon ? 9. If a man travel 3 miles in one hour, how far will he travel in 97 hours? Ans. 347. QUESTIONS. Art. 157. How do you multiply a mixed number by a mixed number? 10. What cost 36111 acres of land at $253 per acre? Ans. $9167113. 11. How many square rods of land in a garden, which is 97 rods long, and 494 rods wide? Ans. 4810 rods. DIVISION OF VULGAR FRACTIONS. ART. 158. DIVISION of Vulgar Fractions is the process of dividing fractions by fractions, or whole numbers and fractions by each other. ART. 159. To divide a fraction by a whole number. Ex. 1. Divide FIRST OPERATION. 8÷4 2 9 9 = In this operation we divide the numerator of the fraction by 4, and write the quotient 2 over the de nominator. It is evident this process divides the fraction by 4, since the number and size of the parts into which the whole number is divided remain the same, while only of the number of parts is expressed by the fraction. Therefore, Dividing the numerator of a fraction by any number divides the fraction by that number. Ex. 2. Divide SECOND OPERATION. 5 = 5 63 We multiply the denominator of the fraction by the divisor, 9, and write the product under the nu merator. 10 7X9 It is evident this process divides the fraction, since multiplying the denominator by 9 makes the number of parts into which the whole number is divided 9 times as many as before, and consequently each part can have but of its former value. Now, if each part has but of its former value, while only the same number of parts is expressed by the fraction, it is plain the fraction has been divided by 9. Therefore, Multiplying the denominator of a fraction by any number divides the fraction by that number. 10 RULE I. Divide the numerator of the fraction by the whole number, when it can be done without a remainder, and write the quotient over the denominator. Or, RULE II.. Multiply the denominator of the fraction by the whole number, and write the product under the numerator. QUESTIONS. Art. 158. What is division of vulgar fractions? Art. 159. How is the fraction divided by the first operation? How does it appear that this process divides the fraction? What inference may be drawn from this operation? How is a fraction divided by the second operation? Will you explain how this process divides the fraction? What inference is drawn from this operation? What is the first rule for dividing a fraction by a whole number? The second? |