contained in 3 fourths, reduce them to the same denomination, twelfths: are 8 twelfths, and are 9 twelfths; and 8 twelfths are contained in 9 twelfths, 9÷8=1}. No use is made of the common denominator, the numerator of the divisor being multiplied by the denominator of the dividend, and the numerator of the dividend by the denominator of the divisor. This is easily performed by inverting the terms of the divisor, then proceeding as in Multiplication of Fractions, Art. 153. 3 4 ANOTHER SOLUTION.-If be divided by 2, the quotient, (Art. 133), is 4×2; but, since 2 is 3 times, the divisor used is 3 times the given divisor; hence, multiply this quotient by 3 to obtain the true quotient: this gives 3×3=2=1} Ans. *3. At of a dollar each, how many knives can you of a dollar? buy for Ans. 2. *4. At of a dollar per yard, how many yards of ribbon can be purchased for of a dollar? Ans. 33. Rule for Case III.-Invert the divisor; multiply the numerators together for a new numerator, and the denominators for a new denominator. NOTE.-Reduce compound to simple fractions, and mixed numbers to improper fractions, before commencing the operation. REVIEW.-155. How do you divide a whole number by a fraction? 156. What is the object in dividing a fraction by a fraction ? ART. 157. The rules in the three preceding Articles may be embraced in this General Rule for Division of Fractions.-Express the divisor and dividend in the form of a fraction; invert the divisor; cancel all the factors common to both terms; then multiply together the numbers remaining in the numerator for a new numerator, those in the denominator for a new denominator. 1. Divide of by of. OPERATION. 1X 2X5×3= 5 In this operation, cancel the factors 2 and 3 on both sides-above and below the 2×3×3×4=12 line—then multiply together the factors remaining on each side. REDUCTION OF COMPLEX TO SIMPLE FRACTIONS. ART. 158. A complex fraction is the expression of an anexecuted division (Art. 125), in which the divisor or dividend, or both, are fractions. 11 Thus, indicates that 1 is to be divided by 21. 21 Hence, to reduce a complex to a simple fraction, Regard the numerator as a dividend, the denominator as a divisor, and proceed as in Division of Fractions, Art. 157. In this operation, after reducing the mixed numbers to improper fractions, the numerator of the result is the product of the REVIEW.-156. When two fractions have a common denominator, how obtain their quotient? How find how often 2 inches are contained in 3 feet? How often two-thirds are contained in three-fourths? How divide a fraction by a fraction? 157. What is the General Rule? extreme terms, 5 and 3; the denominator, the product of the mean terms, 4 and 7. Hence, TO REDUCE A COMPLEX TO A SIMPLE FRACTION, Reduce mixed numbers to improper fractions, then multiply together the extreme terms for a numerator, and the mean terms for a denominator. Reduce these complex to simple fractions : Complex fractions may be multiplied together, or, one divided by another, by first reducing each to a simple fraction. Indicate the operation, and cancel. ART. 158. MISCELLANEOUS EXAMPLES. 1. At a dollar per yd., how many yards of silk can be bought for $31? Ans. 64 yd. 2. At of a dollar per pound, how many pounds of tea can be purchased for $2,3? Ans. 35 lb. 3. Find the quotient of divided by 2; by ; by 1; Ans. to last, 500. by ; by ; by 1000. 4. At 33 dollars per yard for cloth, how many yards can be purchased with $421? Ans. 11 yd. 5. By what must be multiplied, that the product may be 10? Ans. 263. REVIEW.-158. What is a complex fraction? How reduced to a simple fraction? ART. 159. EXAMPLES IN U. S. MONEY. 1. Add $16.061; $9.121; $5.433; $2.811 Ans. $33.433 2. I paid for books $9.121; paper $4.433; a slate $0.37; quills $1.62: what did I pay? 3. Having $50.25, I paid a bill of $27.183: how much had I left? Ans. $15.561 Ans. $23.061 10. 11. 12. 15 yd. of cloth at $3.18 per yd. 13yd. of calico at 163 cts. per yd. Ans. $3.433 Ans. $2.343 Ans. $2.25 10 yd. of cloth at $3.37 a yd. Ans. $34.59g 13. 17 dozen books at $3.75 per doz. Ans. $66.25 14. At 183 cts. per yd., how many yards of muslin can be purchased for $2.25 ? Ans. 12 yd. 15. At 37 cents per bu., how many bushels of barley can you buy for $5.811? Ans. 15 bu. 16. If five yards of cloth cost $11.561, what cost one yard? Ans. $2.311 17. Seven men share $31.061 equally: what is the share of each man? Ans. $4.433 5 yards 11 halves, 20 yards=40 halves. 11)40 = 3 rd. 7 half yd. left, = 31 yd. Ans. 3 rd. 31 yd. SUGGESTION. In reducing numbers from a lower to a higher denomination, when the divisor is a fractional number (Art. 155), reduce both divisor and dividend to like parts of a unit. The remainder being of the same denomination as the dividend, (Art. 38), will be a fraction, which reduce to a whole or mixed number. Here, the remainder, 7 half yd., reduced, makes 31 yd. 21. Reduce 15875 ft. to miles. Ans. 3 mi. 2 rd. 2 ft. 22. 142634 in. to miles. Ans. 2 mi. 2 fur. 2 yd. 2 in. 23. How many steps, of 2 ft. 8 in. each, will a man take in walking 2 miles? Ans. 3960. 24. How many revolutions will a wheel, of 9 ft. 2 in. circumference, make, in running 65 mi.? Ans. 37440. EXAMPLES IN SQUARE MEASURE. 25. Reduce 1 A. 3 R. 16 P. 25 sq. yd. to square yards. Ans. 8979 sq. yd. 2 R. 8 P. 4 sq. yd. Ans. 4078 in. 28. How many acres in a field 40 rd. long, and 32 rd. wide? Ans. 8 A. 16 P. EXAMPLES IN TIME MEASURE. In these examples, the year is supposed to be 365 days. 29. Reduce 4 years to hours. . Ans. 35064 hr. 30. 914092 hr. to cen. Ans. 1 cen. 4 yr. 101 da. 4 hr. 31. In what time will a body move from the earth to the moon, at the rate of 31 miles per day, the distance being 238545 miles? Ans. 21 yr. 243 da. |