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 3 of 1. In this operation, cancel the factors 2 and 3 on both sides—above and below the OPERATION. 1X2X5X3= 5 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 unexecuted division (Art. 125), in which the divisor or dividend, or both, are fractions. Thus, 11 21 indicates that 14 is to be divided by 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 frac tion 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 $34? Ans. 64 yd. 2. At of a dollar per pound, how many pounds of tea can be purchased for $23? Ans. 31⁄2 lb. 3. Find the quotient of divided by 2; by ; by ; by by by 1000. Ans. to last, 500. 4. At 33 dollars per yard for cloth, how many yards can be purchased with $42? Ans. 111 yd. 5. By what must be multiplied, that the product may be 10? Ans. 263. REVIEW.-158a. What simple fraction? a complex fraction? How reduced to a ART. 159. EXAMPLES IN U. S. MONEY. 1. Add $16.061; $9.121; $5.43§; $2.811 2. I paid for books $9.12; paper $0.37 quills $1.62: what did I pay? 3. Having $50.25, I paid a bill much had I left? 4. From $32.311 take $15.121 5. From $5.81 take $1.183 Ans. $33.433 $4.433; a slate Ans. $15.561 of $27.183: how Ans. $23.061 Ans. $17.183 Ans. $4.621 ་ Find the cost of 6. 9 yd. of muslin at 12 cts. a yd. Ans. $1.121 7. Ans. $1.311 9. 21 lb. of sugar at 61 cts. a lb. 8. 15 yd. of cloth at $3.183 per yd. 5 yd. of linen at $0.621 per yd. 12 yd. of ribbon at 183 cts. per yd. 13 yd. of calico at 163 cts. per yd. 12. 101 yd. of cloth at $3.37 a yd. 173 dozen books at $3.75 per doz. 10. 11. Ans. $47.814 Ans. $2.34§ Ans. $2.25 Ans. $34.598 13. Ans. $66.25 14. At 18cts. 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 $11.564, what cost one yard? Ans. $2.311 17. Seven men share $31.064 equally: what is the share of each man? Ans. $4.433 EXAMPLES IN LONG MEASURE. 18. Reduce 5 mi. to inches. 19. 2 mi. 2 rd. 2 ft. to feet. 20. 20 yd. to rods. OPERATION. Ans. 316800 in. Ans. 10595 ft. 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 31⁄2, 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 Ans. 3960. in walking 2 miles? 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. 7506 sq. yd. to A. Ans. 1 A. 2 R. 8 P. 4 sq. yd. Ans. 4078 in. 28. How many acres in a field 401 rd. long, and 32 rd. wide? 26. Ans. 8 A. 16 P. EXAMPLES IN TIME MEASURE. In these examples, the year is supposed to be 3651 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. REDUCTION OF FRACTIONAL COMPOUND NUMBERS. CASE I. ART. 160. To reduce a fraction of a higher denomination, to a fraction of a lower. 1. Reduce of a peck to the fraction of a pint. 4 SOLUTION. To reduce pecks to pints, we multiply by 8 to reduce them to pk. quarts, then by 2 to reduce them to pints. In like manner, multiply the fraction 3 or, of a peck by 8, to reduce it to the frac-X8X2= pt. Ans tion of a quart, then by 2, to reduce it to the fraction of a pint. Hence, fractional numbers may be reduced from a higher to a lower denomination, (Art. 81), by multiplying by that number of the next lower order which makes a unit of the higher. *2. Reduce of a bu. to the fraction of a qt. Ans. §. Rule for Case I.-Multiply as in Reduction of Whole Numbers, Art. 81, according to the rules for the multiplication of fractions. REM. The work in Cases I and II may often be shortened by Cancellation. 3. Reduce lb. Av. to the fraction of an oz. Ans. . it of a lb. Troy, to the fraction of an oz. Ans. 3. 4. 5. of a yd. to the fraction of a na. 6. To of an A. to the fraction of a P. 7. 3 of a dollar to the fraction of a ct. 8. 1384 of a da. to the fraction of a min. 9. 330 of a bu. to the fraction of a pt. REVIEW.-160. How reduce pecks to pints? How the fraction of a peck to the fraction of a pint? How are fractional numbers reduced from a higher to a lower denomination? |