6. Reduce,,,, and, to fractions having 30 for their common denominator. 30. A denominate fraction is a fraction of a number of a particular denomination. Thus, of a foot, of a & yard, of a dollar, and of a shilling are denominate fractions. 5 Reduction of denominate fractions is the changing of them from one denomination to another, without alteringtheir values. 1 360 31. Suppose we wish to reduce of a pound sterling to an equivalent fraction of a farthing, we proceed as follows: Since there are 20 shillings in 1 pound, it follows that of a pound is the same as 20 times 1 36 0 1 360 of a shilling, and this is also the same as 12 times 20 times of a penny; which, in turn, is 4 times 12 1 360 Hence, to reduce fractions of one denominate value to equivalent fractions of other denominate values, we have this RULE. I. When the given fraction is to be reduced to a higher denomination, multiply it by a compound fraction whose terms are the reciprocals of the successive denominate values, included between the denomination of the given fraction, and the one to which it is to be reduced. II. When the given fraction is to be reduced to a lower denomination, then multiply it by a compound fraction whose terms have units for their denominators, and for numerators the successive denominate values included between the denomination of the given fraction and the one to which it is to be reduced. EXAMPLES. 1. Reduce of an inch to the fraction of a mile. In this example, the different denominate values between an inch and a mile are 12 inches, 16-33 feet, 40 rods, and 8 furlongs; .. our compound fraction is of of 4% of ; which, multiplied by the given fraction, produces ofofofof; canceling the 3 and 2 of the numerators, against a part of the 12 of the denominators, we get, 33 0 Therefore, of an inch is equivalent to 1960 of a mile. 3 2. Reduce of a solar day to an equivalent fraction of a second. In this example, the successive denominate values between a solar day and a second, are 24 hours, 60 minutes, and 60 seconds; therefore, our compound fraction. is 24 of 50 of ; which, multiplied by the given frac13 of 24 of 60 of 0; this becomes, 60 tion, becomes 11520 1 after canceling like factors, 45 of a second. 2 3. Reduce of a yard to the fraction of a mile. 360 Ans. 400. 4. Reduce of a gill to the fraction of a gallon. Ans. 896. 5. Reduce of a pound to the fraction of a ton. 195 1 Ans. 3560. 6. Reduce of a mile to feet. Ans. 1760 feet. of of a yard to the fraction of a Ans. 36360. 7. Reduce of mile. 28 8. Reduce of of 24 of a gallon to the fraction of a gill. 1 2 Ans. 4. 9. Reduce of of a hogshead of wine to the fraction of a gill. Ans. 1722-5974 gills. 10. Reduce of of 4 yards to the fraction of an inch. Ans. 243-34 inches. 11. Reduce of of a farthing to the fraction of a shilling. Ans. 1816. 59 3 43 12. Reduce of an ounce to the fraction of a pound avoirdupois. Ans. 14. 9 32. To find what fractional part one quantity is of another of the same kind, but of different denominations. Suppose we wish to know what part of 1 yard 2 feet 3 inches is; we reduce 1 yard to inches, which gives 1 yard 36 inches; we also reduce 2 feet 3 inches to inches, which gives 2 feet 3 inches=27 inches. Now it is obvious that 2 feet 3 inches is the same part of one yard that 27 is of 36, which is 74. Hence, we deduce this RULE. Reduce the given quantities to the lowest denomination mentioned in either; then divide the number, which is to become the fractional part, by the other number. EXAMPLES. 1. What part of £3 4 s. 1 d. is 2 s. 6 d.? 30 d.; there In this example, the quantities, when reduced, become £3 4 s. 1 d.=769 d.; and 2 s. 6 d. fore, is the fractional part which 2 s. 4 s. 1 d. 6 d. is of £3 2. What part of 3 miles 40 rods is 27 feet 9 inches? 3. What part of a day is 17 minutes 4 seconds? 4. What part of $700 is $5.30? 6. What part of $3 is 2 cents? 7. What part of 10 shillings 8 pence 1 penny? 8. What part of 100 acres is 63 acres, 2 roods, and 7 rods of land? Ans. 18107. 33. To reduce a fraction of any given denomination to whole denominate numbers. Suppose we wish to know the value of of a yard; we know that of a yard equals of of a quarter= of a quarter=1 quarter + of a quarter. Again, of a quarter equals 3 of 4 of a nail-2 nails. Therefore, of a yard equals 1 quarter 2 nails. Hence, we deduce this RULE. Multiply the numerator by the units in the next inferior denominate value, and divide the product by the denominator; multiply the remainder, if any, by the next lower denominate value, and again divide the product by the denominator; continue this process until there is no remainder, or until we reach the lowest denominate value. The successive quotients will form the successive denominate values. EXAMPLES. 3 1. What is the value of of an hour? 15 In this example, of an hour =12 minutes. 5 of of a minute 2. What is the value of of 1 yard? 1 Ans. 1 quarter, 2 nails. 3. What is the value of of of one mile? Ans. 1 furlong, 20 rods. 丘 4. What is the value of of 5 of 1 cwt.? 5. What is the value of Ans. 1 quarter, 12 pounds. of 14 miles, 6 furlongs? Ans. 2 miles, 3 furlongs, 26 rods, 11 feet. |