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90. 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 ob vious that 2 feet 3 inches is the same part of a yard that 27 is of 36, which is =*.
Hence, we deduce this
Reduce the given quantities to the lowest aenomination mentioned in either. Then take the number which expresses the quantity of which the other is to be the fractional part, for a denominator, and the other number for a numerator, and the fraction thus formed will denote the fractional part, sought.
EXAMPLES. 1. What part of £3 4s. 1d. is 2s. 6d. ?
In this example, the quantities, when reduced, become £3 4s. 1d.=769d.; and 2s. 6d.=30d.; therefore, sig the fractional part which 2s. 6d. is of £3 4s. 1d. 2. What part of 3 miles, 40 rods, is 27 feet 9 inches?
Ans. 7720 3 What part of a day is 17 minutes 4 seconds ?
Ans. 675 4. What part of $700 is $5:30 ?
Ans. 760 5. What fractional part of 2 hogsheads is 3 pints?
Ans. 26. 6. What part of $3 is 21 cents ? Ans Iho.
7. What part of 10 shillings, 8 pence, is 3 shillings 1 penny ?
8. What part of 100 acres is 63 acres, 2 roods, 7 rods of land ?
Ans. 18167. 9. In the Eagle there are 232} grains of pure gold, and 12 grains of silver, and the same quantity of copper. The silver and copper is each what part, by weight, of the gold? And the silver and copper together is what part of the gold?
Silver and copper are each Ansi) to of the gold. Silver
and copper together are
[ of the gold. 10. In the United States standard silver coin of one dollar, there are 3713 grains of pure silver, and 414 grains of copper. What fractional part is the copper of the silver ?
Ans. š. 11. The silver in standard gold coin is what part of the silver in the same value of standard silver coin ?
Ans. TATE. 12. The pound Troy contains 5760 grains, the pound Avoirdupois contains 7000 grains. A pound Troy is what part of a pound Avoirdupois ?
Ans. 145. 13. The imperial gallon contains 2771 cubic inches, nearly; the old wine gallon contains 231. What part of the imperial gallon is the old wine gallon ? Ans. 9345.
14. The solar year is 365 days, 5 hours, 48 minutes, 48 seconds. By what part of a day does this exceed 365 days?
91. To reduce a fraction of any given denomination to whole denominate numbers.
Suppose we wish to know the value of f of a yard; we know that if of a yard equals of 4 of a quarter= of a quarterzl quarter +} of a quarter. The $ of a quarter may be considered as a remainder.
Again, } of a quarter equals # of 4 of a nail=2 nails Therefore, t of a yard equals 1 quarter and 2 nails. Hence, we deduce this
Multiply the numerator by the number expressing how many of the next lower denomination make one of the denomination of the fraction, and divide the product by the denominator; multiply the remainder, if any, by the number expressing how many of the next lower denomination make one of that remainder, 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 whole denomirate numbers required.
1. What is the value of of an hour ?
In this example, is of an hour equal is of fe of a minute, equals 12 minutes. 2. What is the value of # of 1 yard ?
Ans. 1 quarter, 2% nails. 3. What is the value of 1 of $ of 1 mile ?
Ans. 1 furlong, 20 rods. 4. What is the value of of of 1 cwt. ?
Ans. 1 quarter, 12 pounds 6 What is the value of 1 of 14 miles, 6 furlongs ?
Ans. 2 miles, 3 furlongs, 26 rods, 11 feet. 6. What is the value of of of 2 days of 24 houn each ? .
Ans. 9 hours, 36 minutes. 7. What is the value of of of of an hour?
Ans. 5 minutes, 374 seconds.
8. What is the value of 168 of a solar day?
Ans. 5h. 48m. 48sec. 9. What is the value of 14 of a pound Avoirdupois ?
Ans. 13oz. 214dr. 10. What is the value of to of a bushel? Ans. 31 quarts. 11. What is the value of any of a year of 365 days?
Ans. 30 days. 12. What is the value of Fof of of an acre ?.
Ans. 25 rods.
ADDITION OF DENOMINATE FRACTIONS.
92. So long as fractions are of different denominate values, they cannot be added, any more than integers can of different denominate values. Hence, before seeking to add, it is necessary to reduce them to the same denomination, then, to a common denominator, and apply the rule under Art. 43. What is the Rule for the Addition of Denominate Fractions ?
1. Add } of a shilling to of a pound.
I. } of a shilling equals } of zo of a pound=tbo of a pound, which added to of a pound=id of a pound, gives 2 =1% of a pound for the sum.
II. of a pound=+of ze of a shilling=5 shillings, which, added to } of a shilling, gives 5}=24 of a shilling for the sum.
If our work is right, these two results ought to be of the same value, that is, 13 of a pound must equal 57 shillings.
We know that få of a pound=b3 of 4 of a shilling= qe of a shilling.
2. Add } of a yard, & of a foot, and of a mile.
These fractions, before adding, might be reduced to frac tions of a yard, or of a foot, or of a mile, or of any of the denominate values of Long Measure. But a better way would be to reduce each to its integral denominate value, by Rule under Art. 91. Thus: } of a yard=} of 4 of a foot=1 foot.
of a foot of 12 of an inch=10 inches.
of a mile= of $ of a furlong=3 furlongs Therefore, the sum is 3 furlongs, 1 foot, 10 inches. 3. Add 1 of a week, 7 of a day, 1 of an hour.
of a week= of 7 of a day=31 days=3 days+of 44 of an hour=3 days, 12 hours.
& of a day=1 of 2 hour=4 hours. į of an hour=1 of Ge of a minute=15 minutes. Hence, the sum is 3 days, 16 hours, 15 minutes. 4. Add of a year, † of a week, te of a day, together i
Ans. 75da. 2hr. 5. What is the sum of £ of a cwt., $ of a gr., of a 16. ?
Ans. 2qr. 916. 9oz. 54dr. 6. What is the sum of to of a bushel, → of a peck, 7 of a quart?
Ans. 53oqt. 7. What is the sum of to of a yard, and of a foot ?
Ans. 7 inches. 8. What is the sum of g of a week, † of a day, and I of an hour ?
Ans. 4da. 21hr. 8m. 9: What is the sum of $ of a bushel, } of a peck, and of a quart?
Ans. 3pk. Oqt. 04pt.