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214. To reduce a compound denominate number to a fraction of a higher denomination.
1. What part of a week is 5 da. 14 h. 24 min. ?
SOLUTION. To find what
5 da. 14 h. 24 min. = 8064 min. part one compound number
1 wk. = 10080 min.
is of another, both must be of the same denomination. In 5 da. 14 h. 24 min. there are 8064 minutes, and in one
week there are 10080 minutes. Since 1 minute is rʊʊ of a week, 8064 minutes is 8064 10080 = of a week.
RULE. - Reduce the given number to its lowest denomination for the numerator, and a unit of the required denomination to the same denomination for the denominator of the required fraction.
If the given number contains a fraction, the denominator of this fraction must be regarded as the lowest denomination.
2. What part of a mi. is 266 rd. 3 yd. 2 ft.?
3. What fraction of a £ is 13s. 7d. 3 far. ? 4. Reduce 10 oz. 10 pwt. 10 gr. to the fraction of a pound troy. Ans. 5 lb.
5. Reduce 2 cd. ft. 8 cu. ft. to the fraction of a cord.
6. Reduce 1 bbl. 1 gal. 1 qt. 1 pt. 1 gi. to the fraction of a hogshead. Ans. 11 hhd. 7. What part of 2 rods is 4 yards 14 feet? Ans. 22. 8. Reduce 13 pecks to the fraction of a bushel.
9. What part of 9 feet square is 9 square feet? 10. From a piece of cloth containing 8 yd. a tailor cut 24 yd.; what part of the whole piece did he take? Ans. 4.
11. What fraction of a week is 5 da. 4 h. 30 min. ?
215. To reduce a denominate decimal to integers of lower denominations.
1. Reduce .78125 of a pound troy to integers of lower denominations.
9 oz. 7 pwt. 12 gr., Ans.
We multiply by 12 to reduce the given number from pounds to ounces, and the result is 9 oz. and the decimal .375 of an oz. Multiplying this decimal by 20 to reduce it to pennyweights, we get 7 pwt. and .5 of a pwt. This last decimal we multiply by 24, to reduce it to grains, and the result is 12 gr. The answer is 9 oz. 7 pwt. 12 gr.
RULE.-I. Multiply the given decimal by that number in the scale which will reduce it to the next lower denomination, and point off as in multiplication of decimals.
II. Proceed with the decimal part of the product in the same manner until reduced to the required denominations. The integers at the left will be the answer required.
2. Change .217° to integers of lower denominations. Ans. 13 1.2".
3. Change .659 of a week to integers of lower denominations. Ans. 4 da. 14 h. 42 min. 43.2 sec. 4. Reduce .578125 of a bushel to integers of lower denominations. Ans. 2 pk. 2 qt. 1 pt. 5. Reduce .125 bbl. to integers of lower denominations. Ans. 3 gal. 3 qt. 1 pt. 2 gi. 6. Reduce £.628125 to integers of lower denominations.
7. Change .22 of a hogshead of molasses to integers of lower denominations. Ans. 13 gal. 3 qt. 3.52 gi.
8. Change .67 of a league to integers of lower denominations. Ans. 2 mi. 101 rd. 6 ft. 6.24+ in.
9. Change .42857 of a month to integers of lower denominations. Ans. 12 da. 20 h. 34 min. 13 sec. 10. Reduce 5.88125 acres to integers of lower denominations. Ans. 5 A. 141 P. Ans. 11 lb.
11. Reduce .0055 T. to pounds.
12. Reduce .034375 of a bundle of paper to its value in lower denominations. Ans. 1 quire 9 sheets. 13. Reduce .35 lb. apoth. to integers of lower denominations.
14. Reduce 4.75 lb. troy to integers of lower denominations.
15. Reduce 2.55 lb. avoir. to integers of lower denominations.
16. Reduce .625 of a fathom to integers of lower denominations.
216. To reduce a compound denominate number to a decimal of a higher denomination.
1. Reduce 3 pk. 2 qt. to the decimal of a bushel.
8 2.00 qt.
4 3.2500 pk.
.8125 bu., Ans. Or, 3 pk. 2 qt. = 26 qt. 1 bu. = 32 qt. 3.8125 bu., Ans.
SOLUTION. Since 8 quarts make
1 peck, and 4 pecks 1 bushel, there will be as many pecks as quarts and as many bushels as pecks.
Or we may reduce 3 pk. 2 qt. to the fraction of a bushel (as in 214), and we have of a bushel, which, reduced to a decimal, equals .8125.
RULE. -Divide the lowest denomination given by that number in the scale which will reduce it to the next higher, and annex the quotient as a decimal to that higher. Proceed in the same manner until the whole is reduced to the denomination required. Or,
Reduce the given number to a fraction of the required denomination, and reduce this fraction to a decimal.
2. Reduce 3 qt. 1 pt. 1 gi. to the decimal of a gallon. Ans. .90625 gal. 3. Reduce 10 oz. 13 pwt. 9 gr. to the decimal of a pound troy. Ans. .8890625 lb. 4. Reduce 1.2 pints to the decimal of a hogshead.
Ans. .00238+ hhd.
5. What part of a bushel is 3 pk. 1.12 qt.?
Ans. .785 bu.
6. What part of an acre is 132.56 P. ?
8. Reduce .32 of a pint to the decimal of a bushel.
Ans. .005 bu.
9. Reduce 47 feet to the decimal of a fathom.
Ans. .8125 fathom.
10. Reduce 150 sheets of paper to the decimal of a Ans. .3125 rm.
11. Reduce 47.04 lb. of flour to the decimal of a barrel.
12. Reduce .33 of a foot to the decimal of a mile.
13. Reduce 5 h. 36 min. 57 sec. to the decimal of a day.
14. What decimal of a league is 2 mi. 3 rd. 1 yd. 3 in.?
15. Change 188. 5d. 2 far. to the decimal of a £. 16. Reduce 11 oz. 10 pwt. 9 gr. to the decimal of a lb. troy.
17. Reduce 126 A. 4 sq. ch. 12 P. to the decimal of а Тр.
18. Reduce 5 meters 10 dekameters to the decimal of a hektometer.
19. Reduce 1 dekagram to the decimal of a lb. avoir. 20. What decimal of a lb. troy is 2 oz. 14 pwt. ?
Ans. .225 lb.
217. 1. A banker sold at one time £10 4s. 6d. 3 far. of English money; at another time, £2 9s. 3d. 1 far. ; at another, 11s. 2 far.; and at another, £25 10s. 11d. 3 far. How much did he sell in all?
8. d. far.
SOLUTION. Arranging the numbers in columns, placing units of the same denomination under each other, we first add the units in the right hand column, or lowest denomination, and find the amount to be 9 farthings, which is equal to 2d. 1 far. We write the 1 far. under the column of farthings, and add the 2d. to the The amount of the second column is 22d., which Writing the 10d. under the column of pence, we add the 1s. to the next column. Adding this column the amount is 35s. £1 15s. Placing the 15s. under the column of shillings, we add the £1 to the column of pounds. Adding the last column, the amount is £38.
Ans. 38 15 10
column of pence.
is equal to 1s. 10d.
RULE. -I. Write the numbers so that those of the same unit value stand in the same column.
II. Beginning at the right hand, add each denomination as in simple numbers, carrying to each succeeding denomination 1 for as many units as it takes of the denomination added, to make one of the next higher denomination.
It is customary, in business, to reduce the lower denominations to a common or decimal fraction, and to perform the operation with only one denomination.