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REDUCTION. REDUCTION teaches to change numbers from one denomination to another, without altering their value.
When the given numbers are to be reduced to lower denominations, multiply the highest denomination given by so many of the next less, as make one of the larger, adding in those of the less denomination, (if any,) and thus, continue until it is brought as low as the question requires. To reduce low numbers to higher denominations, divide the given number by as many of that denomination as make one of the next highest, and set down what remains; so proceed till you have brought it into that denomination which your question requires.
Proof. Reduction is proved by reducing the answer back again to the original denomination.
TROY WEIGHT. By this weight, jewels, gold, silver, and liquors, are weighed; and the denominations are pounds, ounces, pennyweights, and grains. 24 grains, gr.
equal 1 pennyweight, dwt. 20 pennyweights
ounce, 12 ounces
1. In 441b. 8oz. 16dwt. 22gr. of gold, how many grains ?
lb. oz. dwt. gr.
In 257686 grains, how many pounds ?
6 1st divisor.
X3 last remainder. 10736
2,0)1073,6+3 +4 İst remainder. 42966 21472
12)536 16 22 22 true remainder.
257686 gr. Ans.
Ans. 441b. Boz. 16dwt. 22gr. Some Arithmeticians term this
Some call this Reduction ascending. Reduction descending.
Note.-When we use two such numbers, as, being multiplied together, produce the divisor, and remainders occur, multiply the first divisor by the second remainder, and to the product add the first remainder; and so on till you have gone through all the divisors and remainders, for the true remainder.
Questions. What is Reduction? How do you reduce high numbers to lower denominations? How do you change low denominations to higher? When you use factors for a divisor, and remainders occur, how do you find the true remainder ? &c.
3. How many grains in a silver bowl that weighs 2lbs. 10oz. 6dwt. ?
Ans. 16464grs. 4. Reduce 184320 grains to pounds. Ans. 32lbs. NOTE.—1lb. troy=5760grs. and llb. avoirdupois=7000grs.
175oz, troy=192 avoirdupois oz.
AVOIRDUPOIS WEIGHT. By this weight are weighed things of a coarse drossy nature; and all metals except silver and gold. The denominations are, ton, hundred weight, quarter, pound, ounce, and dram. 16 drams dr.
1 ounce, 16 ounces
1 pound, 25 pounds*
1 quarter, 4 quarters
1 hundred weight, 20 hundred weight * The old method of quartering the hundred, was by the quantity of 28lbs., referring to the hundredweight of 112lbs. generally used in Europe, and partially in the United States.
1. In 16 tons how many hundred weight, quarters, pounds, ounces, and drams?
Ans. 320H. 1280qrs. 32000lbs. 512000oz. 8192000dr. 2. In 10240000 drams how many tons ?
Ans. 20T. 3. Reduce 67200 pounds to tons. Ans. 33 T. 12 H.
4. One of the stones in the walls of Balbec is said to have weighed 765T. SH. 16lbs.--required its weight in pounds.
Ans. 530816lbs. APOTHECARIES WEIGHT. By this weight apothecaries mix their medicines; but they buy and seli by avoirdupois weight. The denominations are, pound, ounce, dram, scruple, and grain. 20 grains gr.
equal=1 scruple, 5. 3 scruples
1 dram, 3. 8 drams
1 ounce, 12 ounces
EXAMPLES. 1. In 17lb). how many ounces, drams, and scruples?
Ans. 2043, 16323, 4896».
2. In 1332005 grains, how many pounds ?
Ans. 23116, 33, 5grs. 3. In 5lbs. of drugs, how many parcels, each 16 drams?
Ans. 30 parcels. 4. In 188496 grains, how many pounds ?
Ans. 321b, 83,53,19, 16gr.
CLOTH MEASURE. By this measure, cloth, tapes, &c. are measured. The denominations are, French ell, English ell, yard, Spanish var, Scotch ell, Flemish ell, Hamburgh ell, quarter of a yard, nail, inch.
2.25 inches in. equal=1 nail,
1 ell Hamburgh, E. H. 3 quarters, or 27 in. 1 ell Flemish, E. Fl. 27.2 inches
1 ell Scotch, E. Sc. 33 inches
1 var Spanish, 4 quarters, or 36 in.
yd. 5 quarters, or 45 in.
1 ell English, E. E. 6 quarters, or 54 in.
1 ell French, E. Fr. The yard is used in measuring all kinds of piece goods, in the United States.
1. In 127 yards, how many quarters and nails?
Ans. 508qr. 2032na. 2. Change 9173 nails to yards. Ans. 573yds. lqr. Ina. 3. Reduce 7 English ells to quarters.
Ans. 35. 4. Reduce 729 inches to ells Flemish.
Ans. 27. 5. 1 ell Flemish equals how many nails ?
Ans. 12. 6. Change 3 ells French to inches.
Ans. 162. 7. Reduce 25 ells Scotch to inches.
This measure is applied to distance, or length. The denominations are, degree, mile; furlong, pole, yard, foot, and inch.
equal = 1 foot, 3 feet, or 36 in.
yd. 53 (5.5) yards, or 16.5 ft. 1 pole or perch, P. 40 poles, or 220 yds.
fur. 8 fur. or 320 P. or 1760 yds. 1 mile, 69(69.5) miles
1 degree, deg. 360 degrees
1 circle, cir. or the circumference of the earth.
Nore.4 in. equal - 1 hand.* 2 yards, or 6 feet, 1 fathom, fm. 3 miles, i league, l. 60 geographic miles, 1 degree.
li. 25 links
1 pole or perch, P. 4 poles or perches, or 100 links 1 chain, ch. The American or English mile = 5280 feet. French mile
5566 German «
26400 Dutch, Spanish, or Polish
7920 Indian mile, about three Americanı. Irish mile
EXAMPLES. 1. Reduce 273 miles to inches. Ans. 17297280in. 2. In 34594560 inches, how many miles ? Ans. 546m. 3. Required the earth's circumference in yards.
Ans. 44035200yds. 4. In 1 mile how many
Ans. 8000 li. 5. Required the number of miles a wheel, 18 feet 4 inches in circumference, would run, to make 86400 turns.
Ans. 300 miles.
TIME. The denominations of time are, year, month, week, day, hour, minute, and second.
* Hund is a term used in measuring the height of horses; fathom, the depth of water; and a league, distance at sea.
43 60 seconds sec. equal=1 minute, M. 60 minutes
1 hour, H. 24 hours
D. 7 days
1 week, W. 4 weeks
1 month, M. 13 months, 1 day, and 6 hours, or 12 M. of 30.4375 D. each, or 365,25 days
1 solar year.
is also divided into 12 calendar months, as fol. lows:
1st month January, hath 31 days.
Till leap-year gives it twenty-nine. Note.--To know when it is leap-year, divide the years over even centuries by 4: if no remainder, it is leap-year; but if any remains, it is so many years after leap-year.
1. Change 28 days to hours.
Ans. 672 hours. 2. How many hours, minutes, and seconds, are there in a Julian year? Ans. 8766h. 525960m. 31557600sec.
3. Was the year 1784 leap-year?
4. How many hours has a boy 11 years old lived, allowing 365 days 6 hours to each year?
Ans. ! 5. In 14 weeks, how many days?