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Ex. (1.) What is the price of one hide, if 127 cost £84 13s. 4d. ?
Ans. 13s. 4d. (2.) Divide £420 14s. 9d. between 347 persons.
Ans. £1 4s. 3d. (3.) Divide £8 188. 6d. by 126.
Ans. 1s. 5d. (4.) Bought 14 load of hay for £54 5$. ; what is the price per load?
Ans. £3 178. 6d. (5.) If 1174 gallons of gin cost £73 58.7d., what will one gallon cost ?*
Ans. 12s. 6d.
WEIGHTS AND MEASURES.
(6.) If 383 hhd. of sugar weigh 1887 cwt. 2 qrs. 16 lb., what is the weight per hhd. ? Ans. 4 cwt. 3 qrs. 20lb. (7.) Divide 9111 oz. O dwt. O grs. by 72.
Ans. 126 oz. 10 dwt. 20 gr. (8.) Divide 1896 tb. 43. 73. 1 9. 15 gr. by 7.
Ans. 270lb. 10z. 73. 29. 15 gr. (9.) Divide 461 lg. 2 ml. 7 fur. 27 poles, 2; yd. by 6.
Ans. 76 lg. 2 ml. 7 fur. 37 per. 5 yd. (10.) Divide 204 yd. 3 qr. 1 nl. į in. by 5.
Ans. 40 yds. 3 qrs. 3 nls, 1 in. (11.) Divide 759 la. 4 qr. 7 bush. 1 pk. 1 gal. 1 qt. by 19.
Ans. 39 la. 4 qr. 7 bush. 3 pk. 1 gal. 3 qt.
* When there is a fractional part in the divisor, as }, , ,, reduce it into halves or quarters, as the case may be, taking care also to multiply the dividend by a like figure; you then proceed as before directed.
To prove Division, multiply the quotient by the divisor, and the product will be the dividend, if the work is correct.
We have now gone through Arithmetic, both Simple and Compound, and from the knowledge the learner has acquired of the use of numbers to express quantities, whether of money, weights, or measures, it would be only wasting his time to enter again into a detail of the usefulness and beauty of the application of figures to express our thoughts.
One thing is essential to know, that it is often necessary to reduce quantities that are expressed in several denominations, into equal quantities in some other denomination.
For example, bought 4 cwt. 2 qrs. 26 lbs. of sugar for £12. 2s. 11d., what did it cost per pound?
To ascertain this you will find it not only necessary to reduce the cwts. qrs. and lbs. into pounds, but the pounds, shillings, and pence, into pence, and this being to reduce things from higher to lower denominations, is called Reduction Descending, and is performed thus :
The cwts. multiplied by 4, and the quarters added in, will reduce it to quarters; the quarters multiplied by 28, adding in the pounds, gives 530 lbs., or the number of pounds contained in 4 cwt. 2 qrs. 26 lbs. Again, the pounds sterling being multiplied by 20, reduces it to shillings, the shillings being multiplied by 12, reduces it to pence, or 2915, the number of pence contained in £12. 2s. lld. Then to ascertain the cost per pound, we divide 530 into 2915, which gives a quotient of 5, and a remainder of 265; this remainder we take for a nume. rator, and the divisor for a denominator, which, when reduced, (see Reduction of Fractions,) gives į, thus shewing that 5 d. is the price per pound.
Examples in Reduction Descending.
Ans. 19200 farthings. (2.) Reduce £270. 4s. 6d. to pence.
Ans. 64854 pence. (3.) Reduce £21. 158. 4 d. to farthings.
Aus. 20898 farthings. (4.) In 10 tons 2 cwt. 3 qrs. 14 lbs. avoirdupois weight, how many pounds?
Ans. 22722 lbs. (5.) In 4 cwt. 3 qrs. 27 lbs. how many oz.?
Ans. 8944 oz. (6.) How many grains in 12 lbs. troy?
Ans. 69120 gr. (7.) How many ounces, scruples, drams, and grains in 20 lbs. apothecaries' weight?
Ans. 240 oz. 1920 dr. 5760 sc. 115200 gr. (6.) In 27 yards, cloth measure, how many nails ?
Ans. 432 nails. (9.) In 57 miles how many furlongs and poles ?
Ans. 456 fur. 18240 poles.
(10.) How many minutes in 20 years of 365 days each?
Ans. 10512000 min. (11). In 27 a. 3 r. 19 p. how many perches? Ans. 4459. (12.) In a field containing 18 acres how many gardens may be made, each containing 500 yards ?
Ans. 174 gardens, and 120 yards remain.
REDUCTION ASCENDING. As Reduction Descending is performed by Multiplication, Reduction Ascending is performed by Division. For instance, in the first example we had two quantities, having several denominations in each, to reduce to their lowest denomination; we had money under the different denominations in which it is current amongst us ; we also had weights under the different denominations used to express their quantity, and in which we keep our accounts; you also perceived the necessity there was for so reducing them.
We will now reverse the question, and ask, In 530lbs. how many hundreds, quarters, and pounds; and at 5 d. per pound, how much will it cost ?
By first multiplying the 530 lbs. 530 lbs. by 5 d. we have the price of it in
5) pence; but the amount being more 2650 easily understood, and indeed more
265 convenient to express, in pounds, 12) 2915 shillings, and pence, we reduce it 20) 242
11 to these denominations, by divid- 12. 2. 11. ing by 12 and 20, the pence in a shilling and the shillings in a pound; first by 12, and we find the quotient 242, with a remainder .)f 11; now as we were dividing pence, and this 11 is undivided, we write it on the right of the 242, and call it ild.; we next divide 242 shillings by 20, and find our quotient is 12, with a remainder of 2, which
two we call shillings, and write on the right of the twelve; our amount then is £12 2s. 11d., the answer to one part of the question. Next we have to reduce 530 lbs. to cwts. qrs. and lbs.
4 cwt. 2 qrs. 26 lbs.
For this purpose we divide the 530 lbs. by 28 lbs. the number of lbs. contained in a quarter, the quotient 18 is quarters, the remainder 26 is pounds; we divide the 18 by 4, the number of quarters in a cwt., and we have a quotient of 4 with a remainder of 2, and this last quotient, with both remainders, being written in a line, thus, 4 cwt. 2 qrs. 26 lbs. is the number of cwt. qrs. and lbs. contained in 530 lbs.; and if you look to the first example you will find that the answers in this sum were the terms of our proposition in that, which proves the work to be correct.
Ex. (1.) In 19200 farthings, how many pound sterling?
Ans. £20. (2.) Reduce 64854 pence to pounds, shillings, and pence.
Ans. £270 4s. 6d. (3.) Reduce 20898 farthings to pounds, shillings,
Ans. £21 15s. 4 d, (4.) In 22722 lbs. avoirdupois how many cwt. qrs. and lbs.?
Ans. 202 cwt. 3 qrs. 14lbs. (5.) In 8944 ounces avoirdupois, how many cwts. qrs. and lbs.?
Ans. 4 cwt. 3 qrs. 27 lbs. (6.) In 69120 grains troy how many lbs.?
Ans. 12 lbs.