8th. Reduce 4 lb. 8 oz. 10 dwt. 15 grs. troy weight to grains : =48 oz. 1 lb. 12 oz... 4 lbs. =(4 × 12) oz.= and 4 lbs. 8 oz.= 48 oz.+8 oz.=56 oz. 1 oz.=20 dwt... 56 oz.=(56 × 20) dwt.=1120 dwt. 1 dwt. 24 grs... 1130 dwt.=(1130×24) grs.=27120 grs. The reduction may be more concisely made by multiplying the lbs. by 12, and simultaneously combining the oz. with the product; multiplying this result by 20, and combining the dwts. with the product, thus, lbs. oz. dwts. grs. 4 8 10 15 12 56 oz. 4 lbs. 8 oz. 20 1130 dwts. =4 lbs. 8 oz. 10 dwts. 24 4525 2261 27135 grs.=4 lbs. 8 oz. 10 dwts. 15 grs. 301. Whence, to reduce a compound number to a lower denomination, Rule. Multiply the number which expresses the highest denomination of the given compound quantity by a multiplier which expresses the number of parts into which one unit of the highest denomination is divided to form one unit of the second; to this product add the units of the second denomination. Multiply the sum by the number expressing the parts into which one unit of the second denomination is divided to form one unit of the third; to the product add the units of the third denomination, and proceed in this manner to multiply and add until the given compound number is reduced to units of the denomination required. The reduction of a multiple or part of any unit of a weight or measure may, in like manner, be carried by repeated reduction through the units intermediate between the given and the required denominations; or it may be made at once, by multiplying the given number by as many of the required denomination as make one of the given denomination. 302. Exercises in the reduction of compound numbers from higher denominations to lower: 1st. Reduce 23 £. to pence?. 2d. 3d. 4th. 5th. 7th. 8th. 9th, 351 £. 13 sh. 02 d. to farthings? Ans. 5520 pence. Ans. 337587 farthings. 17lbs. apoth. wt. to scruples?... Ans. 4896 scruples. 72 leagues to yards? Ans. 380160 yards. 2 m. 1 fur. 8 po. 3yd. 2 in. to inches?...Ans.136334 in. 27 lb. 7 oz. 2 dr. 1 sc. 2 gr. apoth. weight to grains? Ans. 159022 grains. Ans. 3024 lbs. 27 cwt. to lbs. ?.......... 4 tons 3 cwt. 2qrs. 1lb. to drams?..Ans. 2394368 dr. 5738 lbs. troy to grains?..... Ans. 33050880 grains. 10th. Reduce 69lbs. 11 oz. 12 grs. troy to grains? Ans. 402732 grains. 697 acres to square yards?...Ans. 3375900 sq. yds. 3 acr. 2ro. 12 po. 10yds. to square yards? Ans. 17313 square yards. 519 barrels beer meas. to pints?...Ans. 149472 pts. 17 hhd. 3 gal. 2 qts. 1 pt. wine measure to pints? Ans. 8853 pints. 275 yards to nails?.... ........ ..Ans. 4400 nails. £. to farthings ?.................Ans. 384 farthings. 7 cwt. to oz. ?..... Ans. 784 oz. T6 5 12 13lbs. troy to grains ?.......... Ans. 4992 grains. acre to square feet?.......Ans. 18150 square feet. I square poles to square links? Ans. 4625 sq. links. 34 furlongs to inches?...............Ans. 25740 inches. 9 quarter to cubic inches?...Ans. 11830·024 cub. in. 25 chains long meas. to inches?... Ans. 198 inches. 2-3 yards, cloth measure, to nails?...Ans. 36.8 nails. 1-28 sq. chains to square feet?...Ans. 5575'68 sq. ft. 2.6 cubic feet to cubic inches?... Ans. 4492.8 cubic in. 7 gallons to gills?......... ..... Ans. 22.4 gills. 328 weeks to seconds ?.......Ans. 198374·4 seconds. 625 shillings to farthings?.........Ans. 30 farthings. 00945 hogsheads to pints ?... ....Ans. 4·7628 pints. 0125 lbs. troy to grains ?.. ...........Ans. 72 grains. •199325 miles to links?...... ..Ans. 1594.6 links. 513-6 poles in length to yards?...Ans. 2824-8 yards. 625 cwt. to lbs. ?.... .Ans. 70 lbs. 009943 miles to inches?.....Ans. 629.98848 inches. 3375 acres to square yards? Ans. 1633 5 square yds. fe of 5 cwt. 3 qrs. 10 oz. to drams?... Ans. 72198 dr. of 3 hhd. 12 gal. 1 pt. to pints?...Ans. 8938 pints. of 2 £. 12 sh. 4 d. to farthings? Ans. 558% farthings. 14 of 4 acr. 2 ro. 19 po. to square yards? 6 Ans. 1577914 square yards. 303. A fractional expression of any unit of a weight or measure being given, it may be required to reduce the given fraction to a compound number expressed in terms of the lower denominations of that weight or measure. The compound number obtained as a result is called the value of the fraction. What is the value of 3 £. ? Since this value is to be expressed in terms of the lower denominations, the reduction of £. to shillings must form the first step of the calculation: Now (2) sh.= For the same reason the reduction of 4 sh. to pence must be the second step: 12 48 6 sh.=(x2)d.=4d And that of d. to farthings, the third : Farthings being the lowest denomination of money, the reduction cannot be carried lower, Whence the value of £. is 8 sh. 6 d. 33 qrs. The process may be represented more compactly thus, = 384 cwt. (384 × 4) qrs.=1.536 qrs. Reserving 15 lb., and reducing '008 lb. to oz. The value contains 0 oz.; reducing 128 oz. to drams, = With drams the reduction terminates, and it is found that 384 cwt. 1 qr. 15 lb. 0 oz. 2·048 drams. Without detail the calculation may appear thus, .384 cwt. 4 1.536 qrs. 28 4288 1072 15.008 lb. 16 128 oz. 2.048 And the result, as before, is 1 qr. 15 lb. 0 oz. 2.048 dr. 304. Whence, to find the value of a fractional part of any unit of a given weight or measure as a compound number expressed in terms of the inferior denominations of that weight or measure, Rule. Reduce the given fraction to an equivalent fraction expressed in terms of the next lower denomination; bring this fraction to a mixed number; reserve the integral part of the mixed number for the highest denomination of the value; reduce the fractional part to an equivalent fraction expressed in terms of the next lower denomination; bring this fraction to a mixed number, reserve the integral part, and reduce the fractional part, as before; and thus proceed till there is no remainder, or till the lowest denomination of the given weight or measure is attained. The compound number formed of the integral parts reserved from the successive reductions, and of the result of the last reduction, is the value sought. 305. Exercises on the preceding rule: 1st. Find the value of £. ?........... 2d. .........Ans. 15 sh. of a crown (the crown=5 sh.)? Ans. 3 sh. 10 d. 23 qrs. 1 lb. troy?.............. Ans. 6 oz. 15 dwt. 4th. Find the value of 1 cwt. ?....Ans. 3 qr. 17 lb. 13 oz. 1 dr. mile?........Ans. 7 fur. 1 po. 1 yd. 6 in. 72 barrel?............... Ans. 19 gal. 3 qts. 5th. 6th. 7th. 8th. 9th. 10th. 11th. 12th. 13th. 14th. 15th. 16th. 17th. 18th. 19th. 20th. 21st. 22d. 23d. 24th. acre?..Ans. 3r. 22 po. 16 yds. 4ft. 72 in. bushel?.......Ans. 1 qt. 019 pt. 1 fathom?............Ans. 1 yd. 1 ft. 11 in. 17hhd.?...........Ans. 27 gal. 1 qt. 13 pt. 19 yd. ?..... ......Ans. 57 in. 15 year?...... Ans. 195 d. 12 h. 51'. 25". 625 fathom?.................Ans. 1 yd. 9 in. 08 mile?.........Ans. 25 po. 3 yds. 10.8 in. •142857 hhd. ?......... Ans. 9 gal. 8665 £.?.........Ans. 17 sh. 3 d. 3·84 qrs. 425 half-crown?............Ans. 1 sh. Og d. 249825 pole, in length? Ans. 1 yd. 1 ft. 1·4653 in. 024 day?............Ans. 34 min. 33.6 sec. -4694 lb. troy?. Ans. 5 oz. 12 dwt. 15.744 gr. 076 cub. yds.?......... Ans. 2 ft. 89.856 in. 2.354 acres? Ans. 2 ac. 1 ro. 16 p. 19 yds. 3.24 ft. 4.4226 £.?...Ans. 4 £. 8 sh. 5 d. 1.696 qr. 065 acre?............. Ans. 10 po. 12.1 yds. 306. A number which expresses units of a given denomination of any weight or measure is reduced to a number expressing the equivalent units of any other denomination of that weight or measure, if the given number is divided by a divisor expressing the number of units of the given denomination, which make one unit of the required denomination (Art. 298). To apply this principle, let it be required to reduce 135 sh. to pounds sterling: 135 20 Since 20sh.=1£., it follows that £. is the unreduced expression of the required result. When the numerator in such a case is a multiple of the denominator, the same result, and in the same form (namely, a whole number), is obtained, in whatever manner the reduction of the fraction is made. But 135 is not a multiple of 20. Now, it may be required to reduce 135 sh. to a fractional expression, vulgar or decimal, of a pound, or to a compound number consisting of pounds and shillings. a. If to a vulgar fraction, the calculation is 27 3 135 sh.=(135+20).£.=(*** × 1) £. = 247 €. = 62 €. b. If to a decimal fraction, it is 20)135 sh. 6.75 £. c. If to a compound number consisting of pounds and shillings; dividing 135 by 20, the quotient is pounds, and the remainder (which is composed of units of the same value as the dividend) is shillings; now, the quotient arising from the division of 135 by 20 is 6, and the remainder 15; therefore 6 £. 15 sh. is the result required. 63 L. 307. In this example the reduction is from one denomination to that which is next higher. If several denominations are interposed between that of the given number and the higher denomination to which it must be reduced; then, if it is required to express the result in terms of the highest denomination only, the reduction may be made, either by successive steps, each like the last example, through all the intermediate denominations; or by dividing at once by the number of units of the given denomination which make one unit of the required denomination. But if the result is to be expressed as a compound number, the reduction must be made step by step. In this, as in the similar cases of Articles 285, 286, 287, the remainders, taken in order, and the last quotient form the ascending orders of units of the compound number. 2d Example. Let it be required to reduce 4665 qrs. to the vulgar and to the decimal fraction of a pound sterling; also to a compound number expressed in pounds, shillings, &c. The denominations intermediate between qrs. and pounds are pence and shillings. The reduction of 4665 qrs. to pounds may, therefore, be made thus: The reduction of 4665 qrs. to the decimal of a pound is made as follows: 4)4665 qrs. |