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written under the column of farthings, and 1d. is carried to the column of pence the sum of the column of pence is 26, and 26+1 carried=27d.= 2s. 3d.; 3 is written under the column of pence, and 2s. are carried to the column of shillings: the sum of the column of shillings is 50, and 50+2 carried=52s.= £2 12s.; 12 is written under the column of shillings, and £2 carried to the pounds, which are added as whole numbers. The sum of the proposed compound numbers is thus found to be £197 12s. 3 d.

The rule for reducing numbers from a lower to a higher denomination is required in addition of compound numbers. That for reducing a fractional part of any denomination to a compound number expressed in terms of the lower denominations of the weight or measure to which the proposed fraction belongs, may sometimes be necessary; and also the rule for reducing fractions to a common denominator.

As an example requiring these reductions, let it be proposed to find the sum of £; 3s.; and 4d.

£, s., and 4d. may be reduced to the same denomination, and added together as fractional parts of pounds, or of shillings, or of pence, or of farthings. But if it is required to express the result in the form of a compound number, then

7£=17

S. d.
6

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7 >adding.

the sum = 18 17

that is £+3s.+4d.=18s. 137d.

Following with the two preceding examples, the course recommended in Art. 147., it is found that

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and £90738095=188. 137d., it is concluded that these results are correct.

With money expressed in pounds, florins, cents, and mils, the reductions are unnecessary; the number of decimal figures is fewer (only three); and the whole calculation is shortened to

that of finding the sum of, for example, 35.882, 86 788, and 74.943.

a. The rule for the addition of whole numbers may, with some slight changes, be also made the rule for the addition of compound numbers, thus

Having found the values of fractional parts of any other denomination than the lowest, in terms of the lower denominations, and reduced fractional parts of this denomination with different denominators to equivalent fractions having a common denominator; and having written the numbers under each other in horizontal lines so arranged that the same denomination in all the compound numbers falls in the same vertical column; add together the numbers in the right vertical column; reduce the sum to a compound number expressed in terms of the last and the preceding denomination; write the units of the last denomination under the last column and carry the number expressing the units of the next higher denomination to the second vertical column. Add together the number carried and the numbers in the second column. Reduce the sum to a compound number expressed in terms of this and the next higher denomination. Write the number of units of the second denomination under the second vertical column; carry the units of the third denomination to the third vertical column: and in this manner proceed till the sum of all the columns has been found.

149. Examples. Add together,

1. £82 7s. 5d.; £59 13s. 8d.; and £196 8s. 6d.

2. £54 17s. 6d., £39 8s. 4d., £27 18s. 9d., and £9 5s. 11d.

3.

4.

5.

6.

7.

8.

9.

10.

11.

15 mi. 7 fur. 25 po. 4 yd., 32 mi. 5 fur. 38 po. 3 yd., and 146 mi. 3 fur. 18 po. lyd.

42 ac. 3 ro. 17 po. 25 yd., 18 ac. I ro. 35 po. 16 yd., and 27 ac. 2 ro. 29 po. 8 yd.

5 hhd. 28 gal. 3 qt., 9 hhd. 57 gal. 1 qt., and 17 hhd. 38 gal. 2 qt. 1 pt.

8 t. 17 cwt. 1 qr. 18 lb., 4 t. 9 cwt. 3 qr. 24 lb., and 12 t. 8 cwt. 2 qr. 27 lb. 23 lb. 7 oz. 18 dwt. 21 gr., 52 lb. 9 oz. 12 dwt. 18 gr., and 35 lb. 12 dwt.

23 gr.

39 d. 14 h. 25 m. 39 s., 28 d. 23 h. 48 m. 50 s., and 19 d. 17 h. 37 m. 54s. £372 8s. 43d., £568 17s. 11d., £796 9s. 8d., and £937 13s. 93d. 15 cwt. 3 qr. 17 lb. 8 oz., 9 cwt. 27 lb. 13 oz. 15 dr., 6 cwt. 1 qr. 19 lb. 12 dr., and 2 qr. 24 lb. 14 oz.

25 mi. 3 fur. 25 po. 4 yd. 1 ft. 9 in., 37 mi. 7 fur. 2 yd., 51 mi. 34 po. 2 ft. 7 in., and 9 mi. 5 fur. 28 po. 4 yd. 21 ft.

12. 8 qr. 6 bu. 3 pk. 1 gal. 2 qt., 15 qr. 7 bu. 2 pk. 3 qt., 9 qr. 4 bu. 1 pk. 1 qt., and 6 bu. 3 pk. 2 qt.

13.

14.

53 ac. 1 ro. 35 po. 26 yd. 8 ft., 79 ac. 29 po. 18 yd. 5 ft., and 154 ac. 3 rò. 17 po. 25 yd. 7 ft.

£20 78. 41d., 53 guineas, and 85 half-crowns.

15. 25 cwt. 2 qr. 17 lb., 6 cwt. 86 lb., and 15 cwt. 3 qr. 12.7 lb.

16. 321 ac., 5067 sq. yd., and 50000 square links.

17. 57.5£, 29-34£, 87-176£, and 153-942£.

18. 137.05 yd., 496.346 yd, 874-849 yd., and 682-574 yd.

19. 314-74 lb., 592-819 lb., and 476.59 lb. (av. wt.)

20. 53.25 lb., 29-36 lb., 19-542 lb., and 8·425 lb. (troy: the sum to be given

in lb. av.)

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SUBTRACTION OF COMPOUND NUMBERS.

150. The difference of two compound numbers is found in the same manner as the difference of two whole numbers; excepting that, when the number expressing any denomination of the subtrahend is greater than the number expressing the same denomination of the minuend, the latter is not augmented by 10 but by as many of this denomination as make 1 of the next higher denomination. As an illustration let it be required from 181b. 7 oz. 18 dwt. 15gr. to subtract 9 lb. 8 oz. 14dwt. 20gr. troy weight.

Writing the compound numbers so that, in each, the same denominations may fall under each other, thus

lb. oz. dwt. gr. 18 7 18 15

9

8 14 20

8 11 3 19

and subtracting the numbers in the lower line from the corresponding numbers in the upper line,

(15-20) grs. To render the subtraction possible, 1 dwt. (=24 gr.) is borrowed. 15+24=39 gr., and 39-2019 gr.=the gr. of the remainder. The remainder, 19, is written under the column of grains, and 1 dwt. for that borrowed is carried to the dwt. of the subtrahend.

14+1 carried = 15 dwt., and 18–15- =3 dwt. the dwt. of the remainder.

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The remainder, 3, is written under the column of dwt.

(7-8) oz, To render the subtraction possible 1 lb. (12 oz.) is borrowed. 7+12=19 oz. and 19-8-11 oz. the oz. of the remainder.

=

11 is written under the column of oz., and 1 lb. for that borrowed is carried to the lb. of the subtrahend.

9+1 carried 10 lb., and 18-10-8 lb. the lb. of the remainder.

=

8 is written under the column of lb. and the difference of the proposed numbers is found to be 8 lb. 11 oz. 3 dwt. 19 gr.

If the lowest denomination of the proposed compound numbers contain fractions with different denominators, these must be reduced to equivalent fractions with a common denominator; and if the denominations which contain fractional parts are not the same in both subtrahend and minuend, the fractional part annexed to the higher denomination must be reduced to a compound number, expressed in terms of the lower denominations. If, for example, it be required from £37 8s. 94d. to take £207, either 8s. 94d. may be reduced to the fraction of £1; and the difference of the proposed numbers found by the rule for subtraction of vulgar fractions; or the value of £7 may be found by reduction and the subtraction made as in the last example.

Now £15s. 6d. ... £207=£20 15s. 63d.

F

Whence, if from 37

8. d.

8.91

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Reducing the compound numbers, of the two preceding examples, to decimal fractions,

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a. Rule for the subtraction of one compound number from another :- Having reduced fractional expressions of any denomination as in addition of compound numbers, write the several denominations of the subtrahend under the corresponding denominations of the minuend. Then if the number expressing the lowest denomination of the subtrahend is less than the number expressing the same denominations of the minuend, subtract the former from the latter and write the remainder under this denomination: but if the number in the subtrahend is greater than that in the minuend augment the latter by as many units of this denomination as make one of the next higher denomination, and subtract the number in the subtrahend from the sum.

Write the remainder under its proper denomination. Carry 1 for that borrowed to the next denomination of the subtrahend: subtract the number thus augmented from the number in the corresponding denomination of the minuend, if the subtraction is possible; if not, augment the number in the minuend by as many units of this denomination as make one of the next higher : subtract the number in the subtrahend from this sum. Proceed in this manner till all the partial subtractions are made. The compound number composed of the partial remainders is the difference of the proposed compound numbers.

151. Examples. Find the difference between

1. £179 13s. 5d, and £98 17s. 9d.

2. £432 88. 34d. and £174 13s. 7 d.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

54 mi. 2 fur. 21 po. and 19 mi. 4 fur. 34 po. 2 yd.
81 ac. 11 po. 21 yd. and 59 ac. 1 ro. 34 po. 25 yd.
183 yd. 17 ft. 154 in. and 97 yd. 25 ft. 378 in.
1 ton 3 cwt. qr. 15 lb. and 12 cwt. 3 qr. 26 lb. 13 oz.
5 hhd. 56 gal. 2 qt. and 1 hhd. 60 gal. 1 pt.

42 lb. 3 oz. 7 dwt. 15 gr. and 36 lb. 8 oz. 14 dwt. 20 gr.
30 d. 17 h. 25 m. 18 s. and 18 d. 21 h. 47 m. 53 s.
479.15£ and 398.346£

65.342 yd. and 59.853 yd.
1574.3 lb. and 986-532 lb.

856.271 gal. and 794-837 gal.
£ and £.

15. 5 lb. av. and 5 lb. troy.
16. 1 cubie ft. and 5 gal.

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MULTIPLICATION OF COMPOUND NUMBERS.

152. The product of a compound number by any whole, number may be obtained by finding the sum of certain repetitions of the compound number. Also the sum of these repetitions is obtained by adding together the numbers in each denomination separately from right to left, carrying from each sum to the next higher, one for every time it contains that number of its own denomination which make one of the next higher denomination, and writing the surplus under the denomination to which it belongs. (Art. 148.)

But the sum of these repetitions of each denomination can be

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