COMPOUND ADDITION, ETC.] 



MATHEMATICS. ARITHMETIC. 



441 



quarters, with 21 Ib. over. This number, divided by 



4, gives the number 



7) 591241 of cwts. namely, 5278 



and 3 qrs. over: and 



4) 844C3 lastly, dividing by 20, 



the number of cwt. in 



4) 21115... 21 Ib. 1 ton, we get finally 



263, the number of 

 2. 0)527. 8... 3 qrs. tons: so that there 



are 263 tons 18 cwt 



263 t 18 cwt. 3 qr. 21 Ib. 3 qr. 21 Ib. in 591241 



Ib. 



All this is so easy and obvious that we need not occupy 

 space with any more worked-out examples. We shall 

 merely give one cautionary direction it is this : that 

 when you have to divide by 5', bring both this divisor 

 and the dividend into halves; that is, double both; 

 making the divisor 11, instead of 5^ ; but remember that 

 the remainder will be so many halves. In like manner, 

 when you have to divide by 30 j, bring all into quarters; 

 that is, divide 4 times the dividend by 121, which is 4 

 times 30j ; remembering, however, that the remainder 

 will be quarters ; so that a fourth part of the number, 

 which is the remainder, will be the 

 number of wholes. See the operation Square yards, 

 in the margin, where the factors of 121, 2463 



viz., 11, 11, are used to get the 4 



quotient by short division. This 

 quotient shows that there are 81 square 11) 9852 



js, and 51 quarter-yards over; 



that ig, 12 J square yards: the result 11) 895... 7 



would therefore be written, 81 square 

 perches, 12 J square yards. 81... 51 



The examples given at page 440, may 

 be employed for exercise in this rule, by taking in each 

 the quantity on the right of the sign of equality, and 

 converting it into that on the left ; but others are added 

 here : 



28635 seconds -7 h. 57m. 15 sec. 

 100H5760 gr. =1751 Ib. troy. 

 633600 inches =10 miles. 

 :;'.';"-' 4 yds. =225 mi. 4 fur. 26 per. 1 yd. 

 91476 sq. ft. =2 ac. rds. 16 per. 

 100000 cub. in. 2 cub. yds. 3 ft. 1504 in. 

 The four fundamental operations of arithmetic may 

 now be applied, in order, to compound quantities. 



Additian a/ Cotii/i'i'iiut (jii'iHiitie*. To add together a 

 set of concrete quantities of different denominations, the 

 rule is as follows : 



RULE. Arrange the quantities to be added one under 

 another, so that all in the same vertical column may be 

 of the same denomination. 



Add up the quantities of lowest denomination : find 

 how many of the next denomination are contained in tin: 

 sum : put the remainder under the columu, and carry 

 the quotient to the next column. 



Proceed in this way, from column to column, till all 

 have lieen added up. 



The principle of this rule is too obvious to require any 



explanation ; the carryings merely transfer 



i. d. the quantities of advanced denominations 



17 9 3| to the columns in which those denomina- 



42 13 4} tions are arranged, just as in the addition 



16 10 2L of abstract numbers. 



7 2 9J Thus the sums of money in the margin 



1 18 10J are arranged so that the denomination 



farthings forms one column, the dennini- 



85 14 6} nation pence the next, shillings the next, 

 and pounds the next. The sum of the 

 farthings' column is 10 farthings, in which are contained 

 2 pence, and there are 2 farthings, or | over ; this is 

 therefore put down, and the 2 pence carried to the pence 

 column ; the sum of this column is 30 pence, that is, 

 2s. M. : the 6d. is put down, and the 2s. carried to the 

 shillings' column, the sum of which is 54s., that is, 

 2 14s. ; we therefore say 14, and carry 2 ; and this 2 

 added in with the pounds' column, makes the amount of 

 that column 85 ; therefore the sum of the whole is 

 85 



VOL. I. 



!) 

 (2.) 

 (3.) 



(4-) 

 (5-) 

 (6.) 



It may be noticed, that in adding up the shillings' 

 column of an account, the best way is to disregard the 

 tens in that column till all the units have been added ; 

 then, having reached the top unit-figure, to proceed 

 downwards, taking in every ten that appears. Tims, in 

 the present example, the sum of units' column of shillings 

 is 24 ; so that proceeding downwards, taking in each ten 

 as we meet with it, we say 34, 44, 54 ; so 

 that the sum is 54s., or 2 14s. u>. oz. dr. 



Two other examples are here annexed; 8 13 11 



the one in Avoirdupois weight, and the 9 10 13 



other in Time. In the former the sum of 469 



the drams is found to be 56 dr. ; by reduc- 11 11 15 

 tion, we find that in these drams there are 7 3 



3 oz. 8 dr. ; we therefore put down 8 dr. and 



carry the 3 oz. to the next column, which 41 14 8 

 gives 46 oz., or 2 Ib. 14 oz. ; writing down 

 the 14 oz. , and carrying the 2 Ib. to the column of Ibs. , 

 we get 41 Ib. for the sum of this column : therefore the 

 whole sum is 41 Ib. 14 oz. 8 dr. 



In the next example the column of 

 d. h. m. . seconds amount to 110 seconds ; that 

 34 13 9 15 is, to 1 minute 50 seconds: the 50 

 18 9 37 seconds is put down, and the 1 minute 

 27 21 11 19 carried to the next column, the amount 

 14 18 23 4 of which is 79 minutes ; that is, 1 hour 

 10 7 14 16 19 minutes, 19 and carry 1 : the hour 

 13 14 21 19 column amounts to 83 hours, or 3 days 



11 hours ; 11 and carry 3 to the day's 



119 11 19 50 column, the amount of which is 119 : 

 therefore the whole amount is 119 

 days, 11 hours, 19 minutes, 50 seconds. 



Subtraction f Compound Quantities. The subtraction 

 of concrete quantities, of different denominations, is 

 effected by the following rule : 



RULE. Place the less of the two quantities under the 

 greater, arranging the denominations as in addition. 



Commence with the lowest denomination, and subtract, 

 if the upper number be sufficiently great ; if not, increase 

 it by as many as will make 1 of the next denomination, 

 and then subtract, taking care afterwards to carry 1, as 

 in the subtraction of abstract numbers : and proceed in 

 like manuer with each denomination till the subtraction 

 is finish fl. 



In this way the difference between . rf. 

 124 16s. 9^<i. and 75 19s. 3J A is found, 124 16 9$ 

 as in the margin. Since 3 farthings can- 75 19 3J 



not be taken from 2 farthings, we in- 



crease the 2 farthings by 4 farthings, or 48 17 5J 



Id. , and say 3 from 6 and 3 remain ; that 

 is, 3 t d. : carry 1 : 4 from 9 and 5 remain : 19 from 36 (in- 

 creasing the 16s. by 20s. *) and 17 remain ; carry 1 : 6 

 from 14 and 8 remain ; carry 1 : 8 from 12 and 4 remain ; 

 therefore the difference is 48 17s- 5jt/. 



Again, suppose we have to subtract 24 miles, 6 furlongs^ 



21 perches, 2 yards from 43 inilos, ] 



m. fnr. per. yd. perch, 1 yard. Then, having plaivd 



43 1 1 the quantities as in the margin, and 



24 C 21 2 seeing that the 1 yard is too small, wo 



^ ^ increase it by 1 perch, that is by 5| 



18 1 19 4J- yards, and subtract 2 yards from 6^ 



yards; we thus get the remainder 4^ 

 yards ; and carry 1 : and as 40 perches make 1 

 furlong, we subtract 22 from 41, and get 19 for re- 

 main ing 1 to the 6 we subtract 7 from 8 the 

 furlongs in 1 mile and get 1 for remainder : and carry- 

 ing 1 to the 4, it merely remains to subtract 25 from 43 : 

 the complete remainder is therefore 18 miles, 1 furlong, 



19 perches, 4^ yards. 



If the complete remainder in an operation of this kind 

 be added to the compound quantity immediately above 

 it that is, to the subtractive quantity the sum will be 

 equal to the upper row ; that is, to the quantity which 

 has been diminished : so that we may prove in this 

 way the correctness of the subtraction. The following 



