ARITHMETIC. 
lower taking in the given number of that 
denomination, and continue the division. 
When the divisor is not greater than 12, we 
proceed as before in short divison. 
Examples. 
£. 
s. 
d. 
£. 
s. 
d. 
5) 
84 .. 
3 .. 
9 
11) 
976 
.. 13 
-n 
Ans. 
16 ., 
. 16 .. 
9 
Ans. 
88 
.. 15 
.. 91—8 
lb. 
oz. 
diets. 
cwt. 
qr. 
ib. 
oz. 
8) 994 
.. 4 
.. 8 
12) 45 
. 18 . 
. 8 
124 
.. 3 
.. 11 
3 . 
. 3 .. 
6 .. 
3-5—4 
When the divisor is greater than 12, the 
operation is performed by long division. 
Example. 
£. s. d. 
Divide 8467 .. 16 .. 8 by 659. 
£. s. d. 
659) 8467 .. 16 .. 8 (12 
659 
1877 
- 1318 
7559 
20 
659) 11196 (16 
659 
4606 
3954 
. 652 
12 
659) 7832 (11 
7249 
583 
4 
659) 2332 (| 
1977 
■ 353 
Ans. 12 .. 16 .. 114 IH- 
In connection with the rule of Division, 
we may notice another kind of Reduction, 
so called, though improperly, as by it is 
meant to bring smaller denominations into 
larger ; as pence into pounds, or drams into 
hundred weights, &c. ; for which the rule 
is : divide by the parts of each denomina- 
tion from that given to the highest sought : 
the remainders, if any, will be ot the same 
name as the quantity from which they were 
reduced. 
Examples. 
1. In 415684 farthings how many pounds 
sterling. 
4 ) 41 5684 
12 ) 103921 
2 , 0 ) 866 , 0 - 1 
Ans. ^ 433 .. 0 ■■ 1 
2. How many pounds troy are there in 
67890 dwts. 
2,0) 6789,0 
12) 3394- 
-10 
28 °2 . 
. 10 .. 
, 10 . 
ib. 
OZ. 
dwts. 
Ans. 282 . 
. 10 
.. 10. 
Before we conclude this article we may 
observe, that in computations which require 
several steps, it is often immaterial what 
course we follow. Some methods may be 
preferable to others, in point of ease and 
brevity ; but they all lead to the same con- 
clusion. In addition, or subtraction, we 
may take the articles in any order. When 
several numbers are to be multiplied toge- 
ther, we may take the factors in any order, 
or we may arrange them into several classes ; 
find the product of each class, and them mul- 
tiply the products together. When a num- 
ber is to be divided by several others, we 
may take the divisors in any order, or we 
may multiply them into one another, and di- 
vide by the product ; or we may multiply 
them into several parcels, and divide by the 
products successively. Finally, when mul- 
tiplification and division are both required, 
we may begin with either ; and when both 
are repeatedly necessary, we may collect 
the multipliers into one product, and the 
divisors into another ; or we may collect 
them into parcels, or use them singly ; 
and that in any order. To begin with 
multiplication is generally the better mode, 
as this order preserves the account as clear 
as possible from fractions. 
We have hitherto given the most ready 
and direct method of proving the foregoing 
examples, but there is another which is 
very generally used, called “ casting out 
the 9’s,” which depends on this principle : 
That if any number be divided by 9, the re- 
mainder is equal to the remainder obtained, 
when that sum is divided by 9. For instance, 
if 87654 be divided by 9, there is a remain- 
der of 3 ; and if 8, 7, 6, 5, 4, be added toge- 
ther, and the sum 30 be divided by 9, there 
will be likewise a remainder of 3. 
