ARITHM'ET I C. 
DIVISION of DECIMALS. 
Rule. Proceed as in whole numbers till the work be 
done, then cut oft' fo many decimal places in the quotiont 
as the dividend hath more than the divil'or; but, if there 
is not a fufficient number of figures in the quotient to an- 
fwer the excels, the deficiency mult be made up by pre¬ 
fixing cyphers to the left hand. 
Ex. i. Ex. 2. 
56-7)279-8712(4-936 j-326)-2274202(-0427 
226-8 21304 
5307 14380 
5103 • 10652 
2041 37282 
1701 37 2 8 2 
3 4 °2 * * * . 
34Q2 
* * # 
If any thing remain, annex a cypher to it, and divide as 
before, and fo on; and for every cypher added there rauft 
be a decimal in the quotient. If the number of decimal 
places in the divifor and dividend be equal, all the quotient 
will be whole numbers; and, if any thing remain, annex 
cyphers, and proceed as above diredted. Ex. Divide 
381-764 by -438. 
-438)381-764(871 quotient 
3504 
3 r 3 6 
3066 
A? 
704 
438 
266 remainder 
The truth of this is evident from hence : If you multi¬ 
ply both the divifor and dividend by 1000, they will be¬ 
come whole numbers; and, if the whole number 438 di¬ 
vide the whole number 381764, the quotient nmftof courfe 
be whole numbers. When there are not fo many decimal 
places in the dividend as there are in the divifor, annex 
cyphers to the dividend to make them equal (which is in 
reality reducing both to whole numbers), then will the 
quotient be whole numbers. 
Thus -536)93-800(175 quot. Or 536)93800(175 quot. 
When a decimal is to be divided by an integer with cy¬ 
phers on the right, cut oft' the cyphers, and remove the 
feparating point in the dividend fo many places to the left 
as there are cyphers cut off from the right of the divifor, 
then divide as before, prefixing cyphers if needful. Thus, 
52-74 divided by 10 — 5-274. 
Large divifions may be contracted as follows : 
Ex. Divide 315-46439479 by 46-1062514. 
46-10625! 14 )-315-464301479 ( 6-842117 quotient 
•. 27663750 
3882689 
3688500 
194189 
184425 
Here 6 is multiplied in¬ 
to 46-10625, then 8 is mul¬ 
tiplied into 46-1062, car¬ 
rying 4 ; then 4 is multi¬ 
plied into 46-106, carrying 
1 ; then 2 is multiplied in- 
9764 
9221 
543 
4 - 6 1 to 46-10, carrying 1 ; then 
82 1 is multiplied into 46-1 ; 
46 then 46- by 1, and laftly 
^6 4 by 7, carrying 4. 
32 
J4 
Find, by the general rule, what place of decimals or in¬ 
tegers the firft figure of the quotient will poffefs; then 
take as many of the left-hand figures of the divifor as you 
judge necefiTary for your firft divifor, and juft fo many of 
the left-hand figures of the dividend as will yield the firffc- 
figure of tire quotient. Having found the firft figure, the 
following may be found thus; take each laft remainder 
for a new dividend, and for a new divifor prick oft' one fi¬ 
gure from the right of each preceding one; continue the 
operation till the divifor is exhaufted. In multiplying the 
quotient-figure and divifor^ leave out thofe figures pricked 
oft'; only particular regard muft be had to the increafe 
that would arife from the laft figure fo pricked off, and 
for fuch increafe carry, as in the foregoing rule for mul¬ 
tiplication. 
REDUCTION of DECIMALS. 
To reduce a vulgar fraction to a decimal. Add cyphers 
to the numerator at pleafure, reprefenting fo many places - 
of decimals; and then divide by the denominator, conti¬ 
nuing the operation to what length you pleafe. 
Ex. Reduce a, 4-, 4 £ 4 * 5 and-—, each to a deci¬ 
mal fraction. 
Thus i-oo -2-42=0-25 =J f '| 
1- o -22=0-5 -L I 
3-00 -2-4=0-75 — 3 - r 
2- 000 - 2 - 2 — 0 - 6664 - 2=4 I 
4 0 -f- 5=0-8 ' = I'l 
6-ooo -2- 72=0-8571-4- = 4 I 
5-0000-2- 9=0-5555.4- — _5? r 
8.0000-2-11=0-7272727-4-=^- J 
To reduce decimals, or integers, to equivalent decimals 
of a greater denomination. If the decimals or integers 
propofed be fimple, divide continually by all the denomi¬ 
nations, from the given one to that fought, as in Reduc¬ 
tion of integers before fhewn. 
Ex. 1. Reduce 4 d. = o-75d. to the decimal of a pound 
fterling. Thus 0-75-2-1 2=0-06253. -2-20=0-003 1 25I. the 
anfwer. 
Ex. 2. Reduce 54 grains to the fraction of an ounce troy. 
Firft 5-f = 5-375 grs. -2-24 = 0-223958 dwts. -4-20 = 
o-oii 1979 oz. -4- 
Ex. 3. Reduce nd. to the decimal of a pound fterling. 
Thus 11-ooo- 2 -i2=0-91665. -2-20=0-04583331. anfwer. 
Ex. 4. Reduce 8-564 pints of wine to the decimal of a 
hogfhead. Thus 8-564-4-8=1-0705gall. -2-63=0-016992 
hogfheads, the anfwer. 
If the given part confift of feveral denominations, re¬ 
duce them to the lead for a numerator, and the integer to 
the fame for a denominator; then annex cyphers to the 
numerator, and divide as before. 
Ex. 1. Reduce 18s. 9-Jd. to the decimal of a pound. 
Thus iSs. 9^-d.= 903 farthings for a numerator, and then 
|£§1. = -9406251. the anfwer. 
Or thus, 4 3-00 
12 9 " 75 ^- 
20 18-8125 
0-9406251. the fame as before. 
Ex. 2. Reduce 6oz. 19 dwts. 7^grs. to the decimal of 
a pound troy. 
6 7*5 grs. 
4 1-25 
20 19-3125 dwts. 
12 6-965625 oz. 
0-58046875^. the anfwer. 
To reduce a decimal of fuperior denomination to its va¬ 
lue in the inferior or known parts of the integer. Multi¬ 
ply the given decimal by the number of parts contained 
in the next^nferior denomination, cutting off the decimals 
from the product (as taught in multiplication), then mul¬ 
tiply the decimals fo cut off by the next lower denomina¬ 
tion; thus proceed to the lowed denomination required, 
or till the decimals pointed off are all cyphers; then the 
numbers on the left of the points will exprefs the value of 
the decimals. 
Ex, 
24 
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