ARITHMETIC. 
151 
Examples. Subtract 3 from ±1 : and 5 from £. 
- X_ 5 
TT T 
11 X 7 = 77 
3 X 12 = 36 
41 
'*• ii 
3 X 11 =33 
5 X 4 = 20 
13 rem. 
12 X 7 = 84 4 X 11 = 44 
1'he answers are 4J and .13. 
8 4 4 4 
To subtract a fraction from an integer: Sub- 
tract the numerator from the denominator, and 
place the remainder above the denominator; 
prefix to this the integer diminished by unity. 
Ex. Subtract JL from 12. Remainder 11*. 
5 3 
Multiplication of Vulgar Fractions. 
Rule. “ Multiply the numerators of the fac- 
tors together for the numerator of the product, 
and the denominators together for the denomi- 
nator of the product.” 
Ex. 1st. §■ X i- = 4-5- 
2 x 5 = 10 numerator 
3x? = 21 -denominator. 
82 . — 4 _2 
b b 
74 — 3 T 
42 X 
5 X 
31 = 1302 
4 = 20 
To Multiply 5. by | is the same as to find 
what two-third parts, of 4 comes to ; if one-third 
part only had been required, it would have been 
obtained by multiplying the denominator 7 by 3, 
because the value of fractions is lessened when 
their denominators are increased ; and this comes 
to JL : and, because two-thirds were required, 
2 1 
we must double that fraction, which is done by 
multiplying the numerator by 2, and comes- to 
JL2. Hence we infer that fractions of fractions, 
2 i , 
of compound fractions, such as ■*. of , are re- 
duced to simple ones by multiplication. The 
same method is followed when the compound 
fraction is expressed in three parts or more. 
If a number be multiplied by any integer, its 
value is increased. If it be multiplied by' 1, or 
taken one time, it undergoes no alteration. If 
it be multiplied by a proper fraction, or taken 
for one half, two-thirds, or the like, its value is 
diminished, and the product is less than the 
number multiplied. 
Division of Vulgar Fractions. 
Rule. (1) “ Multiply the numerator of the 
dividend by tile denominator of the divisor. The 
product is the numerator of the quotient.” (2) 
« Multiply the denominator of the dividend by 
the numerator of the divisor. The product is 
the denominator of the quotient.” 
Ex. Divide J. by 1. Quotient 
2X9 = 18 
5X7 = 35 _ 
To explain the reason of this operation ; sup- 
pose it required to divide -f by 7, or to take one- 
seventh part of 2. This is obtained by multi- 
plying the denominator by 7 (for the value of 
fractions is diminished by increasing their deno- 
minators), and it comes to J-_, Again, because 
7. is nine times less than seven, the quotient of 
any number divided by .* will be nine times 
greater than the quotient of the same number 
divided by 7. Therefore we multiply JL. by 9,, 
and obtain JJL. 
. .3 5 
If the divisor and dividend have the same de- 
nominator it issufficient to divide the numerators. 
Ex. divided by _3_ quotes 4. 
The quotient of any number divided by a 
proper fraction is greater than the dividend. It 
is obvious, that any integer contains more 
halves, more third parts, and the like, than it 
contains units; and, if an integer and fraction be 
divided alike, the quotients will have the same 
proportion to the numbers divided; but the 
value of an integer is increased when the divisor 
is a proper fraction; therefore the value of a 
fraction in the like case is increased also. 
The foregoing rule may be extended to every 
case, by reducing integers and mixed numbers to 
the form of improper fractions. 
Decimal Fractions. 
A decimal fraction is a fraction whose denomi- 
nator is ten or some power of ten : but instead of 
writing down the denominator, a comma is 
placed before the number, to mark the fraction’; 
and whatever number of figures follows the 
comma, the same is the index of the power of 
ten in the denominator, or there must be as many 
cyphers after unit in the denominator, as figures 
after the comma. 
Thus 4,7 is 4 and _Z_. 
. . 1 ° 
.47 signifies Forty-seven hundredth parts. 
.047 
Forty-seven thousandth parts. 
.407 
Four hund. and 7 thousandth parts. 
4.07 
Four, and seven hundredth parts. 
4.007 
Four, and seven thousandth parts. 
To reduce vulgar fractions to decimal ones : “An- 
nex a cypher to the numerator 
and divide it 
by the denominator, annexing 
a cypher con- 
tinually to the remainder.” 
EXAMPLES- 
44 = 
-Lj. = .078125 
| = .6 66 8cc. 
75)120(16 
6 64)500(078125 
3)20(666 
75 
448 
18 
450 
520 ' 
20 
450 
512 
18 
0 
80 
20 
64 
18 
160 
20 
328 
320 
320 
The reason of this operation will be evident, 
if we consider that the numerator of a vulgar 
fraction is understood to be divided by the de- 
nominator ; and this division is actually per- 
formed when it is reduced to a decimal. 
Some vulgar fractions may be reduced to de- 
cimals, and are called finite decimals. Others 
cannot be exactly reduced, because the division 
always leaves a remainder; but, by continuing 
the tfivision, we shall perceive how the decimal 
may be extended to any length whatever. These 
are called infinite decimals. 
Lower denominations may be considered as 
fractions of higher ones, and reduced to decimals 
accordingly. 
The value of decimal places decreases like 
that of integers, ten of the lower place in either 
being equal to one of the next higher; and the 
same holds in passing from decimals to integers. 
Therefore, all- the operations are performed in 
the same way with decimals, whether placed by 
themselves or annexed to integers, as with pure 
integers. The only peculiarity lies in the ar- 
rangement and pointing of the decimals. 
In addition and subtraction, “ Arrange units 
under units, tenth-parts under tenth-parts, and 
proceed' as in integers.” 
Add 32.035 
from 
13348 
136.374 
160.63 
12.3645 
take 
9.2993 
4.0487 
341.4035 
In multiplication, “ Allow as many decimal 
places in the product as there are in both factors. 
If the product has not so many places, supply 
them by prefixing cyphers on the left hand,” 
Ex. 1st. 1.S7 2d. .1572 
1.8 ' -12 
1096 .018864 
137 
' 2.466 
The reason of this rule may be explained, by 
observing that the value of the. product depends 
on the value of the factors : and since each de- 
cimal pkjpe in either factor diminishes its value 
ten times, it must equally diminish the value of 
the product. 
To multiply decimals by 10, move the deci- 
mal point one place to the right; to multiply by 
100, 1000, or the like, move it as many places to 
the right as there are cyphers in the multiplier. 
In division, “ Point the quotient so that there 
may be an equal number of decimal places in the 
dividend as in the divisor and quotient together. 
Ex. Divide 14 by .7854. 
.7654) 1,4.-000000(1 7.82 &c. 
7854 
61460 
54978 
64820 
62832 
^ 19880 
Therefore, if there be the same number 
of decimal places in the divisor and dividend, 
there will be none in the quotient. 
Ex. Divide .75 by .25. Answer 3, which is a. 
whole number. 
If there be more in the dividend, the quotient 
will have as many as the dividend has more 
than the divisor. See above. 
If there be more in the divisor, we must annex 
(or suppose annexed) as many cyphers to the 
dividend as may complete the number of deci- 
mals in the divisor, and all the figures of the quo- 
tient are integers. 
Ex. Divide 8 by .125. 
.125)8.000(64 Answer in whole numbers. 
7.50 
500' 
500. 
0 
To reduce numbers of different denominations to their 
equivalent decimal values: Rule, (1) Write the 
given numbers perpendicularly under each 
other, beginning at the least.. (2) Opposite each 
dividend, on the left hand, place such a number 
for a divisor as will bring it to the next superior 
name, and draw a perpendicular line between 
them. (3) Begin with the highest’; and write 
the quotient of each division, as decimal parts, 
on the right hand of. the dividend next below 
it, and so proceed to the. last ; and the last quo- 
tient is . the decimal sought. 
Ex. Reduce 15s. 9gd. to the decimal of a £. 
4 ) 3. 
12 j 9.75 
20 j 15.8125 
.790625 the decimal. required. 
To fnd the value of any given decimal in terms of . 
the integer. Rule. Multiply the decimal by the- 
number of parts in the next less denomination, 
and cut ofFas many places for a remainder, to 
the right hand, as there are places in the given 
decimal ; and so proceed with the rest. 
Ex. What is the value of .875 of a Jf. 
.375 
20 
s. 7.500, 
12 . 
d. 6.000 Answer 7s. C d. 
If the divisor leave a remainder, the quo- 
tient may be extended to more decimal. pH es ; 
