Encyclopaedia Britannica > Volume 3, Anatomy-Astronomy

ARITHMETIC,
599
Vulgar 3d, To multiply a mixed number by a fraction, we may
1* ractions. multiply the integer by the fraction, and the two fractions
together, and add the products.
Ex. 151x1=3-^
15’ X|=3| =3^
Y2 —.
1
_LE
4</i, When both factors are mixed numbers, we may mul¬
tiply each part of the multiplicand, first by the integer of
the multiplier, and then by the fraction, and add the four
products.
Ex. 8f by 7f
56
6
91fi-
*2 0
_o_
2 0
by Prob. II.
product 65^y as before.
SECT. IV. DIVISION OF VULGAR FRACTIONS.
Rule I.—^Multiply the numerator of the dividend by the
denominator of the divisor. The product is the numerator
of the quotient.
II.—Multiply the denominator of the dividend by the nume¬
rator of the divisor. The product is the denominator of the
quotient.
Ex. Divide f by quotient £f.
2 X 9 = 18
5 X 7 = 35.
To explain the reason of this operation, let us suppose
it required to divide ‘j by 7, or to take one seventh part
of that fraction. This is obtained by multiplying the de¬
nominator by 7; for the value of fractions is diminished
by increasing their denominators, and comes to ^j. Again,
because 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. There¬
fore we multiply fj by 9, and obtain
If the divisor and dividend have the same denomina¬
tor, it is sufficient to divide the numerators.
Ex. divided by T37 quotes 4.
The foregoing rule may be extended to every case by
reducing integers and mixed numbers to the form of im¬
proper fractions. We shall add some directions for short¬
ening the operation when integers and mixed numbers
are concerned.
1,9/, When the dividend is an integer, multiply it by
the denominator of the divisor, and divide the product by
the numerator.
Ex. Divide 368 by f
7
5)2576(5151 quotient.
2d, When the divisor is an integer, and the dividend a
fraction, multiply the denominator by the divisor, and
place the product under the numerator.
Ex. Divide f by 5 quotient ^
8X5=40
3(/, When the divisor is an integer, and the dividend a
mixed number, divide the integer, and annex the fraction
to the remainder; then reduce the mixed number thus
formed to an improper fraction, and multiply its denomi¬
nator by the divisor.
Ex. To divide 5761,- by 7 quotient 82f£
7) 576 (82 Here we divide 576 by 7, the
56 quotient is 82, and the remain¬
der 2, to which we annex the
fraction 1, , and reduce 2j4r to
an improper fraction fr(, and
multiply its denominator by 7,
11 X 7=77 which gives
Hitherto we have considered the fractions as abstract 1 ulgar
numbers, and laid down the necessary rules accordingly.
We now proceed to apply these to practice. Shillings
and pence may be considered as fractions of pounds, and
lower denominations of any kind as fractions of higher ;
and any operation, where different denominations occur,
may be wrought by expressing the lower ones in the form
of vulgar fractions, and proceeding by the previous rules.
For this purpose the two following problems are neces¬
sary.
Problem V.—To reduce lower denominations to trac¬
tions of higher.
Place the given number for the numerator, and the value of
the higher for the denom inator.
Examples.
1. Reduce 7d. to the fraction of a shilling. Ans. -fa.
2. Reduce 7d. to a fraction of a pound. Ans. _
3. Reduce 15s. 7d. to a fraction of a pound. Ans. ||^.
16
14
2 t
11 1 r
Problem VI.—To value fractions of higher denomina¬
tions.
Multiply the numerator by the value of the given denomi¬
nation, and divide the product by the denominator; if there
be a remainder, multiply it by the value of the next denomi¬
nation, and continue the division.
Ex. 1 st, Required the value 2d, Required the value
of of LI.
of § of 1 cwt.
17
20
s. a.
60)340(5 8
300
40
12
— qrs. lb.
9)32( 3 15§
27
28
60)480
480
9)140
9
50
45
In the first example we multiply the numerator 17 by
20, the number of shillings in a pound, and divide the pro¬
duct, 340, by 60, the denominator of the fraction, and ob¬
tain a quotient of 5 shillings; then we multiply the re¬
mainder, 40, by 12, the number of pence in a shilling,
which produces 480, which, divided by 60, quotes 8d. with¬
out a remainder. In the second example we proceed in
the same manner; but as there is a remainder, the quotient
is completed by a fraction.
Sometimes the value of the fraction does not amount to
an unit of the lowest denomination; but it may be reduced
to a fraction of that or any other denomination by multi¬
plying the numerator according to the value of the places.
Thus jAtj of a pound is equal to of a shilling, or
T2I& of a Penny, or of a. farthing.
Chap. IX.—Decimal Fractions.
SECT. I. NOTATION AND REDUCTION.
Decimal fractions are such as have 10, or some power
of 10 (that is 100, 1000, &c.), for a denominator: such are
these,—
_3_
1 O’
_24_
1 0 0’
-15.-
lOOO’
10000O
They are more simply written thus:
•3, -24, *075, -00462;
the number of figures after the point being always the
same as the number of ciphers in the denominators.
In decimal fractions, as thus written, the figure next