ARITHMETIC.
22 ()
Multiplication Table.
1
2
3
4
5
6
7
8
9
2
4
6
8
10
12
14
16
18
3
6
9
12
15
18
21
24
27
8
.2
16
20
24
28
32
36
5
10
15
20
25
30
35
40
45
0*
12
IS
24
30
36
42
48
54
7
14
21
28
35
42
49
56
63
8
16
24
32
40
48
56
64
72
9
18
27
36
45
54
63
72
81
To Multiply by any of the nine Digits.
Rule 1 . Begin at the right of the mulplicand, and multiply eachfigure in it by the multiplier, writing down the figure in the units’place of the first product, and carrying the tens to the next pro-duct ; set down the figure in the units’ place of this sum, and car-ry the tens as before, and thus proceed till all the figures are mul-tiplied, and write down the last product.
II. When the Multiplier consists of several Figures.
1. Write it below the multiplicand, and find the product foreach figure in it as in the first case, not regarding in what order thelines are found, provided the iirst figure in each stand straight be-low its respective multiplier.
2. Add all the products together in the order they stand, andthe sum will be the whole product required.
Proof. Multiply the factors, making jthe former multiplicandthe multiplier, and the product will be the same as before whenthe work is right. Otherwise, add together the figures in eachfactor, casting out all the nines in the sums as often as they amountto nine. Multiply the two remainders together, and the nines castout of their product will leave the same remainder as the nines castout of the answer when the work is right.
Examples.
8576 2375 multiplicand.
7 38 multiplier.
60032 19000
• 7125
90250 product.
(14.) Contractions.
I. When there are cyphers at the right of one or both factors,proceed as before, neglecting the cyphers; and to the right of theproduct annex the cyphers.
Thus to multiply 1340 by 2700, multiply 134 by 27, and to theproduct 3618 join three cyphers, and we get 3618000 the productrequired.
II. When the multiplier is the product of two or more numbersin the table, it is often of advantage to multiply continually bythose numbers instead of it.
Thus 248 multiplied into 72 is equal to 24Sx9X 8=2232x 8=17856, the product required.
III. When some of the figures of the multiplier may be pro-
duced by multiplying some others of them by any number, it ismuch easier and more concise after having obtained the productof the less, to multiply that product by the same number for theproduct of the greater, than to proceed by the common method.Thus, 546
672
3276
39312
366912
Here the product of 6 is multiplied by 1C for the product of 72.VOL. i.— no. 15.
Simple Division.
(15.) Simple division is the finding how often one simple num-ber is contained in another; or the dividing of any given simplenumber into any proposed number of equal parts.
The containing number, or number to be divided, is called thedividend. The contained number, or the number of parts intowhich the dividend is divided, is called the divisor. The numberof times the dividend contains the divisor, or the number whichexpresses one of the equal parts, is called the quotient, thus:
Dividend.
Divisor 3) 12 (4 Quotient.
Division is a compendious subtraction, the quotient being thenumber of subtractions in the operation.
Rule 1 . Having written down the divisor and dividend in theform above, consider if the divisor be less than, or equal to, the .same number of the left hand figures of the dividend; if so, writethe figure expressing the number of times it is contained, in thequotient ; but if not, take one place more of the dividend figuresthan are in the divisor, and write the number of times they containit in the quotient as before.
2. Multiply the divisor by the quotient figure.
3. Subtract the product from the said dividend figures.
4. To the remainder affix the next dividend figure, and write inthe quotient the number of times the divisor is contained in this-number ; multiply the divisor by the last quotient figure, aud sub-tract the product from the last mentioned number ; then proceedas before from the beginning of this article, till all the dividendfigures are used.
Proof. Multiply the quotient by the divisor; to the productadd the remainder, and the sum will be equal to the dividendwhen the work is right.
Example.
372)58247(156
372
2104
1860
2447
2232
remainder 215(16.) Contractions.
I. Division by a single figure may be expeditiously performedby multiplying and subtracting mentally, and writing down_onlythe quotient below the dividend.
Example. 7)579325
82760f
II. When the divisor has cyphers on flic right of it, strike them-oil) and divide without them ; but the same number of figuresmust be struck off from the right of the dividend, and affixed to<the last remainder.
Example. 8,00)5734,21'
716$ca
III. When the divisor is the product of two or more small num-bers, it is much easier to divide continually by those numbers,than by the whole divisor at once, if there be any remaindersafter such divisions, multiply the last remainder by the precedingdivisor, and to the product add the remainder belonging to thesame divisor; then multiply the sum by the next preceding divi-sor, and to tile product add’ its corresponding remainder: proceedin the same manner through all the divisors and remainders; andthe last sum will be the remainder, (he same as if the dividon hadbeen performed at once.. Alter this operation is begun, it must becontinued according to the description, though some of the pre-ceding divisions should happen to have no remainders.
Example. 8)42901685
12) 53627105- .
Quotient 446892-4
3N
IV. One