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the whole product or answer of the question, reserving the remainders of these last two, which remainders must be equal when the work is right.--Note, It is common to set the four remainders within the four angular spaces of a cross, as in the example below.
Third Method.-Multiplication is also very naturally proved by Division; for the product divided by either of the factors, will evidently give the other. But this cannot be practised till the rule of Division is learned.
Multiply 123456789 by 3. Ans. 370370367. Multiply 123456789 by 4. Ans. 493827156. Multiply 123456789 by 5. Ans. 617283945. Multiply 123456789 by 6. Ans. 740740734. Multiply 123456789 by 7. Ans. 864197523. Multiply 123456789 by 8. Ans. 987654312. Multiply 123456789 by 9. Ans. 1111111101. Multiply 123456789 by 11. Ans. 1358024679. Multiply 123456789 by 12. Ans. 148148 1468. -Multiply 302914603 by 16. Ans. 4846633648. Multiply 273580961 by 23. Ans. 6292362103. Multiply 402097316 by 195. Ans. 78408976620. Multiply 82164973 by 3027. Ans. 248713373271. Multiply 7564900 by 579. Ans. 4380077100. Multiply 8496427 by 874359. Ans. 7428927415293. Multiply 2760325 by 37072. Ans. 102330768400.
CONTRAC CONTRACTIONS IN MULTIPLICATION.
I. When there are Ciphers in the Factors. Ir the ciphers be at the right-hand of the numbers ; multiply the other figures only, and annex as many ciphers to the right-hand of the whole product, as are in both the face tors. When the ciphers are in the middle parts of the multiplier ; neglect them as before, only taking care to place the first figure of every line of products exactly under the figure multiplying with.
2. Mult. 9001635
Mult. 390720400 by 70100
by - 406000
631014613500 Products 158632482400000
3. Multiply 81503600 by 7030. 4. Multiply 9030100 by 2100. 5. Multiply 8057069 by 70050.
II. When the Multiplier is the Product of two or more Numbers
in the Table; then * Multiply by each of those 'parts separately, instead of the whole number at once.
* The reason of this rule is obvious enough; for any number multiplied by the component parts of another, must give the same product as if it were multiplied by that number at once Thus, in the 1st example, 7 times the product of s by the given number, makes 56 times the same number, as plainly as 7 times 8 makes 56. VOL. I.
2. Multiply 31704592 by 36. Ans, 1141365312.
7. There was an army composed of 104 * battalions, each consisting of 500 men; what was the number of men contained in the whole?
Ans. 52000., 8. A convoy of ammunition + bread, consisting of 250 waggons, and each waggon containing 320 loaves, having been intercepted and taken by the enemy; what is the number of loaves lost?
OF DIVISION. Division is a kind of compendious method of Subtraction, teaching to find how often one number is contained in anothier, or may be taken out of it: which is the same thing.
The number to be divided is called the Dividend. The number to divide by, is the Divisor. ---And the number of times the dividend contains the divisor, is called the Quotient. Sometimes there is a Remainder left, after the division is finished.
The usual manner of placing the terms, is, the dividend in the middle, having the divisor on the left hand, and the quotient on the right, each separated by a curve line; as, to divide 12 by 4, the quotient is 3, Dividend
12 Divisor 4 ) 12 (3 Quotient; 4 subtr. showing that the number 4 is 3 times contained in 12, or may be 3 times 8 subtracted out of it, as in the margin.
4 subtr. | Rule.--Having placed the divisor before the dividend, as above direct
4 ed, find how often the divisor is con- 4 subtr. tained in as many figures of the dividend as are just necessary, and place the 0 number on the right in the quotient.
* A battalion is a body of foot, consisting of 500, or 600, or 700 men, more or less.
+ The ammunition bread, is that which is provided for, ana dis tributed to, the soldiers ; the usual allowance being a loaf of 6 pounds to every soldier, once in 4 days. # In this way the dividend is resolved into parts, and by trial is
Multiply the divisor by this number, and set the product under the figures of the dividend before-mentioned. -Subtract this product from that part of the dividend under which it stands, and bring down the next figure of the dividend, or more if necessary, to join on the right of the remainder.-Divide this number, so increased, in the same manner as before; and so on till all the figures are brought down and used.
N. B. If it be necessary to bring down more figures than one to any remainder, in order to make it as large as the divisor, or larger, a cipher must be set in the quotient for every figure so brought down more than one.
TO PROVE DIVISION.
* MULTIPLY the quotient by the divisor ; to this product add the remainder, if there be any; then the sum will be equal to the dividend when the work is right.
found how often the divisor is contained in each of those parts, one after another, arranging the several figures of the quotient one after another, into one number.
When there is no remainder to a division, the quotient is the whole and perfect answer to the question. But when there is a remainder, it goes so much towards another time, as it approaches to the divisor: so, if the remainder be half the divisor, it will
the half of a time more ; if the 4th part of the divisor, it will go one fourth of a time more ; and so on. Therefore, to complete the quotient, set the remainder at the end of it, above a small line, and the divisor below it, thus forming a fractional part of the whole quotient.
* This method of proof is plain enough: for since the quotient is the number of times the dividend contains the divisor, the quotient multiplied by the divisor must evidently be equal to the dividend.
There are also several other methods sometimes used for proving Division, some of the most useful of which are as follow :
Second Method...Subtract the remainder from the dividend, and divide what is left by the quotient; so shall the new quotient from this last division be equal to the former divisor, when the work is right.
Third Method.-Add together the remainder and all the products of the several quotient figures by the divisor, according to the order in which they stand in the work ; and the sum will be equal to the dividend when the work is right.
3. Divide 73146035 by 4. Ans. 182865214
Ans. 47516365. 6. Divide 74638105
Ans. 20172463 7. Divide 137896254
Ans. 142161037 8. Divide 35821649
Ans. 46886744 9. Divide 72091365 by 5201. Ans. 13861-304 10. Divide 4637064283 by 57606.
11. Suppose 471 men are formed into ranks of 3 deep, what is the number in each rank?
Ans. 157. 12. A party, at the distance of 378 miles from the head quarters, receive orders to join their corps in 18 days: what number of miles must they march each day to obey their orders ?
Ans. 21. 13. The annual revenue of a gentleman being 383301; how much per day is that equivalent to, there being 365 days in the year?
CONTRACTIONS IN DIVISION. There are certain contractions in Division, by which the operation in particular cases may be performed in a shorter manner: as follows: