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Erample (2.) The figure 1 being on the right of the multiplier, the multiplicand is merely repeated; the second figure being a cipher, of course with it you cannot multiply; by writing a cipher under it in the second line you advance the product of the two one step, or to the place of hundreds, consequently the second line is the product of 534672, multiplied by two hundred.
Should you want to multiply by ten, one hundred, or one thousand, write as many ciphers on the right of the multiplicand as there are in the multiplier, and the work will be done. Do for practice the following sums.Ex. (12.) Multiply 6478 by 231. (13.)
6487 Х 204. (14.)
79652 Х 857. (16.)
9473 Х 769. (17.)
74627 Х 763. (19.)
827641 Х 913. (20.)
5746927 Х 54137. (22.) There are 8766 hours in the year ; how
many in 20 years ?
(23.) How many seconds in one year? and how many in twenty years ?
(24.) A gentleman dying, left his fortune to his six children; each received £470 ; how much was he worth?
(25.) How many pins will a boy point in six days, working each day ten hours, if he can point 16076 in one hour?
(26.) A clock strikes 156 times in twenty-four hours ; how often will it strike in a year of 365 days?
(27.) How many miles will a person travel in 40 years, supposing he travels 24 miles a day, and there are 365 days in the year?
(28.) A certain island contains 32 counties, each county 26 parishes, each parish 264 families, and each family 12 persons; required the number of parishes, families, and persons on the island ?
(1.) 1476. (2.) 15741363. (3.) 3821255. (4.) 318528. (5.) 146106. (6.) 371656. (7.) 34624275. (8.) 36968805. (9.) 5220952. (10.) 14752630. (11.) 6953676. (12.) 1496418. (13.) 1323348. (14.) 63221592. (15.) 68261764. (16.) 7284737.
one year; 631152000
seconds in 20 years. (24.) £2820. (25.) 964560 pins. (26.) 56940 times. (27.) 350100 miles. (28.) 832 parishes, 219648
families, 2635776 per
Division is the converse of Multiplication; and as Subtraction teaches us to separate a lesser from a greater number, so Division teaches to separate a number into several equal parts.
To point out clearly the value of this rule, imagine that you have twenty pounds to divide between four persons : you might perform your work, as in Example (1), by subtracting four from twenty, and again from the remainder, and so continuing until nothing remains ; then count the number of your subtractions, and it would be the amount for each person.
But in the second Example you see how easily the same sum is worked by this rule. You write down the twenty pounds, and on the left draw a short curved line, from which you draw a straight horizontal line towards the right and under the twenty; on the left of the curved line write four, and proceed with the work. Four is not contained in the first figure 2, so I say four in two I cannot, but four in twenty five times, which is the amount for each. Ex. (1.) 20
Ex. (2.) 4) 20 4 1 1st.
5 amount for each. 16 4 1 2nd.
O 5 amount for each.
These two examples will convince you that without a knowledge of this rule it would be almost impossible to work arithmetic; I need not therefore impress upon you how necessary a knowledge of it is, and how incumbent it is on you to give the study of it your utmost attention.
In this, as in the other rules of arithmetic, there are certain terms used, with which you must make yourself acquainted. There are always two numbers given. One number is passive, and is to be divided ; it is always the largest number, and is called the dividend. The other, the smaller number, is active, and is called the divisor. The number found by dividing is called the quotient; and if any portion of the dividend remain—that is, if there is a portion of it that is less than the divisor—it is called a remainder. The art of working this rule consists in the proper method of stating the case to be worked ; and when the sum of the dividend is too large to be divided at once, in separating it into parts most convenient for the operation.
Example 1st Divide 12 by 3. In this example we see at Divisor 3)12 Dividend. a glance that twelve contains
4 Quotient. three four times; but had we three figures in the dividend instead of two, our sagacity would not then serve us to divide them at once; we therefore take portions of it, as you will see by the following example.
Example 2nd. - Divide 967 by 3. You see the sum
Divisor 3)967 Dividend. written on the right,
Quotient 322...1 Remainder. as described in the example dividing the £20. between four persons. Then not being able to tell at a glance how many threes there are in nine hundred and sixty-seven, we take the left
figure 9, and say three into nine three times, which 3 we write under the 9, and take the next figure, 6, saying three into six twice; we write the 2 under the 6, and say three into seven, the next figure, twice and one over, which one being less than a unit, in fact being only a third part of one, we write to the right of the quotient, leaving a space between them, and call it the remainder; and we find the three is contained in nine hundred and sixty-seven, three hundred and twentytwo times, and that there remains one.
Example 3rd.—Divide 967 by 11. In this example we have taken the same dividend,
Divisor 11) 967 dividend but with a larger divisor ;
Quotient 87...10 rem. and you
will perceive that the divisor being larger than the first figure of the dividend, you cannot divide it into it; but if you take in the second figure, 6, it will then be 96, and we say, eleven into ninety-six, eight times eleven are eightyeight, from ninety-six and eight remains, which eight represents eight tens; we therefore in our own mind place 8 before the right-hand figure 7, and say, eleven into eighty-seven, seven times eleven are seventyseven, from eighty-seven and ten remains, which being a remainder of the figure in the units place, is therefore less than a unit, and is written on the right for a remainder.
Example 4th.--Divide 7201416 by 6. With this example we say six into
6) 7201416 seven once, and one over, which one we
1200236 place in our own mind before the two, and say six into twelve, twice six are twelve, and nothing over; six into nought, nought times; six into one, nought ; six into fourteen, twice six are twelve,