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In 3756 pence how many four-pences ?
It is evident that this answer will be obtained by finding , how many times 4 pence is contained in 3756 pence.
If we would solve this, as we did the first examples, it will stand thus :
3756 : 3600 + 120 + 36 But if we resolve it into parts, as we perform the operation, it will be done as follows: Dividend 3756 (4 divisor
900 = number that 4 is contained in 3600 156 30 do.
120 120 9 do.
Here I take the 37 hundreds alone, and see how many times 4 is contained in it, which I find 9 times, and since it is 37 hundreds, it must be contained 900 times. 900 times 4 is 3600, which I subtract from 3756, and there remains 156. It is now the question to find how many tiines 4 is contained in this. I take the 15 tens, rejecting the 6, and see how many times 4 is contained in it. It is contained 3 times, and since it is 15 tens, this must be 3 tens or 30 times. 30 times 4 is 120. This I subtract from 156, and there remains 36. 4 is contained in 36, 9 times; hence it is contained in the whole 939 times. Ans. 939 four-pences.
If these partial numbers, viz. 3600, 120, and 36, are compared with the resolution of the number above, they will be found to be the same.
This operation may be abridged still more.
In this I say, 4 into 37 9 times, and set down the 9 in the quotient, without regarding whether it is hundreds, or tens, or units, but by the time I have done dividing, if I set the other figures by the side of it, it will be brought into its proper place.
Then I say 9 times 4 are 36, and set it under the 37, as before, but do not write the zeros by the side of it. I then subtract 36 from 37, and there remains 1. This of course is 100, but I do not mind it. I then bring down the 5 by the side of the 1, which makes 15, or rather 150, but I call it 15. Then I say, 4 into 15, 3 times, (this is 30, but I write only the 3;) I write the 3 by the side of the 9. Then I say, 3 times 4' is 12, which I write under the 15, and subtract it from 15, and there remains 3 (which is in fact 30.) By the side of 3 I bring down the 6, which makes 36. Then I say 4 into 36, 9 times, which I write in the quotient, by the side of the 93, and it makes 939. The first 9 is now in the hundreds place, and the 3 in the ten's place, as they ought to be. If this operation be compared with the last, it will be found in substance exactly like it. All the difference is, that in the last the figures are set down only when they are to be used.
A man employed a number of workmen, and gave them 27 dollars a month each ; at the expiration of one month, it took 10,125 dollars to pay them. How many men were there?
It is evident that to find the number of men we must find how many times 27 dollars is contained in 10,125 dollars.
This may be done in the same manner as we did the last, though it is attended with rather more difficulty, because the divisor consists of two figures.
Operation. Dividend 10,125 (27 divisor
300 = the number of times 27 is contained 2,025
in 8,100 1,890 70 do.
1,890 5 do.
135 135 135 375 do.
I observe that there are not so many as 27 thousands, so I conclude that the divisor is not contained 1000 times in the dividend ; I therefore take the three left hand figures, neglecting the other two for the present. The three first are 101 ; (properly 10,100, but I notice only 101 ;). I seek how many times 27 is contained in 101, and find between 3 and 4 times. I put 3 in the quotient, which, when the work is done, must be 3 hundred, because 101 is 101 hundreds, but disregarding this circumstance, I find how much 3 times 27 is, and write it under 101. 3 times 27 are 81 ; this subtracted from 101, leaves 20. By the side of 20 I bring down 2, the next figure of the dividend which was not used. This makes 202, for the next partial dividend. I seek how many times 27 is contained in this. I find 7 times. I write 7 in the quotient. 7 times 27 are 159, which I subtract from 202, and find a remainder 13. By the side of 13 I bring down 5, the other figure of the dividend, which makes 135 for the last partial dividend. I find 27 is contained 5 times in this. I write 5 in the quotient. 5 times 27 are 135. There is no remainder, therefore the division is completed. Ans. 375 men.
The operation in the above example is precisely the same, as in those which precede it; but it is more difficult to discover how many times the divisor is contained in the partial dividends. When the divisor is still larger, the difficulty is increased. I shall next show how this difficulty may be obviated.
In 31,755 days, how many years, allowing 365 days to the
It is evident, that as niany times as 365 is contained in 31,755, so many years there will be.
I observe that 365 cannot be contained in 317, therefore I must take the four left hand figures, viz. 3175. In order to discover how many times 365 is contained in this, I observe, that 365 is more than 300, and less than 400. I say 300 is contained in 3100, or simply 3 is contained in 31, 10 times, but 365 being greater than 300, cannot be contained in it more than 9 times. Indeed if it were contained more than 9 times, it must have been contained in 317, which is impossible. 400 is contained in 3100, (or 4 in 31) 7 times. This is the limit the other way, for 365 being less than 400, must be contained at least as many times. It is contained therefore 7, or 8, or 9 times. The most probable are 8 and 9. I try 9. But instead of multiplying the whole number 365 by 9, I say 9 times 300 are 2700, or simply 9 times 3 are 27; then subtracting 2700 from 3170, there remains 470; I then say, 9 times 60 is 540, or simply 9 times 6 is 54, which being larger than 470, or 47, shows that the divisor is not contained 9 times. I next try 8 times, and say as before, 8 times 300 are 2400, which subtracted from 3170, leaves 770, then 8 times 60 are 480, which not being so large as 770, shows that the divisor is contained 8 times. I multiply the whole divisor by 8 (which is in fact 80,) the product is 2920. This subtracted from 3175 leaves 255. I then bring down the other 5, which makes the next partial dividend 2555. Now trying as before, I find that 3 is contained 8 times in 25, and 4 is contained 6 times. The limits are 6 and 8. It is probable that 7 is right. I multiply 365 by 7, and it makes 2555, which is exactly the number that I want. If I had wished to try 8, I should have said 8 times 3 are 24, which taken from 25 leaves 1. Then supposing 1 to be placed before the next figure, which is 5, it makes 15. 6 is not contained 8 times in 15, therefore 365 cannot be contained 8 times in 2555. The answer is 87 years.
The method of trying the first figure of the divisor into the first figure, or the first two figures of the partial dividend,
generally enables us to tell what the quotient figure must
Proof 436940074 In this example I seek how many times 6, the first figure of the divisor, is contained in 43, the first two figures on the lent of the dividend ; I find 7 times, and 7 is contained 6 times. The limits are 6 and 7. my times 6 are 42, and 42 from 43 leaves 1, which I suppose placed by the side of 6; this makes 16. But 4, the second figure of the divisor, is not contained 7 times in 16, therefore 6 will be the first figure of the quotient.
It is easy to see that this must be 6000, when the division is completed; because there being five figures in the divisor, and the first figure of the divisor being larger than the first figure of the dividend, we are obliged to take the six first figures of the dividend for the first partial dividend ; and the dividend containing nine figures, the right hand figure of this partial dividend, is in the thousands' place. I write 6 in the quotient, and multiply the divisor by it, and write the result under the dividend, so that the first figure on the right hand may stand under the sixth figure of the dividend, counted from the left, or under the place of thousands. This product, subtracted from the dividend as it stands, leaves a remainder 51518; by the side of this I bring down the next figure of the dividend, which is 0, and the second partial dividend is 515180. Trying as before with the 6, and then with the 4, into the first figures of this partial dividend, I find the divisor is contained in it 8 (800) times. I write 8