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51. Add together, eight thousand and ninety-six; nine thousand and one; seven thousand one hundred and eighteen; seven thousand eight hundred and twenty-five; and thirty-four thousand and forty-nine.

52. Add together sixty-four thousand, one hundred and sixteen; ninety-four thousand, eight hundred and ninety-six; fifty-eight thousand, seven hundred and ninety-four; eight hundred and thirty-six thousand, ond forty-seven; five hundred and forty-eight thousand, nine hundred and ninety-nine; and eleven thousand four hundred and thirty-six.

Let the following Examples be added without placing the umnbers under each other, but, commencing at the right-hand, add the units of each sum together regularly from right to left, placing the first figure of the total at a convenient distance, to make room for a figure or two more than the greatest number of figures in any of the given sums to be added; then adding the tens of the amount of the units thus found, to the tens in the first right-hand row which hath tens, add them as before from right to left, which will give a second figure of the total; thus proceed until all the rows are added. Between the last given number and the total amount, place two small parallel lines thus= which is the sign of equality.

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Should the teacher wish for more examples in this form, let the - pupil dispose of any of the preceding in the same manner and thus add then, which will be another method of proof.

SUBTRACTION.

SUBTRACTion teaches to find the difference between two numbers, by taking the less from the greater. The greater number is called the Minuend, the less the Subtrahend, and the answer or number found, the Remainder. This character—Minus, is used to denote subtraction, and is always placed before the number to be subtracted: thus, 9–4 signifies, that the number 4 is to be subtracted from the number 9; which being done, the result is 5. Here 9 is the minuend, 4 the subtrahend, and 5 the answer or remainder. From the greater of two given numbers of the same denomination, to subtract the less, take the following

- - RULE. , Beginning at the right-hand, subtract each figure of the less number, from its corresponding figure in the greater, and set down the several remainders. When the figure of the subtrahend is equal to the corresponding figure of the minuend, the remainder or difference is nothing, for which a cipher must be set down. Ciphers, when they are all together or single, on the left-hand side of the significant figures, as they do not in such place increase or diminish a number, may be wholly omitted. When the digit to be subtracted, is greater than its corresponding digit in the minuend, add 10 to the latter figure, and . from the sum take the figure to be subtracted; set down the remainder, and carry one to the next figure on the left-hand in the subtrahend, and proceed as before. The remainder added to the subtrahend, will produce a total equal to the minuend, if the work is right; or the remainder subtracted from the minuend, will give another remainder equal to the subtrahend, which affords two methods of proof.

Demonstaarrow.

When the figures in the minuend are all greater than those in the subtrahend, it matters not in what order they are subtracted. They might, in such case, be subtracted from right to left, or in any other order; but if any digit in the minuend be less than its corresponding digit in the subtrahend; for example, if we had to subtract 286 from 564, we cannot subtract 6 units from 4 units; nor 8 tens from 6 tens; we may, in such case suppose the minuend resolved into parts of 400, 150,

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and 14, and then subtract the 6 units from:14, the 8tens from 15. tens, and the 2 hundreds from 4 hundreds; and thus, when any

digit of the minuend is less than its corresponding digit of the

subtrahend conceiving an unit prefixed to it and performing the

subtraction, we may proceed to the next left-hand figure of the

subtrahend, and conceive it to be less by one, on account

of I already borrowed from it: but it affords the same result

in practice, to conceive the next digit of the subtrahend

increased by one, and the digit of the subtrahend unaltered ; as it gives the same remainder to subtract 9 from 16, as to

subtract 8 from 15, and hence appears the reason of what is

called carrying, and proceeding from right to left.*

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* An Example in figures will render the above as plain as can be required. From 475841–300000+ 160000+ 14000+ 1700+ 130+ ll 7ake 296.957-200000+ 90000+ 6000+ 900+ 50+ 7 172884-100000+ 70000+ 8000+ 800+ 80+ 4 See JP'alker's Philosophy of .frithmetic, page 7

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Take 6594601. Remains 870.10886 Proof 65.94601 Črampleg. From 64 - 54 75 382 461 895 Take 32 26 44 121 441 675 From 564 496 935 315 637 401 Take 353 395 742 213 371 220 From 768 325 428 730 2001 5764 Take 209 216 284 417 1090 3280 From 7305 46580 47580 64210 435.46 Take 2802 36594 3594 6429 27438 From 75435 45874 36941 8635 6.7394 Take 67494 36095 21896 7368 9697 From 10000 10000 999 999 501000 Take 1 9999 l 998 50101 From 6000 6000 100000 100000 17534831 Take 5964 36 1009] 89809

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Norf–In compliance with custom, I have placed the figures in the

reater,

above Examples, as is generally practised; but a child should be taught

to subtract a less number from a however placed, whether

above or below, or in any manner, which i have found of advantage.

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• In the 14th Example, the difference is 1 lum signifying, that the total of the sums after subtracted from the total of the three preceding sums.

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00; the stroke or vince: the sign minus, is to be

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