Division affords several contractions in Multiplication, the most useful of which are as follow: When the multiplier is 5, annex a cipher and divide by 2, When 15, annex a cipher, divide by 2, and add the two results together. When 75, annex two ciphers, divide by 4 and subtract the result. When 35, add two ciphers, divide by 4, and add the quotient to the multiplicand, as if but one cipher was annexed. When 175, add two ciphers, divide by 2, also divide the quotient by 2, and add the three sums together. When 275, add 3 ciphers, and divide by 4, put down the quotient with a cipher less, and add the two sums together. The third of the product of a series of nines will give that of a series of threes, which doubled, will give that of a series sixes, or by subtracting the third of the product of a series of nines from itself, will give the same. The ninth of a series of nines will give that of a series of ones, which subtracted therefrom, will give that of a series of eights. By Multiplication we may also contract several operations in division, the most useful of which, that I have seen or practised, are the following :— To divide by 5, double the dividend and cut off the last figure or cipher, the half of which is the remainder. To divide by 15, 35, 45, or 55, double the dividend and divide the result in the first case by 30, in the second by 70, in the third by 90 and in the last by 110, and in each case take the half of the remainder for the true remainder. To divide by 25, multiply by 4, and cut off 2 figures, one-fourth of the number expressed by them, will be the true remainder.* To divide by 125, multiply by 8, cut off three figures, and take one-eighth of the figures cut off for the remainder. To divide by 75, multiply by 4, and divide by 300, and take one-fourth of the remainder for the true one. There are some others which the student may easily discover, in general they are only the converse of those given for multiplication by division. To the others I think it unnecessary to give examples. This method of dividing by 25, is particularly useful in calculations of Interest, as will be more particularly noticed in another part of this work. ADDITION OF DIVERS DENOMINATIONS, OR, COMPOUND ADDITION. PLACE the numbers to be added, orderly under each other, having due regard to their several denominations. Then add the numbers of the lowest denomination together and divide the sum by the number of parts of this denomination which compose one of the next higher, set down the remainder, under the column added, and add the quotient as so many ones to the next higher denomination; proceed in like manner through all the denominations to the highest, the sum of which set down as in simple addition. When there are two places of figures in any denomination, first add the units of each together, then the tens, as in simple addition; divide the total sum as before directed: the sum of the highest is to be wholly set down, unless there is a higher than is mentioned in the given sums to be added, in which case it will be sometimes best, to divide its sum by the number of parts of this denomination, which make one of the higher, and place the quotient to the left of the other denominations. The method of proof is the same as in simple addition. NOTE-Compound quantities are those which are of the same kind, but of different denominations.-This rule is self evident. OF MONEY. In Ireland, accounts are kept in pounds, shillings, pence, and farthings, or parts of a penny. The pound and shilling in this table are only imaginary, and until of late there was not a penny, às a current coin in Ireland; the farthing, though sometimes seen, is almost wholly in disuse, and bits of lead and other tokens substituted in the few instances where a farthing is required, Beside the foregoing, the Bank of Ireland issues silver tokens, current at 6s 0d-2s 6df0d and 5d, though their real value is little more than two-thirds of what they pass for. *The weight given above is the least which any of those coins should weigh, but a new guinea weighs 5dwts 9grs and a small fraction more; 441 guineas weighing 1b Troy weight. 19 2 220. 11 0 0 240. |