Division, like Multiplication, may be proved by casting Cast out the nines of the divisor, placing the surplus on the left hand of the cross; do the same with the quotient, setting down the surplus on the right hand: then multiply these two figures together, adding the amount of the remainder, if any, to the product, and place on the top of the cross the surpluss of nines. Lastly, take the nines out of the dividend, and the surplus written in the bottom of the cross should be the same with the figure on the top. But the best way to ascertain the accuracy of Division, is by reversing the operation, that is, as in the above example, if the divisor and the quotient be multiplied into one another, and the remainder be added to the product, the sum will be equal to the dividend. Compound Division is an operation by which we divide sums composed of numbers of different denominations; in which case that of the highest denomination is divided as in the preceding examples of whole numbers; and if there be a remainder, you must multiply it by the number of units of the next lower denomination, composing an integer of the higher, adding to the product the figures of this second denomination, given in the dividend; proceeding in the same manner, as long as there are any figures, of whatever value, in the dividend. For instance, should it be required same to divide £. 323. 07. 09.. 3 equally among 9 men, the operation would be performed in this way. D. Qrs. £. Sh. D. Qrs கி. Sh. 27. • 53 45 .8 20 9)167(18 9. 77. 72 • 5 12 9)69(7 63 6 Write down the sum to be divided, placing thenumber of men, 9, 、 as a divisor; then dividing, as has been shown, the quotient of pounds will be 35, and the remainder will be 8. These 8 pounds can no more be divided by 9; but if we bring them into shillings, they will be divisible. Multiply therefore the remainder 8 by 20, the shillings in a pound, adding the 7 shillngs in the dividend which are likewise to be shared among the 9 men, and we have 167 shillings, which become a new dividend, to which write 9 as a divisor, and the quotient will be 18, with 5 shillings for a remainder. This 5 must next be brought into pence, by multiplying by 12 the pence in 1 shilling, taking down the 9 pence of the dividend, producing 69 pence, which divided by 9, will give 7 for the quotient, and 6 for the remainder. This 6 being multiplied by 4, the farthings in a penny, to bring the whole into farthings, taking down the 3 of the dividend, will produce 27, which divided by 9, will give 3 for the quotient, without any re.mainder. These 4 quotients, therefore, written out in one line, as in the example, will give £. 35. 18. 07. 3. for the share of each man of the 9, among whom the sum given was to be divided. 9)27(3 27 This species of Division may also be proved, by multi plying the quotient by the divisor, and adding the remain der if any, to the product, thus, In the same manner to divide 837 Tuns, 13 Cwt. 2 Qrs. 27 Lbs. 6 Ozs. by 36. After dividing the tuns by 36, the remainder 9 tuns must be brought to hundreds, by multiplying by 20, and taking down the 15 ewt. of the dividend, which will give 193 cwt. to be divided, as before, by 36: and in this manner the division of the whole quantity given is performed, producing a quotient of tuns 23 .. 05 .. 1 .. 14 .. 07 and a remainder of 26, which being of the last denomination, ounces, will be equal to 1 lb. 10 oz. This remainder might be still brought into drachms; but in articles of such magnitude, this accuracy may safely be disregarded. To prove this division, multiply the quotient by 36, employing such factors as will produce that number, as 6 times 6, 4 times 9, 3 times 12; and to the last product add the value of the last remainder of the division, when, if no error has been committed, the total will be equal to the quantity given to be divided. OF REDUCTION. By Reduction we convert units of one denomination into those of another denomination. When it is required to bring units of a higher denomination into others of a lower, as pounds into shillings, pence, &c. tuns into hundreds, quarters, pounds, &c. the operation is performed by Multiplication, and is called Descending Reduction, as proceeding from a higher to a lower denomination; but when it is required to bring units of a low denomination, into those of a higher, as pence into shillings and pounds, ounces into pounds, quarters, hundreds, &c. the operation is performed by Division, and is called Ascending Reduction. 1st. Reduction by multiplication is performed by multiplying the sum or quantity given, by the number of units of the next lower denomination, constituting one of the higher; adding to it the units of this lower denomination, if any, in the number given to be reduced; and repeating this operation until the whole be brought to the lovest denomination required. Example. How many shillings, pence, and farthings, are in £. 63 .. 15 .. 6? £. Sh. d. 20 1275 Shillings. 12 15306 4 Pence. 61224 Farthings. Writing down the given sum, as in the margin, multiply the 63 pounds by 20, the number of shillings in 1 pound; taking in the 15 shillings of the sum given to be reduced; by which the product comes to be 1273 shillings. This sum is next to be multiplied by 12, the pence in 1 shilling, taking down the 6 pence of the given sum; so that the product will be 15306 pence; which is next to be multiplied by 4, the farthings in 1 penny; but as there are no farthings in the given sum, the simple product of this multiplication, will be 61221 farthings, the number contained in . 63.. 15. 6. Again, reduce the following quantity into ounces; Tuns 85, 3 Cwt. 1 Qr. 17 lb. |