plying the quotient by the divisor, and adding the remainder if any, to the product, thus, In the same manner to divide 837 Tuns, 13 Cwt. 2 Qrs. 17 Lbs. 6 Ozs. by 36. T. Cwt. Qrs. lbs. 36)837 13 2 72 117 17 6. (23 .. 05 1 14 . 07 20 36 Rem. 15.. 12 1 10 After dividing the tuns by 36, the remainder 9 tuns must be brought to hundreds, by multiplying by 20, and taking down the 13 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; of such magnitude, this accuracy may safely be disregarded. but in articles 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 lowest denomination required. Example. How many shillings, pence, and farthings, are in £. 63 .. 15 .. 6? 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 1275 shillings. This sum is next to be multiplied by 12, the pence in 1 shilling, taking down the 6. L. Sh. d.. 63.. 15 .. 6 20 1275 Shillings. 12 15306 Pence. 61224 Farthings. 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 61224 farthings, the number contained in £. 63.. 15 .. 6. Again, reduce the following quantity into ounces; Tuns 85, 3 Cwt, 1 Qr. 17 lb. 2d. Reduction by Division is performed by dividing the given number of units by the number constituting an unit of the next higher denomination; observing that the remainder, if any, is always of the same nature with the dividend. For example, let it be required to reduce the number of farthings found by the reduction of £. 63.. 15 6 in the former example, back into pence, shillings, pence remaining, which number is placed as in the example here given: Lastly, divide the shillings thus found, by 20, the number of shillings in 1 pound, when the remainder will be 15 shillings; and the result of this Reduction will be, that 61224 farthings are equal to 15306 pence, 1275 shillings, or £ 63 .. 15 .. 6. Again, agreeably to the 2d. example, before given, reduce 3052496 ounces, back into pounds, quarters, hundreds, and tuns. Dividing the ounces by 16, the ounces in 1 pound, we have a quotient of 190781 pounds, without any remainder : then dividing this number by 28, the pounds in 1 quarter (writing the divisor 28 over the beginning of the dividend, as in this example) we have a quotient of 6813 quarters, and a remainder of 17 pounds: again dividing this number of quarters by 4, those in 1 hundred-weight, we have 1703 hundreds and 1 quarter over: lastly, dividing these hundreds by 20, the number in 1 tun, we have 85 tuns and 3 qrs. over: hence the whole quantity obtained by this Reduction will be 85 tuns, 3 hundreds, 1 quarter, 17 pounds. From these examples we may observe that the two kinds of Reduction are mutually checks upon or proofs of each other. It often happens however in reducing a number of units of one denomination into those of another, whether higher or lower, that both multiplication and division are to be employed. This is the case when the one unit does not contain an entire number of the other, as when it is required to reduce guineas to pounds, or pounds to guineas: for example, if a person's pay be one guinea per day, what is his income in a year, counting 365 days? result comes to be £383 .. 5, equal to 365 guineas. Or to reverse the case, how many guineas are there in £ 383 .. 5? Here |