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Division of divers denominations is performed by dividing each denomination in the dividend by the divisor, placing the quotient of each denomination under the same denomi. nation in the quotient, and carrying the overplus of each denomination to be added to the next, as follows:
Divide 4371, 195, 10d, among 24 men,
In this example, I first divide the 4371. by 24, the quotient of which is 18l. and there remains 5ł, whioh, added to the 195, makes 1 19s. for a new dividend; then the divi is coutained 4. times in 119, wherefore I set down 4s, in the quotient, and 234. remain, which added to the 19d, make 2860. for the next dividend; in which the divisor is contained 11 times, and there remains 22d, or 88 farthings, in which the divisor is coutained 3 times. Thus the quotient is 181, 45.11.de and there remains 16 farthings.
Sums of this nature may also be more expeditiously wrought, if the divisor be a commensurable number, and can be refolved into two parts, as in the following example :
Qu. 1. If the expense of a country feast be 2631. jos. 6cho to be paid by 28 stewards, what must each steward pay?this example I divide by 4, and the quotient thence ariting by 7, which is equal to dividing by. 28, and the answer is found to be gl. 8s. 2d, for each fteward to pay.
24. 2. If a gentleman spend 3471. 155. gdh. in the space of; one year and eight weeks, it is desired to know how much it is per week on an average ?-Here I-divide the whole sum by 5, and the quotient thence arising by 6, and that quotient by 2; which is equal to dividing by 60 (the number of weeks in one year and eight weeks), and the answer is 5h 155. vid. per week, as in the example.
Qu. 3. If the capital stock of 4 tonţine amount to 43721. 145. od. and there be in it 160 shares, what will be the amount of each fhare? - In this example:I-divide by 4, and that quotient by 10, and the quotient thence arifing by 40 which is equal to dividing by 160; as 4 times. 10 is 40, and 4 times, 40 is 160, and the answer is 271. 6s7d. for each fliare.
To find the exact remainder in sums where there are two or more divisors, as in the foregoing ones, the rule is to mule tiply the firft divifor by the last remainder, adding thereto the first remainder, if any, and the product will be the true re. mainder; as if it had been divided by the long method : thus in the first of the foregoing examples, I multiply 4, the first divisor, by 5, the last remainder, which produces 20, to which adding 2, the firft remainder, the true remainder is found to be 22, which may be proved at leisure.
Qu. 4. There is a piece of land, having 4 sides, containing 1398 acres, 3"roods, 35 perches, and 240 feet in breadth, it is defired to know how many feet it is in length?
Qu. 5. There is a piece of timber, the solid contents of which is 600 feet, its' length is 40 feet, and its depth 3 feet; it is required to know its fuperficial contents ?
In the fourth question I divide the contents of the land, Sirft by 4, and that quotient by 6, and the next quotient hy 10, which is the fame as dividing at once by 240; and the answer is found to be 5 acres, 3 roods, and 12 perches, for the length of the piece of land.
In the last question I divide the solid contents of the piece of timber by 3, the depth, and the quotient 200 feet, is the fuperficial contents, which if divided again by 40 feet, the length, would give 5 feet for the breadth.
In the same manner as the foregoing examples are wrought division of other denominations may be performed, having respect to the table of quantity belonging to the fame.
But in this species of division, is the divisor be not a commensurable number, or one which cannot be divided into parts exactly, the divifion must then be performed by one divisor.
Division also teacheth how to bring small denominations into great ones; but as this part more properly belongs to Reduction, I have deferred treating of it till I come to that rule.
Reduction is only the application of the rules of multiplication and division, and teacheth how to bring numbers of one denomination into another denomination without altering their value.
Reduction is either defcending or afcending. Reduction de scending is performed by multiplication, and serves to bring great denominations into small ones; as pounds into thillings, pence, or farthings; hundred-weights into pounds or ounces, &c. Reduction ascending is perforined by division, and brings small denominations into great ones; as farthings into pence, thillings, or pounds; drams or ounces into pounds, hundred weights, &c.
Rule. In reduction descending, multiply the number by the number of units of the next lower denomination which make an unit of the next greater, and multiply such product by the number of units of the next lower denomination which make one of the next greater; and proceed in this manner till the number be reduced to the denomination required.
EXAMPLE 1. Reduce 250l. os. od. into farthings.
250 multiply the 250 by 20, Shillings in 1 pound which is the number of units
Shillings in 2 gol. 5000
Pence in 1 shilling of the next lower denomina.
Pence in 250l. 60000 tion which make an unit of
Farthings in i penny the next higher; that is, the number of thillings contained in a pound, and the product shews the number of shillings contained in 250l. ; which product must be again multiplied by 12, the number of units of the next lower denomination which nake one of the next greater, or the number of pence contained in one thilling, and the product gives the number
of pence contained in 2 gols; which product, again multiplied by 4, the number of farthings in one penny, the product gives the number of farthings contained in 2501.; or the answer.
Rule. In reduction afcending, divide the given number by the rrumber of units of that denomination which make one of the next greater ; and divide that quotient by the number of units of the famie denomination which make one unit of the next biglier, and proceed in this manner till the whole is finished.
Thus, as in the foregoing example I reduced 2 50l, into 240000 farthings ; lo inversely, here I say, in 240000 farthings, how many pounds ?
Farthings in i peony 41240000
In this example, I first divide the given number of farthings by fy the number of units of that denomination which are contained in an unit of the next higher denomination; or the riumber of farthings contained in one penny, and the quotient gives the number of pence contained in 240000 farthings; which quotient is again divided by 12, the number of pence contained in one shilling, and that quotient gives the number of Ahillings contained in the given sum of farthings; and lastly, these thillings are again divided by 20, the number of shillings in one pound, and the quotient is 250l. for the answer,
*Thus it may be seen, that reduction ascending and descending prove each other. For if the sum be performed by reduction descending, it must be proved by reduction ascending, as in the two foregoing examples; and if it be in reduction afcending, it must be proved by reduction descending.
In reduction descending, when the fum consists of several denominations, the number in each denomination, after the first, is to be added to the denomination to which it belongs; as in the following example :