REDUCTION. DEFINITION. REDUCTION is a rule which shows how to convert num bers from one denomination to another, still retaining the same value, and is either ascending or descending. PROBLEM IX. To reduce a number of any kind into another, still retaining the same value, whether ascending or descending. I. In reduction descending, multiply the highest denomination by as many as will make one of the next, and to the product add the next, if any, and proceed in the same manner from one denomination to another, until you arrive at that sought, observing to add to each product, as you proceed, those of the same with itself. II. In reduction ascending, divide the given number by as many as make one of the next higher denomination, and that quotient by as many as make one of the next, &c. till you arrive at the denomination required; then the last quotient, with all the remainders annexed to it, will be the answer. In 141. 15s. 10d. how many shillings, pence, and farthings? In 14202 farthings, how many pence, shillings, and pounds? farthings. 4)14202 12) 3550 2'0) 29'5 10 Answer 14 15-10 EXAMPLE III. In 18cwt. 3qrs. 26lb. 15oz. (avoirdupoise weight), how many quarters, pounds, and ounces? EXAMPLE IV. In 34031 ounces (avoirdupoise weight), how many pounds, quarters, and hundred weights? WHEN four numbers are compared together, if the first be the same part, or parts, of the second, as the third is of the fourth, then the four numbers are said to be proportional, and are generally expressed, as the first is to the second, so is the third to the fourth: thus, 2, 6, 3, 9, are proportional numbers, for 2 is contained in 6 thrice, and 3 is contained in 9 thrice. Corollary. Hence if four numbers are proportional, the quotient arising by dividing the second by the first, is equal to the quotient arising by dividing the fourth by the third. II. In any proportion (2, 4, 6, 12), the two outside terms (2, 12) are called the extremes, and the two middle ones (4, 6), are called the means. NOTATION. The signs of proportion are marked thus,,, which being placed in the following manner, 283 12, signifies that the four terms, or numbers, are proportional; that is, 2 is to 8, as 3 is to 12. PROBLEM X. Three numbers being given, to find a fourth proportional. I. Place the term that is of the same kind with that which asks the question, first; that which is of the same kind with the term sought for, the second; and that which asks the question, the third: then say, if the first term given requires the second, what will the third require? II. Multiply the second and third together, and divide the product by the first, and the quotient is the fourth term, or answer, which must be reduced as required. Note. The first and third terms must be reduced to the same denomination. To prove proportion, multiply the quotient by the first term, and the product will be equal to the dividend, or the two means multiplied together. METHOD I. £ s. d. 20 EXAMPLE I. If 15 feet of wood cost 11. 16s. 6d. what will 25 feet cost? METHOD II. by Compound Multiplication, &c. £ s. d. 15 1 16 6 :: 25 X5 In Method I. the middle number being reduced to pence, the answer will also be in pence, and therefore it is reduced by reduction into pounds, shillings, and pence, as it will admit of them. In an example of this kind, where the terms can be resolved into simple parts, compound multiplication, &c. is the quickest way, as is shown in Method II.; but when the divisor cannot be separated into simple parts, I would prefer the common way, as is shown by Method F. |