Or, in every case, the multiplicand being reduced to a decimal fraction expressed in terms of the principal unit of that weight or measure to which it belongs, and the multiplier being also reduced, if necessary, to a decimal fraction; the product may be found by the rule for multiplication of decimal fractions. Application of this method to the preceding examples of multiplication of compound numbers : 1st. Multiply 3 cwt. 2 qr. 24 lb. 14. oz. by 8. 3 cwt. 2 qr. 24 lbs. 14 oz. =3.72209821428g cwt. Multiplying by 8 The product =29-77678571428 cwt. This product=29 cwt. 3 qr. 2.999. . . lb. If 1 lb. is made the principal unit, 3 cwt. 2 qr. 14 lb.=416.875 lb. and 416.875 x 8=3335 lb. 2nd. Multiply £5 14s. 74d. by 56. £5 14 7=£5.73020833 3rd. Multiply 5 lb. 8 oz. 14 dr. by 123. 5 lb. 8 oz. 14 dr. -5°5546875 lb. e. Rule for the multiplication of a compound number by a whole number: Multiply the number in the lowest denomination of the mul tiplicand by the multiplier; divide the product by the number of this denomination which make one of the next higher denomination; write the remainder under the lowest denomination and reserve the quotient (which is composed of units of the next denomination) for combination with the product of the next denomination by the multiplier. Multiply the number in the next denomination by the multiplier; to the product add the number of units of this denomination reserved from the preceding reduction; divide the sum by the number of this denomination which make one of the next higher, and continue thus to multiply, reduce, and carry till the products of all the denominations of the given compound number by the multiplier are obtained. When the multiplier is greater than 12, it may be resolved, if possible, into factors, each not exceeding 12: then the continual product of the multiplicand by these factors is equal to the product of the compound multiplicand by the whole multiplier. When the multiplier cannot be resolved into such factors, take the nearest inferior number which can be so resolved ; muld; tiply by the factors of this number, and augment the product by as many repetitions of the multiplicand as this number is less than the multiplier. DIVISION OF COMPOUND NUMBERS. 154. The difference between the operation of division when the dividend is a compound number and when it is a whole number, consists in the different manner of passing from the higher denominations, or higher orders of units, to the lower. In the case of a whole number, and also of the highest denomination of a compound number, the reduction of a remainder to the next lower denomination, and the addition of the number of units of this denomination to the result, are both made by simply annexing the proper figure of the dividend to the remainder. But in passing from the higher to the lower denominations of a dividend which is a compound number, it is necessary to multiply each successive remainder by a factor expressing the number of ones of the next lower denomination which make one of the remainder and to add to this product the number in the lower denomination. As an example, let it be required to divide £47 6s. 74d. by 6. Dividing 47£ by 6, the quotient is 7£, and the remainder 5£. = 5£ 100s. and 100s. rem2 + 6s. of dividend = 106s. Dividing 106s. by 6, the quotient is 17s. and the remainder 4s. 48.48d. and 48d. rem2 +7d. of dividend=55d. Dividing 55d. by 6 the quotient is 9d., and the remainder ld. 1d.=4qr. and 4gr. rem3. + 2qr. of dividend=6qr. Dividing 6qr. by 6, the quotient is 1gr., and the remainder 0. Whence the complete quotient is £7 178. 91d. The calculation may be exhibited in either of the two following forms. With a divisor not greater than 12, the second is commonly adopted. S. d. £ 8. d. 1. F In the first form, the details of the calculation are written at length; in the second, the calculation is made mentally, and the results, only, are recorded. If a large divisor can be resolved into factors, each not greater than 12, division of the dividend and successive quotients by these factors may be substituted for division of the dividend by the whole divisor; as in the following example. Divide 2248 lb. 2 oz. 7 dwt. by 112. The calculation by the undecomposed divisor appears longer, all the steps If a large divisor cannot be decomposed into convenient factors, the division must be made by the whole divisor, as in the second form of division, by 112. a. The product of any multiplicand, by a fractional multiplier, is found by multiplying the multiplicand by the nume rator of the multiplier and dividing the product by its denomi As an illustration, let it be required to multiply £25 7s. 6d. by §. b. Conversely, the quotient of any dividend by a fractional divisor is found by multiplying the dividend by the denominator of the divisor and dividing the product by its numerator. (Art. 99. a.) Let it be required to divide 5 tons 14 cwt. 3 qr. 18 lb. by 3. t. cwt. qr. lb. 5 14 3 18 7 4 1 14 3)40 13 8 0 14 c. If the multiplicand is a compound number and the multiplier a mixed number, the sum of the partial products of the multiplicand by the integral and fractional parts of the multiplier, is the product required: thus, the product of £2 16s. 103d. by 123 is equal to the sum of the two products (£2 16s. 10 d.) × 12 and (£2 16s. 10 d.) ×z. d. If the dividend is a compound number, and the divisor a mixed number; the mixed number being reduced to an improper fraction; if the dividend is multiplied by the denominator of this fraction and the product divided by its numerator, the result is the quotient of the given compound dividend and fractional divisor. If it is required to divide 380 ac. 3 ro. 11 po. 13 jd. by 573; 573288; it is, therefore, necessary to multiply the dividend by 5, and to divide the product by 288, or successively by 4, 6, 12, the factors of 288. |