£ 3. d. 7 Exain. 1. Multiply £1 14 73 by 9. In this example the farthings, pence, and shillings are multiplied successively by 9; the products, as they are found, are respectively divided by 4, 12, 20, (or the tens of the shillings by 2;) the several remainders are written down, and the quotients carried. The pounds are multiplied as in Simple Multiplication; and the product is found to be £.5 11 93. £ S. £15 11 92 8. 6 13 4 X9 12. 1 16 7 X 12 Ex.l. 3 14 4. 1 10 4 X 5 Problem 2. To multiply by a number which exceeds 12, but is the product of two or more factors, each less than 13.* £ 3. d. 0 18 31 RULE. (1.) By the preceding problem, multiply the given multiplicand by one of the factors. (2.) Multiply the result by another. (3.) Multiply this last result by another, if there be so many; and thus proceed, whatever is their number. Exam. 2. Multiply 18/3 by 42. In this example the multiplicand is multiplied by 6, and the product is £5 97. This again is multiplied by 7 and the product is £38 7 44. The reason of the operation is sufficiently obvious, since 42 is the product of 6 and 7. The work might be proved by multiplying the multiplicand by 7, and the result by 6. : 5 9 7 7 £38 7 4 £1 13 74 When the multiplicand contains one or more farthings, if one of the factors be even, it is better to use it first, as the farthings may thus disappear, and the rest of the work be easier. But if the multiplicand end in even pence, without farthings, and one of the factors he 3, 6, or 9, it is better to use that factor first, as the pence may thus disappear and in all cases in multiplication of money, when 12 is one of the factors, it should be used first, as part of the operation may be performed by inspection, by setting down 3 pence for each farthing, and carrying to the shillings 1 for every penny in the multiplicand. Thus, in multiplying £1 13 74, by 12, set down 3 pence, and carry 7 to the shillings, saying, 12 times 3 are 36, and 7 are 43, &c. £20 12 3 3 * For the method of finding the factors in the more difficult cases, see page 36, pwał, however, will soon learn to ând them by inspection, in ail useful cases, 101. If a carpenter receive 18/4 52 weeks? Answ. £47 15 6. week, what is his salary in 102. How many pounds are there in 91 guineas? Answ. £95 11 0. 103. What is the amount of the duty on 100 gallons of brandy, at 13/7 gallon? Answ. £67 18 4. 104. What is the duty on 63 gallons of rum, at 10/10 gallon? Answ. £34 2 6. 105. What is the duty on 58 cwt. of raw sugar, at £1 12 6 P cwt.? Answ. £94 5 0. 106. What is the amount of the duty on 149 lbs. of West India coffee, at 74d. It.? Answ. £4 16 23. 107. From 1783 to 1793, both inclusive, the money paid for slaves, imported into the West Indies in Liverpool vessels, was, at an average, £1,380,622 16 41⁄2 each year. What was the entire amount? Answ. £15,186,851 0 11⁄2. It may perhaps be proper to caution learners against the absurdity of attempting to multiply money by money. This caution will not appear unnecessary, if it be considered that whole pages have been filled with instructions how to perform this problem; and it has been attempted to be shown, even with the semblance of geometrical demonstration, that if 2/6 be multiplied by 2/6, the product may be either 32d. or 6/3! Let it be considered, however, that in Multiplication a quantity is simply repeated a given number of times: thus, if 2/6 be repeated 4 times, the amount is 10/; if 5 times, 12/6 &c. To talk, therefore, of multiplying 2/6 by 2/6, or, which is precisely the same, of repeating 2/6 2/6 times, is absolute nonsense. In the Rule of Proportion, indeed, we sometimes appear to multiply such quantities. Thus, in finding the interest of a sum at a given rate, for a year, we multiply by the rate, and divide by 100. In this case, however, both 100 and the rate are divested of their characters as expressions for money, and are merely to be regarded as abstract numbers, used as the terms of a ratio. By multiplying by the rate, suppose 5, we merely repeat the principal 5 times, or find a principal times as great; and then, as there must be one pound of interest for each hundred pounds in this increased principal, we try by Division how often it contains £100; and we thus find the pounds of the interest.-We see from the nature of Division, that there is no absurdity in dividing money by money; that is, in finding how often one sum is contained in another. COMPOUND DIVISION. Problem 1. To divide a number of more denominations than one, by a number not exceeding 12. RULE. (1.) Divide the highest denomination by the given divisor, by Short Division (2.) Reduce the remainder, if there be any, to the denomination next lower, and add to the result what was given of that denomination. (3.) vide the sum by the divisor; and thus proceed to the lowest denomination, or till nothing remains. £ s. d. 10)14 16 74 19 7...7 far. 10 [or 1d. Exam. 1. Divide £14 16 7 by 10. In this example, after dividing £14 by 10, we have remaining £4, or 80 shillings; which, increased by 16, becomes 96 shillings. Hence, we find the next part of the quotient to be 9 shillings, and the remainder is 6 shillings, or 72 pence, which, increased by 7, becomes 79. This being divided by 10, we have the remainder 9 pence, or 36 farthings, to which the odd farthing is annexed; and continuing the division, we find the entire quotient to be £1 972, and the remainder 7 farthings, or lid. The proof is performed as in Simple Division. £14 16 74, proof. 3. 4 5 93÷4 61-÷5 9. 6 ÷10 14. 1 10 08 7 13 10. 5 12 9 ÷11 15. 1 9 39 Problem 2. To divide by a number which is greater than 12, but is the product of two or more factors, each less than 13.. RULE. (1.) Divide the given number, by Short Division, by one of the factors. (2.) Divide the quotient by another factor. (3.) Divide the result thus obtained by another factor, if there be so many: and thus proceed, whatever may be their number. £ S. d. 6)59 13 3 11)9 18 10...2 £0 18 5 Exam. 2. Divide £59 13 3 by 66. In this example the factors are 6 and 11. In the division by 6 the quotient is £9 18 10, and the remainder 2 farthings; and in the division of this quotient by 11, the quotient resulting is 18/03, and the remainder 9. This remainder being multiplied by 6, the first divisor, and the product increased by the former remainder 2, (see page 27,) the true remainder is found to be 56 farthings, or 1/2.-In the proof by Multiplication, this remainder must be added to the final product. 03.56far 6 [or 1/2. 841 11 59 12 1 add. 1 2 £59 13 3, proof In the use of this Rule, if there be no pence in the dividend, or if it end 3, 6, or 9 pence, and if one of the factors be 3, 6, or 12, it is better to use that factor first, as in the division by it there will be no remainder. If one factor be 2, 4, or 8, and there be no farthings in the dividend, it is generally better to begin with that factor. If the shillings and pence of the dividend be any multiple of 10 pence, (as 1/8, 2/6, 3/4, 15/10, &c.) and one of the factors be 5 or 10, it is best to begin with that factor. The same may be observed in relation to multiples of 5d. and 24d. (and, when 5 is a factor, in relation to multiples of 14d. ;) but these are not so easily discovered. Problem 3. To divide by a number which is greater than 12, and is not produced by factors below 13. RULE. The process is to be conducted as in problem 1, except that Long Division is to be employed instead of Short. Exam. 6. Multiply 33 c. 3 q. 22 lbs. by 7.* In this example the pounds, when multiplied, are divided by 28, and the quarters by 4. In long weight the divisors are 3 22 X 86.... cwt. q. lbs. 7 237 2 14, Answ. Answers. cwt. q. lbs. 44 2 11 94. O 95. 2 81 1 16 2 19 (long weight,) × 59......156 3 11 96. 2 27 (long weight,) X 96......165 2 12 97. Required the cost of a chest of tea, containing 97 lbs. at 6/10 ft. Answ. £33 6 104. 98. Required the amount of a box of linen cloth, containing as under: 99. Required the amount of a hogshead of rum containing 66 gallons, at 4/10 gallon, and a puncheon containing 116 gallons, at 4/9 gallon. Answ. £15 19 ̊0, and £27 11 0. 100. What cost a hundred weight of indigo, at 11/4 Answ. £63 14 0. pound? Compound Multiplication is seldom employed, except in relation to money; but if it should be necessary to use it in cases not illustrated here, no difficulty can arise, as the method is similar in vil cases |