CONTRACTION. When the factors contain many decimal places, the product may be limited to any proposed number of decimals by the following RULE. Write the multiplier inverted under the multiplicand, so that its unit's place may stand directly under that decimal place of the multiplicand, which is of the same denomination as you wish the last figure of the product to be. Then multiply by the figures of the multiplier in their order, beginning always with the figure of the multiplicand which stands directly above the figure by which you are -multiplying; rejecting the figures to the right, only taking in the carriage from them; and place the several lines of products so that their right hand figures may stand directly under each other. Note. To have the last decimal place exact, it is best to compute for two fi gures in the product more than would otherwise be requisite. Or, instead of the regular carriage, you may carry to the right hand figure of each line thus: 1 from 5 to 14; 2 from 15 to 24; 3 from 25 to 34; and so on, inclusively; and the last decimal place will generally be true to the nearest unit. 2. When the multiplier is a decimal, write a cipher for its unit's place, and then proceed as the rule directs. EXAMPLE. Mult. 97-2915837 by 186785, so that the product may contain only 4 decimal places. 1. Multiply 237-157643 by 38-37254, so as to have only 1 decimal place in the product. 2. Multiply 6-827195 by 83.7945, so as to have only 2 decimal places in the product. 3. Multiply ·274683 by 3461965, so as to have only 5 decimal places in the product 4. Multiply 17.514563 by 6.59385, so as to have only 4 decimal places in the product. 5. Multiply 01546968 by 11-7145984, so as to have only 4 decimal places in the product. 6. Multiply 58647473 by 0053948, so as to have only 8 decimal places in the product. DIVISION OF DECIMALS. 1st. To divide when the divisor is not a repetend. RULE. Divide as in whole numbers, and for decimals from the right of the quotient point off as many figures as the decimal places in the dividend exceed those in the divisor; when there are not so many figures in the quotient, supply the defect by prefixing ciphers. Rule. When there are more decimal places in the divisor than in the dividend, annex ciphers to the latter, so as to make the decimal places in both equal, before you begin to divide. 2. When the dividend is exhausted, annex ciphers to the remainder; but when it is a repetend, bring down the repeating figures in their order, and continue the division till the quotient is as exact as is necessary. Remarks. The reason for fixing the decimal point as the rule directs is evident; for since, by the nature of multiplication, the number of decimal places in the dividend is equal to those in the divisor and quotient taken together, the quotient must contain as many decimal places as those in the dividend exceed those in the divisor. 2. The first figure of the quotient is (as in whole numbers) of the same de nomination with that figure of the dividend, which stands over units in the first product. 1. Divide 7.935 by ⚫23 9. Divide 58-415420 by 178-5 10. Divide 358-3 by 472.55 2d. To divide when the divisor is a repetend. RULE. Reduce the divisor to a vulgar fraction; then multiply by the denominator, and divide by the numerator. Note. Since the multiplier is one or more 9s, it is best to multiply as in Rule III. page 18. When there are many figures in the divisor, and but few decimal places are requisite in the quotient, the work may be contracted by the following RULE. Consider how many figures the quotient must contain, and point off as many from the left of the divisor, by which divide; and instead of annexing a figure to the remainder, omit one on the right of the divisor; (only take in the carriage from the figures on the right, as in contracted Multiplication ; )—proceed thus, dropping a figure of the divisor at each division, till it be exhausted. EXAMPLE. Divide 7.623185 by 375465, so as to have only 1 decimal in the quotient. 375,465)7-62,3185(20-3 Ans, 751 11 11 EXERCISES. 1. Divide 15.1275 by 9.813275, so as to have only 2 decimals in the quotient. 2. Divide 291-439765 by 27.65138725, so as to have only 3 de cimals in the quotient. 3. Divide 264-23125 by 8213-7169125, so as to have only 5 deci. mals in the quotient. 4. Divide 351-7 by 4125-6539725, so as to have only 6 decimals in the quotient. 5. Divide 594-3271325 by 29675-362175, so as to have only 7 decimals in the quotient. COMMISSION AND BROKERAGE. Commission and Brokerage are allowances of so much per cent. to Bankers, Agents, or Brokers, for transacting the business of others. RULE. When the rate is under £1 per cent., divide the sum by 100, the quotient is the amount at £1 per cent., of which take aliquot parts for the given rate. 2. When the rate exceeds £1 per cent., either multiply the sum by the rate per cent., and divide the product by 100, or compute by taking aliquot parts of 100 for the rate. EXAMPLE. Find the commission on £484: 12: 6, at 24 per cent. 1. £573: 16:10 at EXERCISES. Find the brokerage on per cent. 3. £412:6:8 at per cent. 2. 756:19:84 at per cent. 4. 1219:15: 6 at 3 per cent. Interest is an allowance of so much per cent. given for the loan or use of a sum of money; and it is always proportional to the time. CASE I. To find the interest of any sum for any number of years. RULE. Find the interest for a year, as in Commission, zohich multiply by the given number of years. EXAMPLES. Find the interest of £850 for 4 years, at 5 p. cent. and at 4 p. cent. 2,0)85,0 0 0 £8.50 4 34 4 years, £136 at 4 per cent. Remarks. The sum lent is called the principal; and the sum of principal and. interest is called the amount. 2. To check the power of accumulating wealth without industry, and to give encouragement to trade, by enabling adventurers in it to borrow at a moderate rate, it is, in most countries, established by law, that no individual shall demand above a certain rate of interest, which rate is called the legal interest. In Britain, the legal rate is 5 per cent.; in Ireland and America, it is 6 per cent.; in India, 12 per cent. On the continent of Europe, the interest of money is not limited, but rises and falls according to the demand for money in the market Interest above what the law allows is now called usury; but formerly interest and usury were synonymous.. 3. Our ancient legislators, misunderstanding the spirit of the Mosaic law, pronounced it criminal to receive interest for money: all such transactions are styled by Henry VII. " dampnable bargains;" in an act (1571) fixing the legal interest at 10 per cent. it is declared, that "all interest, being forbidden by the law of God, is sin and detestable;" and in another act (1624) reducing the legal rate to 8 per cent., it is declared, that "no words, in this law contained, shall be construed to allow the practice of receiving interest, in point of religion or conscience." It is true, that when a Hebrew lent money to one of his countrymen, he was not allowed to take interest for it; and, among a pastoral and agricultural people, that law was equitable and humane; for the money borrowed was not employed to gain more, but to supply the pressing wants of the borrower. Among the Hebrews, none but a poor man would borrow; and his poverty, it was reasonably presumed, rendered him incapable of paying interest. In a commercial state, the case is very different. There, money is often borrowed by the rich, to be employed in enterprises which yield great profits. In this case, nothing is more reasonable than that the lender should receive a share of the profit made by the borrower by means of his money.. |