11. What is the method of proof in each ? 12. How would you proceed to multiply by 5? to divide by 5? 13. What is meant by Reduction of Decimals? 14. How would you proceed to find the value of a decimal in integers of a lower denomination? how to reduce compound numbers to decimals of a higher denomination? 15. How many days are commonly reckoned to a month, in computing interest? (145) How are days and months reduced to a decimal of a year? 16. What is Compound Addition ? -the Rule?-Proof? 17. What is compound Subtraction -the Rule ?-Proof? 18. If you wish to subtract one date from another, how would you proceed? (152) 19. What is Compound Multiplication ?-the Rule? What is Compound Division ?-the Rule What relation have these two rules to each other? Of what contractions are these rules susceptible? 20. What are the several contractions of Simple Multiplication? (90, 91, 92, 93,) of Division? (108, 109, 110, 111.) 21. What is meant by a simple number? What is the distinction between a simple and a compound? 22. How would you proceed to take quantities of several denominations, each an equal number of times, from a given quantity? SECTION V. PER CENT. 161. Per Cent. is a contraction of per centum, Latin, signifying by the hundred, and implies that calculations are made by the hundred. Per Annum signifies by the year. Interest. ANALYSIS. 162. If I lend a neighbor 25 dollars for one year, and he allow me 6 cents for the use of each dollar, or 100 cents, how much must he pay me in the whole at the end of the year? 25 .06 1.50 25. 26.50 If he pay 6 cents .06 of a dollar (132) for the use of 100 cts. or 1 dollar, he must evidently pay 25 times .06, or (86) .06 times 25-$1.50 for the use of 25 dollars. Hence, 25+1.50 26.50 is the sum due me at the end of the year. The $25 is called the principal, the .06 is called the rate per cent., the $1.50 is called the interest, and the $26.50 is called the amount. Hence the following DEFINITIONS. 163. Interest is a premium allowed for the use of money. The sum of money upon interest is called the principal. The rate is the per cent. per annum agreed on, or the interest of one dollar for one year, expressed decimally. The principal and interest added together are called the amount. Interest is of two kinds, Simple and Compound. 164. The rate per cent. is expressed in hundredths of a dollar. Decimals in the rate below hundredths are parts of one per cent. The rate of interest is generally established by law. In New-England legal interest is 6 per cent., in New-York 7 per cent., and in England 5 per cent. Where the rate is not mentioned in this work, 6 per cent. is understood. SIMPLE INTEREST. 165. Simple Interest is that which is computed on the principal only. FIRST METHOD. ANALYSIS. 166. 1. What is the interest of $38.12 for 2 years, 8 months and 21 per cent. per annum? days, at 6 $38.12 $2.2872 114360 45744 160 104 45744 $6.2326200 Multiplying the principal by the rate gives the interest for one year, (162) and the interest for one year multiplied by the number of years, is evidently the interest for the whole time. Twenty-one days are of a month 0.7, and 8 mo. 21d. 8.7 mo. But months are 12ths of a year, hence 8.7m. mo. 0.725 year (142), and 2yr. 8mo. 21d.2.725 years; we therefore multiply 2.2372, the interest for one year, by 2.725, the number of years, and the produci, $6.232, is the interest for the whole time. Hence, 167. To compute the interest on any sum for any time. RULE.-Multiply the principal by the rate expressed as a decimal of a dollar, and the product will be the interest for one year. Multiply the interest thus found by the number of years, reducing the months and days, if any, to the decimal of a year) (145) and the product properly pointed (106, 116) will be the interest required. NOTE. In solving the following questions, the decimal of a year, when has not terminated sooner, has been carried to four places of figures, and that will give the interest sufficiently correct for common practice. When great accuracy is required, find the number of days in the given months and days, and divide these by 365, the number of days in a year, and the quotient will be the true decimal of a year. QUESTIONS FOR PRACTICE. 2. What is the amount of $175.62 for one year and six | months, at 6 per cent.? 175.62 prin. .06 rate. 10.5372 one yr. int. The decimals 526860 below mills are 105372 omitted in the 10. What is the interest of £86 10s. 4d. for 1 year and 6 months, at 6 per cent.? 86.5166 If the principal .06 be English money, the shillings, pence, €5.190996 &c. must be reduc1.5 ed to the decimal of a pound, (143), 25954980 then proceed as in 5190996 Federal money. The interest will be Ans. £7.7864940 in pounds and decimal parts, which must be re Int. 15.80580 and the follow-duced to shillings, &c. (144). SECOND METHOD. ANALYSIS. 168. 1. What is the interest of $60, for 5 months and 21 days, at 12 per cent. per annum? 60 prin. .057 rate. If the interest of $1 be 12 cents for 12 months, the interest of $1 for 1 month will be 1 cent, for 2 months 2 cents, for 3 months 3 cents-and generally the number of months written as so many cents, or hundredths of a dollar, will be the interest for that time. And as the interest of $1 for 1mo. (30 days) is 1 cent, the interest for any number of days is so many 30ths of a cent, or 3ds of a mill. In the present example we write the 5 months as so many cents, or hundredths of a dollar, and dividing the days by 3, find of them to be 7, which we write in the place of mills in the multiplier; and $60 multiplied by $0.057, (the interest of $1 for the given time,) the product, $3.42, is evidently the interest of $60 for that time. 4.20 300 $3.420 Ans. 169. 2. What is the interest of $60 for 5 months and 21 days, at 6 per cent. per annum? Since interest at 12 per cent. (168) is found by multiplying by the whole number of months and of the days, interest at 6 per cent. being of 12, may evidently be found by multiplying by half the former multiplier, that is, by half the months written as cents, and one sixth of the days written 2) 60 .028 480 120 30 at the right hand. In the present example, half the months is 2, and if there were no odd days, we should write down 2 cents, 5 mills, or 0.025 for the multiplier; but when there is an odd month and days, as in the present case, it is as well to call the odd month 30 days, and adding thereto the odd days, divide the whole by 6, the quotient (30+21-68) will be mills. $0.028 then is the interest of $1 for 5 months 21 days, and 60 times $0.028, or $0.028 times 60, (86) $1.71, is the interest of $60 for the same time. To multiply 60 by, we take of 60, or divide 60 by 2, and in general for the odd days, less than 6, we take such part of the multiplicand as the odd days are part of 6. Hence, $1.710 Ans. 170. To compute the interest at 6 per cent. per annum upon any sum for any time. RULE. Under the principal write half the even number of months, for a multiplier, (pointing them as so many cents, or hundredths of a dollar.) If there be an odd month, call it 30 days, to which add the odd days, if any, and, dividing them by 6, write the quotient in the place of mills in the multiplier. Multiply the principal by this multiplier, and the product, properly pointed, (122) will be the interest for the given time. NOTE.-Odd days less than 6 are so many 6ths of a mill, and to multiply by these, proceed as follows: and add the quotient, or quotients, to the product of the principal by half the months. QUESTIONS FOR PRACTICE. 3. What is the interest of $75, for 4 months and 2 days, at 6 per cent.? 3)751 Here the months is .020.02, and as 6 is not con tained in the days, we 1500 write a cipher in the 25 place of mills, that the quotient, in dividing by Ans. $1.525 3, may fall in its proper place. There being 3 decimal places in the factors, there must be 3 pointed off in the product. 4. What is the interest of $215 for 1 month and 15 days? 1mo. 15d.—45d.; 6 in 45, 7 times and 3 over. 2) 215 As there is no even .007 1505 107 number of months, the two first decimal places must be supplied with ciphers, and 7 must take the place of mills. The use of the ciphers is to guide us in pointing the product. Ans. $1.612 |