11. Sold 14 pairs of stockings, the first at 4 cents, the second al 12 cents, and so on in a geometrical progression ; what did the last pair bring him, and what did the whole bring him ? Ans. last, D63772.92 ; whole, D95659.36. 12. If our ancestors who landed at Plymouth, A. D. 1620, being 101 in number, had increased so as to double their number in every 20 years, how great would have been their population at the end of 1840 ? Ans. 206747. 13. A sum of money is to be divided among 10 persons ; the first to have D10, the second, D30, and so on, is a threefold proportion ; what will the last have? Ans D196830. 14. A man bought a horse, and by agreement was to give lc. for the first nail, 2c. for the second, 4c. for the third, &c. ; there were 4 shoes, and 8 nails in each shoe ; what was the cost of the horse ? Ans. D42949672.95. REVIEW What is Geometrical Progression ? How do you form a geometrical progression? What is the common ratio ? What is the difference between arithmetical and geometrical progression? What is an ascending series? What is a descending series? What are the several numbers called ? In every geo. - metrical progression how many things are to be considered? What are they? What are the first and last terms called ? What are the intermediate terms called ? 15. If the ratio be 4, the number of terms 6, and the greatest term 3072; what is the sum of the series ? Ans. 4095. Divide the last term by the 5th power of the ratio, &c. PERMUTATION. Permutation is the method of finding how many different ways any number of things may be changed. Thus take the first three letters of the alphabet, a b c; they will admit of six changes, a b c, a c b, b a c, b c a, c b a, c a b, and so on, according to the given number of terms. RULE. Multiply all the terms of the natural series constantly from 1, or unity, to the given number, inclusive; the last product will he the number of changes required. 1. In how many different positions can five persons be placed at a table ? Thus : 1 X2 X3 X4 X5=120. Ans. 2. What time will it require for 8 persons to seat themselves differently every day at dinner? Ans. 110 years, 142 days. 3. How many variations can be made of the English alphabet, it consisting of 26 letters? Ans. 403291461126605635584000000. 4. How many changes may be rung on 15 bells ; and in what time may they be rung, allowing three seconds to every round? Ans. 1307674368000 changes ; 3923023104000 seconds. 5. How many variations may there be in the position of the nine digits ? Ans. 362880. 6. A man bought 25 cows, agreeing to pay for them 1 cent different order in which they could all be placed ; how much did the cows cost him? Ans. D155112100433309859840000. 7. Christ's church, in Boston, has & bells; how many changes may be rung upon them. Ans. 40320. for every COMBINATION. COMBINATION teaches how many different ways a less number of things may be combined out of a larger; thụs out of the letters a b c d, are six different combinations of two, namely, a b, a c, a d, d c, d b, b c. Thus : 4x3=12; 1x2=2)12(=6. Ans. RULE, Take a series, proceeding from and increasing by a unit, up to the number to be combined ; then take a series of as many places, decreasing by unity, from the number out of which the combinations are to be made ; multiply the first continually for a divisor, and the other for a dividend; the quotient will be the nswer. 1. How many combinations of five letters in ten? Thus : 10 X 9 X8*7*6=30240, dividend ; 1x2x3x4x5 =120, divisor. Ans. 252. 2. How many combinations of ten figures may be made out of twenty? Ans. 184756, 3. How many combinations may be made of seven letters out of twelve. Ans. 792. 4. How many combinations can be made of six letters out of The 24 letters of the alphabet ? Ans. 134596. REVIEW. What is Permutation? What is the rule? What is combination? What is the rule ? '5. How many changes may be rung with 4 bells out of 8 ? Ans. 1680. 6. How many variations may be made of the letters in the word Zaphnathpaaneah (15) ? 2x6x1x2x120=2880 divi Ans. 454053600. 7. How many different numbers can be made of the following figures : 1223334444 ? Ans. 12600. sor. COMPOUND INTEREST, BY DECIMALS. COMPOUND INTEREST is that which arises from interest and principal added together annually, as the interest becomes due, by the continued multiplication of the new principal by the ratic or rate per cent. ; thus, if I owe A. D100, payable on demand, and neglect to pay either interest or principal for several years, he would be justified in adding the interest to the principal annually, and computing the interest on this amount: D100+6 =106, first year; then D106 x6=6.36 interest second year, which 16.00+6.36D.=12.36 interest or D112.36 amount for the second year; D112.36 x 6=6.74 interest for the third year, which 16.00+6.36+6.74=19.10 interest for the third year, or D119.10.160 amount at compound interest for three years at 6 per cent. In many cases it is considered illegal to receive compound interest, therefore it is seldom computed; but when a note, bond, or obligation, is given with a credit of several years, on condition that the interest shall be paid annually, if the interest is not paid until the obligation becomes due, it is no more than just and right that compound interest should be paid, as well as principal; for the person having the use of the princi. pal has likewise the use of the interest, which of right belongs io another; therefore interest should be computed accordingly. The following table is computed by the continual multiplication of 1, or D100 by the ratio as above; which may sometimes be found useful where the higher powers are required, or o test the accuracy of calculations. TABLE I. A table showing the amount of Di, or D100, for :5 years at 4, 5, 6, and 7 per cent. at compound interest. Years. 4 per cent. 5 per cent. 6 per cent. 7 per cent. A table showing the amount of D1 or D100 for one year, and for quarters of a year, at compound interest, and simple interest for one month. To compute interest by the use of the tables, find the amount or tabular number, under the rate per cent., and opposite the number of years; multiply this number by the principal, and you have the amount, from which subtract the principal, and the remainder will be the compound interest. If quarterly payments are required at compound the 4th root of the ratio ; for half-yearly, the square root; and for three quarters, the product of the quarterly and half yearly; thus, the tabular number at 6 per cent., will be for one quarter 1.014674; for two quarters, 1.029536; for 3 quarters, 1.044671, at 6 per cent. Or find the powers by the tables, which multiply together, and the product will be the amount; thus, for 4.25 years at 6 per cent. per annum, 1.2624769 x 1.014674=1.2810024860306 amount of D100, &c., or D28.10 interest for 44 years. It may be found very nearly by finding the compound interest for the years, and then compute the simple interest on that amount for quarterly or parts of a year. Again, 1/1.06=1.014674 quarterly amount : and /1.06= 1.029536 half-yearly amount: then, 1.014674 X 1.029536= 1.044671 : then, for 3 years at 6 per cent., 1.191016 x 1.0114674=1.2184929, amount for 37 yrs. at 6 per ct. per ann. 1. What is the compound interest of D100 for 4 years, at 6 per cent. per annum ? By the table, 1.2624769 x 100= 1.262476900-1.00=126.24,7,6900. Ans. 2. What is the amount of D1500 for 12 years at 3.5 per ct. per annum? Thus, tabular number, 1.5110686 X 1500= D2266.60.29. (The tabular number is not in the table.) Multiply the principal by the ratio, which will give the simple interest for 1 year; add this interest to the principal and calculaté as before for the second year, and so continue up to the number of years required, and the last product will be the amount, from which subtract the principal and the remainder will be the compound interest. 3. What is the compound interest of D100 for 4 years at 6 per cent. per annum ? Thus, D100 principal, D100 x 6 ratio= 6 interest 1st year, RULE Il. 126.24.7.696 amount 4th year. Principal -100 D26.24,7.696. Ans. |