payable yearly for a limited period, but not to commence until after the expiration of a certain time. RULE. Find the present worth, to commence immediately, and this sum, divided by the power of the ratio denoted by the time in reversion, will give the answer. 10. What is the present worth of a reversion of a lease of $40 per annum, to continue 6 years, but not to commence until the end of three years, allowing 6 per cent. to the purchaser ? Present worth, 196.69280 Third power of the ratio, 1.19101 =165.147. Answer, $165.147. The same result may be obtained by finding the present worth of the annuity to commence immediately, and to continue the whole time. Thus 3+6=9 years, and from the present worth for this time subtract the present worth of the annuity for the time of reversion, 3 years. Or, by the table, find the present worth of $1 for the whole time; from the sum subtract the present worth of $1 for the time of reversion, and multiply the difference by the given annuity. Thus : The whole time, The time of reversion, 6.80169 2.67301 $165.14720, Answer. 11. What is the present worth of $50, payable yearly for 4 years, but not to commence until 2 years, at 6 per cent.? Answer, $154.1965. 12. What is the present worth of the reversion of a lease of $70 per annum, to continue 20 years, but not to commence until the end of 8 years, allowing 6 per cent. to the purchaser ? Answer, $503.7459. 13. What is the present worth of a lease of $200, to continue 30 years, but not to commence until the end of 10 years, allowing 6 per cent? Answer, $1513.264. QUESTIONS. 8. Rule for finding the present worth of an annuity to continue forever? 9. Rule for finding the present worth of a freehold estate in reversion at compound interest? To find the present worth of an annuity to continue forever. Divide the annuity by the rate per cent., and the quotient will be the present worth. 14. What is the present worth of a freehold estate whose yearly rent is $60, allowing 6 per cent. to the purchaser ? It is evident, that the estate is worth as much money, as, at the given rate per cent., would give interest equal to the rent. 15. What is $300 annuity worth, to continue forever, allowing 5 per cent, to the purchaser? Answer, $6000. To find the present worth of a freehold estate, in reversion at compound interest. RULE. Find the value, as though it were to be entered on immediately, by the foregoing rule, and divide this value by that power of the ratio denoted by the time of reversion, and the quotient will be the present worth of the estate in reversion. 16. Suppose a freehold estate, of $48 per annum, to commence two years hence, be put on sale. What is the value, allowing 6 per cent. to the purchaser ? 48 $711.997, Answer. 17. Which is the more valuable, a term of 16 years, in an estate of $100 per annum, or the reversion of such an estate for ever after 16 years, computing at the rate of 5 per cent., compound interest? Answer, The term of 16 years by $167.551+ PERMUTATION. PERMUTATION is the method of finding how many changes may be made upon the order of any given number of things? 1. How many changes can be made of the first three letters of the alphabet? The letter a can occupy but one position; a and b can change places, and occupy 2 positions, ab and ba; 1X2 2. The three letters, a, b and c, can, any two of them, leaving out the third, have two positions 1×2=2, consequently, when the third is taken in, there can be 1×2×3 6 positions which may be thus expressed. Thus, abc, acb, bac, bca, cba, cab. The same may be shown of any number of things. Hence, to find the number of changes which can be made of any given number of different things: او RULE. Multiply all the terms of the natural series of numbers, from 1 up to the given number, continually together, and the last product will be the answer required. 2. Christ Church, in Boston, has 8 bells. How many changes may be rung upon them? 1×2×3×4×5×6×7×8=40320, Answer. 3. Six men met at a public house, and agreed to remain so long as they could occupy different situations at the dinner table. How long did they remain, and what was the price of their board, at 25 cents for each dinner? POSITION teaches to find the true number by the use of false, or supposed, numbers. It is of two kinds-Single and Double. Single Position is so called, because the true number is obtained by the use of one supposed number. 1. A, B and C travelled. C paid a certain part of the expense; B paid double, and A treble the sum which C paid. The amount of their expenses was $60. What did each one pay? Suppose C's expense was $8; then, by the conditions of the question, B's expense 8X2-$16; and A's 8X3=$24; and the sum of their expenses $8+$16+$24=$48. As the ratios, in the true and supposed, are the same, it fol QUESTIONS. 1. What is permutation of quantities? 2. Rule for finding the number of permutations? 3. What is Position? 4. What is Single Position? lows, that the true sum of their expenses will have the same ratio to the true expense of each individual, that the sum of their supposed expenses has to the supposed expenses of each individual. Thus: 48: 860: 10 C's expense; and RULE. Suppose any number, and proceed in the operation as though it were the true; then, as the result of the operation, or sum of the errors, is to the supposed number, so is the given number to the true number required. EXAMPLES. 2. A person, after spending and of his income, had $30 left. What had he at first? Then 10: 60:30: 180, Answer. Or by fractions: =, and ; then +, the income spent, and remains $30; then =30×6=$180, as before. 3. A certain sum of money is to be divided between 5 men, in such a manner that A shall have 4, B, C, D , and E the remainder, which is $40. What is the sum? Answer, $100. 4. A schoolmaster being asked how many scholars he had, replied, if he had as many more, and as many more, he would have 11 less than 99. How many had he? Answer, 32. Answer, $108. 5. A man bought a horse, chaise and harness for $216. The horse cost twice as much as the harness, and the harness one third as much as the chaise. What was the cost of the chaise ? 6. What number is that whose 1, §, 1, 1, 1 and make 127? Answer, 90. 7. A man being asked his age, said, If you add to its double,,and of my age, it will be 122. What was his age? Answer, 45. 8. A certain sum of money is to be divided among 4 persons, in such a manner that the first shall have 3 of of the second of 11 of 2; the third of 45; the fourth has $110. What is the sum divided? Answer, $240. 9. A and B, having found a purse of money, disputed who should have it. A said that, and of it amounted to $35, and if B would tell him how much was in it, he should have the whole; otherwise he should have nothing. How much did the purse contain? Answer, $100. DOUBLE POSITION. DOUBLE POSITION teaches to discover the true, by the use of two supposed, numbers. RULE. I. Suppose two numbers, and proceed with each according to the conditions of the question, as in Single Position, noting the error. The difference between the result and the given sum is the error. II. Multiply the first supposition by the second error, and the second supposition by the first error. III. If the errors are alike—that is, both too great or both too small, divide the difference of the products by the difference of the errors. IV. If the errors are unlike—that is, one too large, and the other too small, divide the sum of the products by the sum of the errors. EXAMPLES. 1. A man being asked what his carriage cost, replied, If it had cost twice as much as it did, and $20 more, it would have cost $370. What was the cost of the carriage? QUESTIONS. 5. Rule? 6. What is Double Position? 7. On what supposition is this rule founded? 8. Rule for the operation? |