It is evident, that the amount of $2 annuity is 2 times as much as one of $1; and one of $3, 3 times as much. Hence, To find the Amount of an Annuity, at 6 per Cent.;— Find, by the Table, the amount of $1, at the given rate and time, and multiply it by the given annuity, and the product will be the amount required. 3 What is the amount of an annuity of $120, which has remained unpaid 15 years? The amount of $1, by the Table, we find to be $23,2759; therefore, $23,2759 X 120 = $2793,108, Ans. 4. What will be the amount of an annual salary of $400, which has been in arrears 2 years?-824. 3 years?-127344. 4 years?-174984. 6 years?-279012. 12 years?-674796. 20 years?-147142. Ans. $28099,56. 5. If you lay up $100 a year, from the time you are 21 years of age till you are 70, what will be the amount at compound interest? A. $26172,08. 6. What is the present worth of an annual pension of $120, which is to continue 3 years? In this example, the present worth is evidently that sum, which, at compo interest, would amount to as much as the amount of the given annuity the three years. Finding the amount of $120 by the Table, as before, we have $382,032; then, if we divide $382,032 by the amount of $1, compound interest, for 3 years, the quotient will be the present worth. This is eviden from the fact, that the quotient, multiplied by the amount of $1, will give the amount of $120, or, in other words, $382,032. The amount of $1 for 3 years, at compound interest, is $1,19101; then, $382,032 $1,19101 = $320,763, Ans. Hence, To find the Present Worth of an Annuity;— Find its amount in arrears for the whole time; this amount, divided by the amount of $1 for said time, will be the present worth required. Note. The amount of $1 may be found ready calculated in the Table of compound interest, T LXXI. 7. What is the present worth of an annual rent of $200, to continue 5 years? A. $842,472. The operations in this rule may be much shortened by calculating the present worth of $1 for a number of years, as in the following TABLE II. Showing the present worth of $1 or £1 annuity, at 6 per cent. compound interest, for any number of years, from 1 to 32. Years. 6 per cent. Years. 6 per cent. Years. 6 per cent. Years. 6 per cent. To find the present worth of any annuity, by this Table, we have only to multiply the present worth of $1, found in the Table, by the given annuity, and the product will be the present worth required. 8. What sum of ready money will purchase an annuity of $300, to continue 10 years? The present worth of $1 annuity, by the Table, for 10 years, is $7,36008; then 7,36008 X 300 = $2208,024, Ans. 9. What is the present worth of a yearly pension of $60, to continue 2 years?-1100034. 3 years?-1603806. 4 years?-207906. 20 years?-6881952. 30 years?-8258898. A. $2364,9624. 8 years?-3725874. 10. What salary, to continue 10 years, will $2208,024 purchase? This example is the 8th example reversed; consequently, $2208,024 ÷ 7,36000300, the annuity required. A. $300. Hence, To find that Annuity which any given Sum will purchase ; Divide the given sum by the present worth of $1 annuity for the given time, found by Table II.; the quotient will be the annuity required. 11. What salary, to continue 20 years, will $688,95 purchase? A. $60+. To divide any Sum of Money into Annual Payments, which, when due, shall form an equal Amount at Compound Interest ; 12. A certain manufacturing establishment, in Massachusetts, was actually sold for $27000, which was divided into four notes, payable annually, so that the principal and interest of each, when due, should form an equal amount, at compound interest, and the several principals, when added together, should make $27000; now, what were the principals of said notes? years, It is plain, that, in this example, if we find an annuity to continue 4 which $27000 will purchase, the present worth of this annuity for 1 year will be the first payment, or principal of the note; the present worth for 2 years, the second, and so on to the last year. The annuity which $27000 will purchase, found as before, is 7791,97032+. Note. To obtain an exact result, we must reckon the decimals, which were rejected in forming the Tables. This makes the last divisor 3,4651056. (The 1st is $7350,915, amount for 1 year, $7791,97032 2d $6934,825, .. 3d $6542,288, .... 4th.. 2 $7791,97032 3 $7791,97032 Proof, $26999,998+. For the entire work of the last example, see the Koy. PERMUTATION. ¶ XCII. PERMUTATION is the method of finding how many different ways any number of things may be changed. 1. How many changes may be made of the three first letters of the alphabet? In this example, had there been but two letters, they could only be changed twice; that is, a, b, and b, a; that is, 1 X 2 = 2; but, as there are three letters, they may be changed 1 x 2 x 3 6 times, as follows: 1 (a, b, c. 2a, c, b. Hence, To find the Number of different Changes or Permu tations, which may be made with any given Number of different Things; Multiply together all the terms of the natural series, from 1 up to the given number, and the last product will be the number of changes required. 2. How many different ways may the first five letters of the alphabet be arranged? A. 120. 3. How many changes may be rung on 15 bells? and in what time may they be rung, allowing 3 seconds to every round? A. 1307674363000 changes; 3923023104000 seconds. 35 4. What time will it require for 10 boarders to seat themselves differently every day at dinner, allowing 365 days to the year? A. 994138 years. 5. Of how many variations will the 26 letters of the alphabet admit? A. 403291461126605635584000000. POSITION Is a rule which teaches, by the use of supposed numbers, to find true ones. It is divided into two parts, called Single and Double. SINGLE POSITION. ¶ XCIII. This rule teaches to resolve those questions whose results are proportional to their suppositions. 1. A schoolmaster, being asked how many scholars he had, replied, "If I had as many more as I now have, one half as many more, one third, and one fourth as many more, I should have 296." How many had he? We have now found that we did not suppose have been 296. But 24 has been increased in the the right number. If we had, the amount would 12 | same manner to amount to 74, that some unknown 8 number, the true number of scholars, must be, to Let us suppose he had 24 as many amount to 296. Consequently, it is obvious, that 674 has the same ratio to 296 that 24 has to the true number. The question may, therefore, be solved 74 by the following statement: As 74 296: 24: 96, Ans. This answer we prove to be right by increasing it by itself, one half itself, one third itself, and one fourth itself; 96 96 48 Suppose any number you choose, and proceed with it in the same manner you would with the answer, to see if it were right. Then say, As this result: the result in the question :: the supposed number : number sought. More Exercises for the Slate. 2. James lent William a sum of money on interest, and in 10 years it amounted to $1600; what was the sum lent? A. $1000. 3. Three merchants gained, by trading, $1920, of which A took a certain sum, B took three times as much as A, and C four times as much as B; what share of the gain had each? A. A's share was $120; B's, $360; and C's, $1440. 4. A person having about him a certain number of crowns, said, if a third, a fourth, and a sixth of them were added together, the sum would be 45; how many crowns had he? A. 60. 5. What is the age of a person, who says, that if of the years he has lived be multiplied by 7, and of them be added to the product, the sum would be 292? A. 60 years. 6. What number is that, which, being multiplied by 7, and the product divided by 6, the quotient will be 14? A. 12. 23 * DOUBLE POSITION. ¶ XCIV. This rule teaches to solve questions by means of two supposed numbers. In SINGLE POSITION, the number sought is always multiplied or divided by some proposed number, or increased or diminished by itself, or some known part of itself, a certain number of times. Consequently, the result will be proportional to its supposition, and but one supposition will be necessary; but, in DOUBLE POSITION, we employ two, for the results are not proportional to the suppositions. 1. A gentleman gave his three sons $10000, in the following manner: to the second $1000 more than to the first, and to the third as many as to the first and second. What was each son's part? Let us suppose the share of the first, 1000) The first error, then, is $4000, and the second $2000. Now, the difference between these errors would seem to have the same relation to the difference of the suppositions, as either of the errors would have to the difference between the supposition which produced it and the true number. We can easily make this statement, and ascertain whether it will produce such a result: As the difference of errors, 2000 500, difference of suppositions :: either of the errors (say the first), 4000 1000, the difference between its supposition and the true number. Adding this difference to 1000, the supposition, the amount is 2000 for the share of the first son; then $3000 that of the second, $5000 that of the third, Ans. For 2000 + 3000 + 5000 10000, the whole estate. Had the supposition proved too great, instead of too small, it is manifest that we must have subtracted this difference. The differences between the results and the result in the question are called errors: these are said to be alike, when both are either too great or too small; unlike, when one is too great, and the other too small. From these illustrations we derive the following RULE. Suppose any two numbers, and proceed with each ac cording to the manner described in the question, and see how much the result of each differs from that in the question. |