arrears 2 years? (824) 3 years? (127344) 4 years? (174984) 6 years. (279012) 12 years? (674796) 20 years? (147142) Ans. $23099,56. 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 compound interest, would amount to as much as the amount of the given annuity for the 3 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 $i, compound interest, for 3 years, the quotient will be the present worth. This is evident 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 roorth required. Note. The amount of $1 may be found ready calculated in the Table of compound interest, ¶ LXXI. 7. What is the present worth of an annual rent of $200, to continue 5 A. $842,472. years? 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 in terest for any number of years, from 1 to 32. 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 con tae 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) 8 years? (3725874) 20 years? (6681952) 30 years? (8258898) A. $2364,9624. 10. What salary, to continue 10 years, will $2208,024 purchase? This example is the 8th example reversed; consequently, $2208,0247,36000 300, 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 Ťable ÍI.; 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 4 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? It is plain, that, in this example, if we find an annuity to continue 4 years, 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. Ans. The 1st is $7350,915, amount for 1 yr. $7791,97032 $6934,825, .... 2d .... .... 2 $7791,97039 4 .... $7791,97032 3d $6542,288, 4th $6171,970, .. Proof, $26999,998+ PERMUTATION. TXCII. 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, 1X2=2; but, as there are three let ters, they may be changed 1X2 X 36 times, as follows: Hence, to find the number of different changes of permutations, 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 num ber of changes required. 2. How many different ways may the first 5 letters of the alphabet be ar ranged? 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. 1307674368000 changes; 3923023104000 seconds. 4. What time will it require for 10 boarders to scat themselves differently every day at dinner, allowing 365 days to the year? A. 99418 years. 5. Of how many variations will the 26 letters of the alphabet admit? A. 403291464126605635584000000. POSITZON 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. TXCIII. 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? Let us suppose he had 24 as many=12 } as many= 8 as many=6 74 We have now found that we did not suppose the right number. If we had, the amount would have been 296. But 24 has been increased in the same manner to amount to 74, that some unknown number, the true number of scholars. ⚫ must be, to amount to 296. Consequently, it is obvious, that 74 has the same ratio to 296 that 24 has to the true number. The question may, therefore, be solved 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 I. Suppose any number you choose, and proceed with it in the same manner you would with the answer, to see if it were right. II. Then say, As this result: the result in the question : : the supposed number: number sought. More Exercises for the Slate. 2 James lent William asam 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. DOUBLE POSITION. YXCIV. 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 sous $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 Then the second = 2000 Third 3000 Total, 6000 This, subtracted from 10000, leaves 4000 The shares of all the sons will, if our supposition be correct, amount to $10000; but, as they amount to $6000 only, we call the error 4000. 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: 1009, 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. I. Suppose any two numbers, and proceed with each according to the manner described in the question, and see how much the result of each differs from that in the question. II. Then say, As the difference* of the errors: the difference of the suppositions either error: difference between its supposition and the number sought. More Exercises for the Slate. 2. Three persons disputing about their ages, says B, "I am 10 years older than A;" says C, "I am as old as you both :" now what were their several ages, the sum of all of them being 100? Ans. A's, 20; B's, 30; C's, 50. 3. Two persons, A and B, have the same income; A saves of his yearly; but B, by spending $150 per annum more than A, at the end of 8 years finds himself $400 in debt; what is their income, and what does each spend per annum? First, suppose each had $200; secondly, $300; then the errors will be 400 and 200. A. Their income is $400; A spends $300, B $450. 4. There is a fish whose head is 8 feet long, his tail is as long as his head and half his body, and his body is as long as his head and tail; what is the whole length of the fish? First, suppose his body 30; secondly, 28; the errors will then be 1 and 2. A. 32 feet 5. A labourer was hired 80 days upon this condition,-that for every day he was idle he should forfeit 50 cents, and for every day he wrought he should re ceive 75 cents; at the expiration of the time he received $25; now how many days did he work, and how many days was he idle? A. He worked 52 days, and was idle 28, MISCELLANEOUS EXAMPLES. 1.. There is a room, one side of which is 20 feet long and 8 feet high, how many square feet are contained in that side? This side is a regular parallelogram ( LXXIX.); and, to find the square contents, we have seen that we must multiply the length by the breadth; thus, 20 ft. 8 ft. =160 sq. ft., Ans. A Hypothenise B But, had we been required to find the square contents of half of this par'allelogram, as divided in the figure on the left, it is plain that, if we should multiply (20) the whole length by of (8) the width, or, in this case, the height, the product would be the square contents in this half, that is, ia the figure BCD; thus, of 84; then, 4 X 20 ==80 sq. ft., which is precisely of 160, the square contents in the whole figure. The half BCD is called a triangle, because it has, as you see, 3 sides and 3 augles, and because the line BC fails perpendicularly on CD; the angle at C is called a right angle; the whole angle, then, BC may properly he called 4 right-angled triangle. D Base. 20. C The difference of the errors, when alike, will be one subtracted from the other; when unlike, one added to the other. |