Hence, to find that annuity which any given sun 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 annui ty 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 actua!ty 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, &t 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. The 1st is $7350,915, amount for 1 yr. $7791,97032 $6934,825, Ans. ...... 4th .. $6171,970, 2 $7791,97039 Proof, $26999,998+ PERMUTATION. TXCII. PERMUTATION is the method of finding how many differen ways any number of things may be changed. 1. How many changes may be made of the three first letters of the al phabet? In this example, had there been but two letters, they could only be changes twice; that is, a, b, and b, a; that is, 1X2=2; but, as there are three letters, they may be changed 1X2 X 36 times, as follows: 1(a, b, c. Hence, to find the number of different changes o permutations, which may be made with any given number of different things; Multiply together all the terms of the natural series, from 1 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 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 $1, 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 worth 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 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 To find the present worth of any annuity, by this Table, we kave only to multiply the present worth of $1, found in the Ta ble, 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 the 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 yeare? (1100034) 3 years? (1603806) 4 years (207906) 8 years? (3725874) 20 years? (6881952) 30 years? (8259898) A. $2364,9624. 10. What salary, to continue 10 years, will $2208,024 purchase? This example is the 8th example reversed; consequently, $2208,024-7,36000 300, the annuity required. ".4. $300. 3. Three merchants gained, by trading, $1920, of which A took a certais 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. TXCIV. This rule teaches to solve questions by means of two sup posed 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 propor-tional 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 Third 3000 Total, 6000 Tuis, 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. Suppose, again, that the share of the first was 1500 Then the second 2500 We perceive the error in this case to be $2000. 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 uifference 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. 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; bu 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 ind 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 hu 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 ille? 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. But, had we been requ..ed to find the A Hypothese B square contents of half of this parallelogram, 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, in the figure B CD: thus, of 8=4; then, 4 X 20 ==80 8. 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 angles, and because the line B C falls perpendicularly on C D; the angle at C is called a right angle; the whole angle, then, B C D may properly be called a 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. The line BC is called a perpendicular, CD the bass, and D B the hypotho Note. Both the base and perpendicular are sometimes called the legs of the triangle. Hence, to find the area of a right-angled triangle ;— Multiply the length of the base by the length of the per pendicular; the product will be the area required. 2. What is the area of a triangular piece of land, one side of which ir 40 rods, and the distance from the corner opposite that side to that side 20 rods' Ans. 20X40 400 rods Note. To find the area of any irregular figure, divide it into triangles. Hypothenuse. 50. Perpendicular. 30. In any right-angled triangle, it has been ascer tained, that the square of the hypothenuse is equal to the sum of the squares of the other two sides. Thus, in the adjacent figure, 403 1600, and 302 900; then, 900+1600=50, the hypothenuse Base. 40. I. Hence, to find the hypothenuse, when the legs are given; Add the squares of the two legs together, and extract the square root of their sum. II. When the hypothenuse and one leg are given, to find the other leg ;— From the square of the hypothenuse subtract the square of the given leg, and the square root of the remainder will be the other. 3. A river 80 yards wide passes by a fort, the walls of which are 60 yards high; now, what is the distance from the top of the wall to the opposite bank of the river? In this example we are to find the hypothenuse. Ans. 100 yards. 4. There is a certain street, in the middle of which, if a ladder 40 feet long be placed, it will reach a window 24 feet from the ground, on either side of said street; what is the width of the street? In this example, we are to find the length of the base of two triangles, and then the sum of these will be the distance required. Ans 64 feet. 5. There is a certain elm, 20 feet in diameter, growing in the centre of a circlar island; the distance from the top of the tree to the water, in a straight line, is 120 feet; and the distance from the foot 90 feet; what is the height of the tree? As the tree is 20 feet in diameter, the distance from its centre to the water is the length of the base, that is, 10+90=100 feet. A. 66,332 ft.+. 6. Two ships sail from the same port; one goes due north 40 leagues, the ther due east 30 leagues; how far are they apart? We are here to find the hypothenuse. A. 50 leaguos |