2. Suppose an estate of $100 per annum to commence 6 years hence, were to be sold, allowing the purchaser 5 per cent what is its value ? Ans. $1492,430.0*m. 3. What is the value of the reversion of an estate of $120 per annum, to commence 15 years hence, at 6 per cent ? Ans. $834,53+ Questions. 1. How do you find the amount of an- of annuities, leases, &c. taken in revernuity at simple interest, by Arithmetical sion, at compound interest ? Progression ? 5. How do you find the present worth 2. How do you find the amount of an of freehold estates or annuities fornuities or pensions in arrears, at com. ever! pound interest ? 6. How do you find the present worth 3. How do you find the present worth of perpetual annuities, or freehold esof annuities at compound interest ? tates in reversion at compound inter 4. How do you find the present worth est ? PERMUTATION. Permutation is a method of finding how many different ways any given number of things may be changed. To find the number of different changes or permutations that can be made of any given number of things, different from each other. RULE. Multiply all the terms of the natural series continually together, from one up to the given number, and the last product will be the answer. EXAMPLES. 1. How many changes can be made of the first three letters of the alphabet ? If there were but 2 letters, we could only change them 1X2=2 ways, thus, a, b, and b, a. But three letters can be changed i x2x3=6 different ways, as follows: 1 r a b b 3 b 4 b b a с a с с a с a с a 2. How many changes can be made with the nine digits? Ans. 362880. 3. Eight gentlemen agreed to dine together so long as they could sit every day in a different position ; now admitting they had fulfilled their agreement, how long must they have tarried together? Ans. 40320 days,=110yrs. 4. How many changes may be rung on 9 bells ?-and how long will it take to ring them, allowing 20 seconds to every change? Ans. to the last, 84 days. 5. Of what number of variations will the twenty-six letters of the alphabet admit ? Ans. 403291461126605635584000000. Questions. 1. What is Permutation ? 2. How do you find the number of different changes or permutations that can be made of any given number of things? POSITION Is à Rule which, by the use of false, or supposed numbers, discovers the true ones required. It is divided into two parts, Single and Double. SINGLE POSITION. Single Position teaches to solve those questions whose results are proportional to their suppositions. RULE. 1. Take any number, and perform the same operations with it as are described to be performed in the question. 2. Then as the result of the operation is to the given sum, so is the supposed number to the true one required. EXAMPLES. 1. A schoolmaster being asked how many scholars he kad, replied : If I had as many more as I now have, half as many, one-third as many, and one-fourth as many, I should then have 222. How many scholars had he? Suppose he had 12 In this example Then as many more 12 Proof. it is evident that, as many 6 72 had we supposed as many 4 72 the true number of as many 3 36 scholars which he 24 actually had, the 37 18 amount would have Then, as 37:222 :: 12 : been 222. But 12 122 our supposed num37)266 4672 Ans. ber has been in creased in the same proportion as the true number, and consequently bears the same proportion to the true number as its result bears to the result of the true number ; that is, 12 bears the same proportion to the true number, as 37 bears to 222, the true result. 2. A man having a certain number of dollars, said that }, }, }, and į of them added together would make 114.How many dollars had he ? Ans. $120 3. Divide 125 dollars among A, B, and C, so that B may have half as much as A, and C three times as mnch as B. Ans. A's share $41%; B's $206; and C's $623. 4. A person having a certain sum of money, said that if of it be multiplied by 9, the product would be $945; how many dollars had he ? Ans. $126. 5. A person lent his friend a certain sum of money, to receive interest for the same at 6 per cent per annum, simple interest, and at the end of twelve years, received, for principal and interest, $774. What was the sum lent? Ans. $450. 6. A, B and C, gained by trading $1360, of which A took a certain sum, B took 3 times as much as A, and c took as much as A and B both ; how much did each have? Ans. A had $160, B $520, and C $680. 7. A man going to market with some sheep, cows and oxen, being asked how many he had of each, replied, that he had three times as many cows as oxen, and twice as many sheep as cows, and ihat one-fifth of his whole num: ber made 18; how many had he of each ? Ans. 9 oxen, and 54 sheep. 27 cows, DOUBLE POSITION Teaches to solve questions by means of two suppositions, or false numbers. In Single Position, the number sought is always multiplied, or divided, by some proposed number, or increased or diminished by itself, or a certain part of itself, and the result is always proportional to the supposition. But in Double Position, the number sought is always increased by the addition, or decreased by the subtraction, of some number which is not a known proportional part of the true number. Hence the results are not proportional to the suppositions. RULE. 1. Suppose any two convenient numbers, and proceed with each according to the conditions of the question, and find how much the results differ from the result in the question. 2. Multiply the first position by the last error, and the last position by the first error. 3. Then if the results be both greater or both smaller than the true number, divide the difference of these products by the difference of the errors, and the quotient will be the answer. 4. But if one result be greater, and the other less, than the true number, divide the sum of the products by the sum of the errors, and the quotient will be the answer. EXAMPLES, 1. A father gave his three sons 15000 dollars in the following manner; to the first he gave a certain sum, to the second he gave $3000 more than to the first, and to the third he gave as much as to the first and second both; how much did he give to each ? Sup. he gave the 1st $1500 then the second 4500 Again, suppose the 1st 2000 and the third 6000 then the second 5000 and the third 7000 the result = 12000 this subtracted from 15000 the result = 14000 subtracted from 15000 leaves the 1st error 3000 last position X 2000 leaves the 2d error 1000 first position x 1500 6000000 1500000 1500000 Diff. of the prod. 4500000; this divided by 2000, the difference of the errors, gives $2250, the share of the first; to which + 3000=5250, the share of the second ; and 2250+ 5250=7500, the share of the third. 2. B, C and D built a house which cost $1000. C paid $100 more than B, and D paid as much as B and C both; how much did each man pay ? Ans. B paid $200, C $300, and D $500. 3. What number is that which being increased by its į, its 4, and 16 more, will be doubled ? Ans. 64. 4. D, E and F, playing at cards staked 324 crowns, but disputing about tricks, each one took as many as he could get. D got a certain number, E got as many as D and 15 more, and F got part of both their sums added together ; how many did each get? Ans. D got 127, E 142, and F 54, 5. A man has 100 acres of land in 3 lots. The second lot contains twice as much as the first, lacking 8 acres, and the third contains three times as much as the first, lacking 15 acres; how many acres does each lot contain ? Ans. the 1st contains 204 acres, the 2d 33, the 3d 46ļa. 6. A and B laid out equal sums of money in trade ; A gained $150, and B lost $225; then A's money was double that of B's. What sum did each lay out? 1st, Supp. A $300 B $300 2d Supp. A $400 B $400 +150 -225 +150 -225 double $75 1st error 450 150 300 75 550 175 double $175=350 2d error = 200 Ans. Each laid out 600 dollars. |