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The errors being alike, are both too small, therefore,
2. A, B and C, built a house which cost 500 dollars, of which a paid a certain sum; B paid 10 dollars more than A, and C paid as much as A and B both; how much did each man pay?
Ans. A paid 120, B 130, and C 250 dols. 3. A man bequeathed 100l. to three of his friends, afte. this manner: the fiest must have a certain portion; the second must have twice as much as the first, wanting 81. and the third must have three times as much as the first, wanting 151, : I demand how much each man must have ?
Ans. The first £ 20 10s. second £35, third £,46 10s.
4. A laborer was hired for 60 days upon this condition ; that for every day he wrought he should receive 4s. and for every day he was idle, should forfeit 2s. : at the expiration of the time he received 71. 10s.; how many days did he work, and how many was he idle?
Ans. Ile wrought 45 days, and was idle 15 days. 5. What number is that which being increased by its to its , and 18 more, will be doubled ? Ans. 72.
6. A man gave to his three sons all his cstate in money, viz.. to F half, wanting 50l. to G one-third, and to Il the l'est, which was 101. less than the share of G; I demand the sum given, and each man's part? Ans. The sum ziren was £360, whereof F had £ 150,
G£120, and II f110
7. Two men, A and B, lay out equal suins of money in trade; A gains 126l. and B looses 871. and A's money is now double to B's : what did each lay out ?
Ans. £,300. 8. A farmer having driven his cattle to market, recived for them all.1301. being paid for every ox 7l. for every cow 51. and fer every call 16. 10s. there were twice as many cows as oxen, and three times as many calves as GUITS;
Ans. 5 oxer, 10 co?c's, and 30 calves. 9. A, B and C, playing at cards, staked 324 crowns; but disputing about tricks, eachi man took as many as he could : A got a certain number; B as many as A and 15 more; C got a fifth part of both their sums added together: how many did cach get?
Alls. A 127, B 143), C 54.
PERMUTATION OF QUANTITLES, Is the showing how many different ways any given number of things may be changed.
To find the number of Perinutations or changes, that can be made of any given number of things, all different lioni cach other.
RULE. Niultiply all the terins of the natural series of numbers, from one up to the given number, continually together, and tire last product will be the answer required.
EXAMPLES. 1. How many changes can be
ila uc maile of the three first letters of the alphabet ?
3 bac Proof,
4 | Ն : մ
5 ca 1 X2 XS=6 Ans.
6 2. Ilow many changes may be rung on 9 bells ?
3. Seven gentlemen, met at an ini, and were so well pleased with their host, and with cach other, that they agreed to tarry so long as they, together with their host, could sit every day in a different position at dinner; how long must they have staid at said inn to have fullilled their agreement ?
Jus. 110475 years.
ANNUITIES OR PENSIONS,
CASE 1. To find the amount of an annuity, or Pension, in arrears,
at Compound Interest.
RULE. 1. Make 1 the first term of a geometrical progression, and the amount of $1 or £1 for one year, at the given rate per cent. the ratio.
2. Carry on the series up to as many terms as tho given number of years, and find its sum.
3. Multipiy the sum thus found, by the given annuity, and the product will be the amount sought.
EXAMPI.ES. 1. If 125 dols. yearly rent, or annuity, be forbornc, or unpaid) 4 year's; what will it amount to, at 6 per cent. per annun, compound interest :
1+1,06+-1,1936+-1,191016=4,374616 sum of the series.* -Then, 4,374613x125=8546,827 the amount . sought.
OR BY TABLE I Multiply the Tabular nuinber under the rate and opposite to the time, by the annuity, and the product will be ilie amount sought.
* The sum of the series thus found, is the amount of 11. or 1 dollar annuity, for the given time, which may be found in Table II. really calculated.
Hence, either the amount or present worth of annuities may be readily found by Tables for that purpose.
2. If a salary of 60 dollars per annum to be paid year.. ly, be forborne 20 years, at 6 per cent. compound interest; what is the amount?
Uniler 6 per cent. and opposite 20, in Table II, you will find, Tabular number=36,78559
Ans. $2207,15540=$2207, 13cls. 5m. + 3. Suppose an Annuity of 1001. be 12 years in arrears,
, it is required to find what is now due, compound interest being allowed at 5l. per cent. per annum?
Ans. f 159.1 14s. 3,024d. (by Table II.) 4. What will a pension of 120l. per annum, payable yearly, amount to in 5 years, at 5l. per cent. conpinund interes: ?
Ans. £578 os. II. To find the present worth of Annuities at Compound
Divide the annuity, &c. by that power of the ratio sig. nified by the number of years, a nd subtract the quotient from the annuity: This remainder being divided by the ratio less 1, the quotient will be the present vakre vf the Annuity sought.
1. What really money will purchase an Annuity of 501. to continue 4 years, at 5!. per cent. compound interest? 4th
power of the ratiu, s
50 Subtract 41,15515
Divis. 1,05-1=05) 8,86487
177,297=5177 58. 111d. Ans.
BY TABLE II
5 Wc have 3,54593=present worth of 1l. lor 4 years, Multiply by 50=Annuity.
Ans. $177,29750=present worth of the annuity.
2. What is the present worth of an annuity of 60 dols. per annum, to continue 20 years, at 6 per cent. compound interest ?
Ans. 8688 19} cts. + 3. What is 30l. per annum, to continue 7 years, worth in ready moucy, at 6 per cent, compound interest ?
Ins. £ 167 9s. 5d.+ II. To find the present worth of Annuitics, Lcases, &c.
taken in REVERSIOx, at Compound Interest? 1. Divide the Annuity by that power of thc ratio denoted by the time of its continuance.
2. Subtract the quotient from the Annuity: Divide the remainder by the ratio less i, and the quotient will be the present worth to commence immediately.
3. Divide this quotient by that power of the ratio denoted by the time of Reversion, (or the time to come before the Annuity coininences) and the quotient will be the present worth of the Annuity in Reversion.
1. What ready money will purchase an Annuity of 501. payable ycarly, for 4 years : but not to commence till tivo years, at 5 per cent. ? 4th power of 1,05=1,215506)50,00000(41,15513
Subtract the quotient=41,15515
2d. power of 1,05:=1,1025)177,297(160,8136=£160 16s. 3d. Igr. present worth of the Annuity in Reversion.
OR BY TABLE III. Find the present value of 11. at the given rate for thre sum of the time of continuance, and time in reversion added together; from which value subtract the present worth of il. for the time in reversion, and multiply the remainder by the Annuity; the product will be the answer.