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7. Two men, A and B, lay out equal sums of money in trade; A gains 1261. and Blooses 877. and A's money is now double to B's; what did each lay out?
8. A farmer having driven his cattle to market, receiv ed for them all 150l. being paid for every ox 71. for every cow 51. and for every calf 11. 10s. there were twice as many cows as oxen, and three times as many calves as cows; how many were there of each sort?
Ans. 5 oxen, 10 cows, and 30 calves.
9. A, B and C, playing at cards, staked 324 crowns; but disputing about tricks, each man took as many as he could: A got a certain number; B as many as A and 15 more; C got a 5th part of both their sums added together; how many did each get?
Ans. A got 1271, B 142), C54.
PERMUTATION OF QUANTITIES, Is the shewing how many different ways any given number of things may be changed.
To find the number of Permutations or changes, that can be made of any given number of things, all different from each other
Multiply all the terms of the natural series of numbers, from one up to the given number, continually together, and the last product will be the answer required.
2. How many changes may be rung on 9 bells?
3. Seven gentlemen met at an inn, and were so well pleased with their host, and with each 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 fulfilled their agreement ? Ans. 110179 years.
ANNUITIES OR PENSIONS,
To find the amount of an annuity, or Pension, in arrears, at Compound Interest..
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 the given number of years, and find its sum.
3. Multiply the sum thus found, by the given annuity, and the product will be the amount sought.
1. If 125 dols. yearly rent, or annuity, be forborne, (or unpaid) 4 years; what will it amount to, at 6 per cent. per annum, compound interest ?
1+1,06+1,1236+1,191016=4,374616 sum of the Then, 4,374616×125-8546,827 the amount
OR BY TABLE II.
Multiply the Tabular number under the rate and op posite to the time, by the annuity, and the product will be the 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. ready 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 yearly, be forborne 20 years, at 6 per cent. compound in terest; what is the amount?
Under 6 per cent. and opposite 20, in Table II, you will find,
Ans. $2207,13540-82207, 13cts. 5m.+\
3. Suppose an Annuity of 100l. be 12 years in arrears, it is required to find what is now due, compound interest being allowed at 56. per cent. per annum ?
Ans. 1591 14s. 3,024d. (by Table III.) 4. What will a pension of 120l. per annum, payable yearly, amount to in 3 years, at 5l. per cent. compound interest ? Ans. £378 6s.
II. To find the present worth of Annuities at Compound
Divide the annuity, &c. by that power of the ratio signified by the number of years, and subtract the quotient from the annuity: This remainder being divided by the ratio less 1, the quotient will be the present value of the Annuity sought.
1. What ready money will purchase an Annuity of 50%. to continue 4 years, at 51, per cent. compound interest? 4th th power of=1,215506)50,00000(41,13515+
BY TABLE III.
cent. and even with 4 years.
We have 3,54595-present worth of it. for 4 years. j 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 Ans. $688, 19 cts.+ 3. What is 30l. per annum, to continue 7 years, worth in ready money, at 6 per cent. compound interest? Ans. 167 9s. 5d.+
III. To find the present worth of Annuities, Leases, &c. taken in REVERSION, at Compound interest.
1. Divide the Annuity by that power of the ratio de noted by the time of its continuance.
2. Subtract the quotient from the Annuity: Divide thé remainder by the ratio less 1, and the quotient will be the present worth to commence immediately.
3. Divide this quotient by that power of the ratio de noted by the time of Reversion, (or the time to come before the Annuity commences) and the quotient will be the present worth of the Annuity in Reversion.
1. What ready money will purchase an Annuity of 501. payable yearly, for 4 years : but not to commence till two years, at 5 per cent.
4th power of 1,051,215506) 50,00000(41,15513 Subtract the quotient 41,13513
Divide by 1,05-1=,05)8,86487
2d. power of 1,05=1,1025)177,297(160,8136-160 16s. 3d. 1gr. present worth of the Annuity in Reversion. OR BY TABLE III.
Find the present value of 1. at the given rate for the 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
2. What is the present worth of 75l. yearly rent, which is not to commence until 10 years hence, and then to continue 7 years after that time at 6 per cent.? Ans. 233 15s. 9d. 3. What is the present worth of the reversion of a lease of 60 dollars per annum, to continue 20 years, but not to commence till the end of 8 years, allowing 6 per cent. to the purchaser ? Ans. $431 78cts. 2m. IV. To find the present worth of a Freehold Estate, or an Annuity to continue forever, at Compound Interest. RULE.
As the rate per cent. is to 100l.: so is the yearly rent to the value required. EXAMPLES.
1. What is the worth of a Freehold Estate of 401. annum, allowing 5 per cent. to the purchaser?
As £5 £100 £40 £800 Ans. 2. An estate brings in yearly 150l. what would it sell for, allowing the purchaser 6 per cent. for his money? Ans. £2500 V. To find the present worth of a Freehold Estate, in Reversion, at Compound Interest.
1. Find the present value of the estate (by the foregoing rule) as though it were to be entered on immediately, and divide the said value by that power of the ratio de noted by the time of reversion, and the quotient will be the present worth of the estate in Reversion.
1. Suppose a freehold estate of 40l. per annum to commence two years hence, be put on sale; what is its value, allowing the purchaser 5l. per cent.?