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CASE II. When a sum of money is to be apportioned among several, in such a manner that each one's share, placed at interest for unequal terms, shall produce equal amounts.
RULE, Find the amount of $100, at the given rate per cent. for the time each one's share is to remain at interest, and multiply the continued product of these amounts by the whole sum for a dividend.
Leave out one of these amounts, and multiply the others continually together; then leave out another of the amounts, and multiply the remaining ones continually together; and thus proceed, leaving out one of the amounts each time, until each of the amounts are respectively left out, and the remaining oncs multiplied continually together; then multiply the sum of these products by 100, for a divisor. The quotient will be the equal amount of each one. The present worth of this equal amount, for the time it remains at interest, will be each one's share.
Examples. 1. A man at his decease left $7200, to be divided among his four sons, whose several ages were 14, 17, 18 and 20 years, in such manner that their several portions, when they respectively would arrive to the age of 21 years, should be equal, reckoning interest at 6 per cent. during their minorities—I desire to know the sum bequeathed to each.
$106 amount of $100 for one year.
seven years. 106x118x124x142x7200 1585734220800 dividend.
106x118x124 = 1550992 here 142 is left out.
7271320.00)15857342208.00(2180.80.7 received by each 14542640
[at the age of 21.
[aged 20 years.
7200.000 whole sum bequeathed. 2. Divide $400 into two such parts, that the amounts will be equal, one being put to interest for one year, and the other for two years, at 6 per cent.
Ans. $205.54 and $194.496. To find the annual rent which any real property should bring, so as to pay debt, interest and costs in 7 years.
RULE. Multiply the given debt by .17913, the product will be the annual rent which the property should rent for.
Note.-If real property will rent for as much as will pay the debt and costs, with interest, in seven years, it cannot be legally condemned and sold for debt.
Examples. 1. What annual rent would be sufficient to pay the amount of debt and costs, upon a property amounting in the whole to 8558.25 cents, in seven years.
558.255 x.17913 100.00+ Answer.
PROOF. 558.255 whole amount of debt and costs.
33.495 interest for one year. 591.750 amount due at the end of the first year. 100.000 rent received the first year.
491.750 balance due after receiving the first year's rent. 29.505 interest on this balance for one year.
521.255 amount due at the end of the second year. 100.000 rent received the second year.
421.255 balance due after receiving the 2d year's rent. 25.275 interest on this balance for one year.
446.530 amount due at the end of the third year. 100.000 rent received the third year.
346.530 balance due after receiving the 3d year's rent. 20.792 interest on this balance for one year.
367.322 amount due at the end of the fourth year. 100.000 rent received the fourth year.
267.322 balance due after receiving the 4th year's rent. 16.039 interest on this balance for one year.
283.361 amount due at the end of the fifth
year. 100.000 rent received the fifth year. 183.361 balance due after receiving the fifth year's rent. 11.001 interest on this balance for one year,
194.362 amount due at the end of the sixth year. 100.000 rent received the sixth year. 94.362 balance due after receiving the sixth year's rent.
5.638 interest on this balance for one year. 100.000 amount due at the end of the seventh year. 100.000 rent received the seventh year. Hence it is evident that the debt, interest and costs are
2. A property which rented annually for $360, was condemned by a court of enquiry, for an amount of debt against it of $2100-Was the condemnation legal ?
[nually. Now, because this sum exceeds $360, the condemnation was legal.
3. What annual rent would be sufficient to pay the amount of debt an: costs, (upon a property) amounting in the whole to $12000, with the interest, in seven years ?
Ans. $2149.56. 4. A property worth an annual rent of $300, has a debt of $1500 against it—Should it be condemned?
$ 1500 x.17913 = $268.693. Now, because this is less than $300, the property should not be condemned.
5. Should a property worth an annual rent of $600, be condemned for a debt of $3400 ?
3400 X.17913=8609.042. Now, because this sum is more than the annual rent, it should be condemned.
ALLIGATION. ALLIGATION is a rule by which we adjust the prices and simples of compound quantities.
CASE I. When the quantities and their prices are given, to find the price of a part of the composition.
Examples. 1. A merchant mixes 3 C. of Sugar, at $8.25; 1 C. at $7.50; 5 C. at $8.00; and 2 C. at $6.50I desire to know what 10 C. of this composition is worth.
85.25 total value. C. C. S $
As 11 : 10 :: 85.25 : 77.50 answer. 2. If 12 bushels of corn, at 50 cents per bushel ; 100 of oats, at 30 cents; 16 of rye, at 60 cents; and 14 of buckwheat, at 55 cents, be mixed together, what will 22 bushels of the mixture be worth ?
Ans. $11.127, nearly. 3. A wine merchant mixes 6 gallons of wine, at $1.00 per gallon, with 4 gallons at $1.05, 7 gallons at $1.40, and 5 gallons at $1.50—what is a gallon of this inixture worth?
CASE II. When the prices of several simples are given, to find how much of each must be taken to make a compound at any proposed price.
RULE. The prices or rates must all be of the same denomination, Set down the prices, one under the other, and the mean rate on the left hand of these. Join or link together the several rates, so that each rate which is less than the mean rate be linked withi some one that is greater, or with as inany greater as you choose; and each of the greater with some one that is less, or as many less as you choose; the difference, or sum of the differences, between each rate and the méąn price, placed opposite the respective rate with which it or they are linked, will be the respective quantities required.
Note 1.-Different modes of linking will produce different
2. Any number of answers may be had, by dividing or multiplying any set of differences, by any common divisor or multiplier; which is evident from the following example.
Examples. 1. A merchant would mix wines at $1.20, $1.50, 82, and 82.50 per gallon, so that the mixture should stand him in $1.80 per gallon-What quantity of each sort must he take?