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What is the present worth and discount of $350, payable in half a year, discounting at 6 per cent. per annum?
Pup. I see nothing very difficult in this, and think that I shall be able to perform any question belonging to this rule.
EQUATION OF PAYMENTS.
Tut. What is equation of payments ?
Pup. Equation of payments is the finding a time to pay, at once, several lells due at different times, so that neither party shall sustain loss.
Tut. You are right. The following is the rule by which payments are generally equated.
Multiply each payment by the time at which it is due ; then divide the sum of the products by the sum of the payments, and the quotient will be the time required.
This rule is founded on the supposition, that the sum of the interests of the several debts, which are payable before the equated time, from their terms to that time, ought to be equal to the sum of the interests of the debts payable aster the equated time, from that time to their terms. But this is not strictly true ; for by keeping a debt after it is due, you save the interest; but by paying a debt before it is due you only lose the discount, wbich is not so much as the interest ; ( 4, Con. IX.) therefore the rule is not true. But the error is so small that this meth: od will always be used as the most eligible in common business.
A. owes B. $400 to be paid in 4 months, and $400 more to be paid in 8 months; what is the equated time for paying the whole ?
It is evident the time must be 6 months, for the two debts are equal, and the equated time must be half the sum of both the times. 400 multiplied by 4, is 1600 ; and 400 multiplied by 8, is 3200, the sum of which is 4800, which, divided by 800. the sum of the two payments, gives 6 months, the equated time.
A. owes B. $380, to be paid as follows ; $100 in 6 months ; $120 in 7 months; and $160 in 10 months, what is the equated time for the payment of the whole debt?
Why do the creditors ought to allow discount for the payment of a debt before it is due ?
What is the rule for discount?
Why should not the discount be the same as the interest on the sum for the given rate and time ?
What is the rule for equation of payments ?
Examples for Practice.
1. What ready money will discharge a debt of $1595, due 5 months and 20 days hence, at 6 per cent ?
Ans. $1541,326. 2. What is the present worth of $960, payable as follows, į at 3 months, į at 6 months, and the rest at 9 months, supposing the discount to be made at 6 per cent?
Ans. $936,70. 3. Bought a quantity of goods for $250, and sold them
$300 payable 9 months hence; what was the gain, supposing the discount to be made at 6 per cent ?
Ans. $37,0811 4. What is the present worth of $426,55 to be paid 8 months bence, at 6 per cent?
5. What is the present worth of $426 payable in 4 years and 12 days, at 5 per cent?
Ans. $354,519. 6. The sum of $164,166 is to be paid, j in 6 months,
in 8 months, and din 12 months ; what is the mean time for the payment of the whole ?
Ans. 7 mo. 7. A debt is to be paid as follows ; at 2 months,
at 3 months, j at 4 months, į at 5 months, and the rest at 7 months ; what is the equated time to pay the whole ?
Ans. 4 months and 18 days. 8. A merchant owes me $900, to be paid in 96 days ; $130 in 120 days, $500 in 80 days, $1267 in 27 days; what is the mean time for the payment of the whole ?
Ans. 63 days, very nearly.
SINGLE AND DOUBLE POSITION.
Tut. What is position ?
Pup. Position is a rule relating to false or supposed numbers.
Tut. Position is a method of performing, by false or supposed numbers, such questions as cannot be resolved by the common direct rules. It is of two kinds, single and double.
SINGLE POSITION Shows how to resolve, by the working of one supposed number, as if it were a true one, those questions, whose results are proportional to their suppositions.
The following is the rule for single position.
Take any number and perform the same operations with it as are described to be performed in the question. Then say, as the sum of the errors is to the supposed number, so is the given sum to the true one required.
A B. and C. talking of their ages, B. said his age was once and a half the age of A.; and C. said his age was twice and one tenth the age of both, and that the sum of their ages was 93; what was the age of each ? Suppose A.'s age is 20 Then as 155 : 20 :: 93 B.'s must be 30
20 and C.'s 105
155) 1860(12 A.'s age.
155 : 30 :: 93
155 : 105 :: 93
A.'s age 12 years.
Bsage 10 years. dus aye wü years.
Pup. It appears very plain that the proportion between the sum of the supposed ages and each of the supposed ages respectively, must correspond with the proportion between the sum of the true ages and each of the true ages, respectively.
Tut. Do you know in what double position differs from single position ?
Pup. I do not ; but expect it is more difficult than single position, and should think from the name that there must be more than one supposition for the same question.
Tut. You are correct ; for it is more difficult, and to every question there must be two suppositions. Double Position shows how to reselve such questions as require two suppositions. The following is the rule for
Take any two numbers, and proceed with them according to the conditions of the question. Place the errors against their respective positions or supposed numbers, and if the error be too large, mark it with † ; but if too small with —. Multiply the first supposed number by the last error, and the last supposed number by the first error. If the errors be alike, that is, both too large, or both too small, divide the difference of the products by the difference of the errors, and the quotient will be the answer ; but if the errors be unlike, that is, one too large and the other too small, divide the sum of the products by the sum of the errors, and the quotient will be the answer,
This rule is founded on the principle, that the first error is to the second, as the difference between the first supposed number and the true number, to the difference between the second supposed number and the true number.