ARITHMETIC.
23Q
In calculating compound interest for parts of times, some au-thors for expedition, calculate for a complete time, and then takea part of the result proportionate to a part of the lime ; but this isnotjust, as the compound interest does not increase in the earlyperiod exactly in proportion to its increase in the late or completeperiod. In all such calculations therefore, where it is necessaryto calculate compound interest, for parts of times, the most expe-ditious and accurate method is by Logarithms ; for though it ispossible to make such calculations without them, the trouble isimmense; whereas, by logarithms it is easy to perform the calcu-lations with parts of times of payment, as with whole ones.
(55.) Discount or Rebate is the difference between a sum ofmoney due at a certain time to come, and its present worth. Thepresent value of any sum or debt, due some time hence, is sucha sum, as if put to interest, would in the time and at the rate forwhich the discount is to be made, amount to the sum or debtthen due.
Rule. As the amount of 100/. for the given rate and time, isto the given sum or debt, so is 100/. to the present worth, or so is theinterest of 10u/. for the given time, to the discount of the givensum.
The method used among the bankers in discounting bills is tofind the interest of the-sum drawn for, from the time tire bill isdiscounted, to the time when it becomes due, (including theMaysof of grace) which interest they reckon as the discount, therebymaking the discount more than it really is. But when goods arebought or sold, and discount is to be made for present payment,at any rate per cent, without regard to time, the interest of thes um as calculated for a year is the discount.
Example. What is the present worth of 700/. due 9 monthshence, discount at 5 per cent, per annum ?
100
3-75 int. for 9 months
103-75 amount of 100/.
L. L. L.103-75 : 700 : : 100100
163-75)70000(674-6988
or Z. 674. 13s. 11 d.present worth sought.
(56.) Equation of payments is the finding of a time, when if asum of money be paid which is equal to the sum of several othersdue at different times, no loss will be sustained by either party.
Rule. Multiply each payment by the time at which it is clue,then dividing the sum of the products by the sum of the payments,the quotient is the equated time.
Ex. A debt of 60/. is to be paid, 30/. at 2 months, and 30/. at4 months: but if it be reduced to one payment, at what timemust it be paid?
2 x 30=604 x 30=120
60 ) ISO (3 months, the time required.
180
Single Fellowship.
(57.) Single Fellowship is a rule by which any sum or quantitymay be divided into any assigned number of parts, which shall beProportional to so many other proposed numbers, each to each.—By this rule are adjusted the gain or loss, or charges of merchantsln company, the effects of bankrupt’s legacies in case of a deficiency°t assets, &c.
Rule. Make the sum of the numbers, to which the requiredParts must he proportional, the first term : the quantity to heparted or divided, the second; and each of the given numbers,|° which the required ones must be proportional, the several thirdmrnis, of so many rule of three questions; the fourth terms of"diich will be the respective parts required.
£x. Two merchants, A and B, trade together -. A puts into the
stock 69/. and B puts in 40/. They gain by trading 24/. Whatare their shares of it ?
60
40 Z. S.
— : 60 : 14 8 A’s share,
v 100:24<
: 40 : 9 12 B’s share.
Proof. 24Double Fellowship.
(58.) When thg shares of partners are continued in companyunequal times, they Occasion the name “ Fellowship with Time,”or “ Double Fellowship which is performed by the followingRule. Multiply each share by the time of its continuance ;then divide the gain or loss in proportion to the products, as insingle fellowship, by saying, as the sum of the products is to thewhole gain or loss, so is each product, to each part of it.
Ex. A had in company 50/. for 4 months, and B 60/. for 5months, at the end of which time, they gained 24/. IIow must itbe divided between them ?
4 X 50 = 200
5 X 60 = 300
500 : 24
L. S.
: -. 200 : 9 12 A’s share.
: : 300: 14 8 B’s share.
(59.) Barter is the exchanging of commodities; and as neitherparty is supposed to sustain any loss, when the commodities ex-changed are not of equal value, the defect is supplied with mo-ney, & c.
Case 1 . When the quantity of one commodity is given, withits value, or that of its integer; as also the value* of the integer,or rate of selling some other commodity to be given for it, to findthe quantity of this ; or having the quantity of the latter given,to find the rate of selling it. If the amount of the given quantityhe unknown, calculate it in the shortest manner you*can, from thegiven value of its integer; then find how much of the other quan-tity this amount will purchase, at the given rate of selling it; or ifthe quantity he given, from thence find the rate of selling it.
Case 2. If the quantities of both commodities, with the rate ofselling them, be given; to find what quantity of some other com-modity or money must be given in case of an inequality of theamount of the first commodities. Calculate the amount of eachof the two given commodities; then their difference is the money,or amount of the third commodity to be advanced; whose quan-tity, from thence or its rate, is easily found.
Case 3. When, in bartering, one commodity is rated above theready money-price ; to find the quantity and bartering price of theother commodity. As the ready money price of the one is to itsbartering price ; so is that of the other to its bartering price ; thenthe quantity of the latter commodity maybe found either fromthe ready-money or bartering prices.
These are the most general cases in barter, and such questionsas are not contained in them, are easily resolved from a little con-sideration of their nature.
Loss and Gain.
(60.) Questions in this rule are such whose solutions determinethe loss or gain upon commodities; of which questions there is agreat variety- ; but they may he all easily solved from a little con-sideration and the following proportion, viz. That the gains orlosses are in proportion as the quantities of goods, &c.
Ex. If, when I sell cloth at Is. per yard, I gain 10 per cent;wiiatwiU be the gain when it is sold lor 8.s. (id. per yard ?
■S'. S. D. Z. Z. 6'. Z>.
7:8 6 : : 110: 13.3 11
100 0 0
33 11 Ans.
Dr. Hutton observes, that questions of this sort are seldom un-derstood or solved rights the gain or loss of 100/. being mat he thethird term instead ot the ainhuut of 100/. Observe, that Is. and-2 8s > 6t/. are