21. When tea is sold at 6s. 4d. I clear 12 per cent. What What must I sell it at to 23. How much per cent. is 41d. per shilling? 24. Cloth, which was bought at 14s. 9d. per yard is sold at 8d. per yard less. What is the loss per cent. ? 25. A person bought 120 oranges at 2 a penny, and double that quantity at 3 a penny, and he sold them again at 5 for 2d. How much did he lose or gain, and how much per cent.? 26. Sold, 15 horses for £675, whereby I cleared as much as I sold 2 horses for. What is my gain per cent.? 27. A grocer had 2 cwt. of tea, of which he sold at 6s. 6d. per lb., and found that he was gaining 10 per cent. At what rate must he sell the remainder to clear 16 per cent. on the whole? 28. The prime cost of 75 dozens of wine is £202. 10s., and 3 dozens are lost by leakage, &c. What must the remainder be sold at per dozen, so as to gain 12 per cent. on the prime cost? 29. Sold, 5 yards of stuff for £1. 3s. 9d., and gained at the rate of 15 per cent.; but had I sold it at 5s. 4d. per yard, what would be the gain per cent. ? 30. A bought 648 yards of stuff, at 7s. 7d. per yard, 68 yards at 11s. 6d., 98 yards at 44d., 487 yards at 103d., 98 yards at 9s. 4d. per yard. The whole was sold at 15 per cent. loss. Find the whole selling price, and also the selling price of each per yard. EQUATION OF PAYMENTS. 303. It is sometimes required, both in commercial and banking business, to find the correct time at which two or more payments, or debts, payable at different times, should be discharged in one payment, so that neither debtor nor creditor will suffer loss. The operation is called Equating the Time, and questions relating to this subject are known under the appellation of Equation of Payments. pay Ex. Suppose a banker has two bills, viz., one of £2500, able in 9 months, and the other of £1600, payable in 16 months. What is the equated time? Evidently, the time of payment lies between 9 months and 16 months, and that time must be such that the interest on £2500, which is paid after its time, should equal the discount on £1600, which is paid before its time. The rate being 5 per cent. For instance: Suppose x to represent the equated time, then the interest of £2500 for (x-9) months must be equal to the discount on £1600 for (16-x) months. .. the interest of £2500 for (x-9) months, at 5 £2500 × 5(x—9) 100 × 12 = per cent. = Also the discount of £1600 for (16-x) months, at 5 per cent. = £1600 × 5(16—x) 304. This method, though correct, is not the one adopted in practice, probably on account of its being complicated, and it becomes more so as the number of payments increase. In the method used, it is supposed that the sum of the interest of each debt for its respective time is equal to the interest of the sum of the debts for the equated time. Thus, interest on £2500, for 9 months, at 5 per cent., is 5 × 9 × £2500 12 × 100 Also, interest on £1600, for 16 months, at 5 per cent,, is 5 × 16 × £1600 12 × 100 And interest on £4100, for equated time, at 5 per cent., is 5×4100 × equated time As every term contains the common factor 5 it may 12 x 100' be omitted, and we have 9×2500+16×1600=4100 × equated time. Hence we infer that the sum of the two products, obtained by multiplying the value of each bill by the number of months that have to elapse before they are due, is equal to the product of the sum of the debts multiplied by the number of months of the equated time. And since equated time × 4100=9× 2500+16×1600=48100, ̧..equated time=48100481=1130 months. The difference of the results, as found by this method and the true one is but small: the difference is in favour of the payer, because he reckons on his side the interest instead of the discount of those debts which he pays before they are due, whilst the other side of the operation is quite correct. 305. EXERCISES. 1. A owes B £1000, to be paid yearly, in five equal payments. What is the equated time to pay the whole? By what has been said in the preceding article, we have: yrs. yrs. 200 × 1 200 2. A debt of £1600 is to be liquidated by three payments, as follows: £600 in 3 months, £700 in 7 months, and £300 in 10 months. When must the whole be paid together? 3. B owes £200, payable in 3 months; £200, payable in 4 months; and £200, payable in 5 months. In what time may the whole be paid, without loss to either party? 4. If a debt of £120 is due in 1 month, of £240 is due in 5 months, of £330 is due in 7 months, and of £480 in 8 months, the equated time for the payment of the four debts is required. 5. If of a debt is to be paid at present, in 4 months, in 9 months, and the remainder in 12 months, find the equated time for the payment of the whole. 6. Find the equated time of three debts, the first of £139. 10s., due at the end of 9 months; the second, of £235, due at 15 months' end; and the third, of £650, due in 1 years. 7. £200 are to be paid in monthly instalments of £20. Find the time when the whole may be paid in one sum. 8. Find the equated time of payment, when of a sum of money is due in 5 months, in 9 months, and the remainder in 13 years? 9. £541. 16s. are due now, and £473. 5s. are to be paid in 5 equal monthly instalments. Find the equated time for the whole. 10. £360 are to be paid as follows: £80 in 80 days, £100 in 100 days, £120 in 120 days, and the rest in 1 year. Find the equated time for paying the whole. Ᏼ Ꭺ Ꭱ Ꭲ Ꭼ Ꭱ. 306. It frequently happens that one person buys goods of another, but instead of paying money for them, gives other goods in return. This kind of transaction is called Barter. Ex. 1. Suppose tea, at 4s. 6d. per lb., is given for 90 yards of linen, at 3s. per yard, what quantity of tea must be given? 1 yard is bought for 3s., ..90 yards are bought for 90 × 3s. = 270s. Now, .4s. 6d. buy 1 lb. of tea, .. 6d. will buy lb. ; ..270s., or 540 sixpences, buy 540 × lb., or 60 lbs. of tea. Ex. 2. Bought, 12 quarters of wheat, at £2. 16s. per quarter, for which I paid in money £13. 12s., and the remainder in stuff, at 5s. per yard. How many yards had I to give? 1 quarter is bought for £2. 16s., ..12 quarters are bought for 12 × £2. 16s., or £33. 12s., And the value paid in stuff is £33. 12s.—£13. 12s., or £20. Now, 5s. buy 1 yard of stuff, ..£20 buy 80 x 1 yard, or 80 yards. Ex. 3. A gave 96 336 yards of calico. quired. yards of cloth, at 12s. 6d. per yard, for The value of the calico per yard is re .1 yard of cloth cost 12s. 6d., ..963 yards cost 963 × 12s. 6d., or 1209s. 44d. Also, 336 yards of calico cost 1209s. 4 d., 1209s. 4 d. 9675s. ..1 yard of calico cost nearly. 336 or or 3s. 7d. 2693 Ex. 4. X has silk worth 16s. per yard, and Y has cloth worth 12s. per yard, which he barters at 14s. 6d. per yard. How much must the silk be raised to be equivalent to the rise of the cloth? 1. How many yards of cloth, at 15s. per yard, must be given in exchange for 450 yards of linen, at 4s. 6d. per yard? 2. Having bought 244 cwt. of sugar, at £3. 16s. 6d. per cwt., and paid in cash £22. 10s., how much coffee, at £4. 13s. 4d. per cwt., is wanted to make up the difference? 3. Bartered, 240 gallons of wine, at £2. 13s. per gallon, for 200 barrels of ale. What was the price of the ale per barrel ? 4. How much must be charged for beef, which is sold for ready money at 10s. 6d. per stone, for mutton at 9s. 4d., which is bartered at 11s 3d. per stone? 5. Bartered 4 cwt. of snuff, at £4. 7s. 6d. per cwt., and got for it eggs at 9d. per dozen, and butter at 1s. 4d. per lb.; and received 4 lbs. of butter to each dozen of eggs. How much did I get of each? 6. Cows, worth £14 each are exchanged for horses, worth £25 each, ready money; but in bartering, at £28. 15s., what ought to be charged for each cow? 7. Received 280 lbs. of tea, at 4s. 4d. per lb., for 360 yards of linen. What was it worth per yard? 8. D has 120 cwt of tallow at 18s. 2d. per cwt., E has 35 cwt. 2 qrs. 18 lbs. of sugar at 53d. per lb., and they barter their goods. What balance must D receive with the sugar ? |