7. If £15-12 pay 16 labourers for 18 days, how many labourers, at the same rate, will £35 - 2 pay for 24 days? Answ. 27. 8. If 36 yards of cloth, 7 quarters wide, cost £25 - 4, what cost 120 yards of the same quality, but only 5 quarters wide? Answ. £60. 9. If a tradesman earn 16 guineas in 108 days, how many sovereigns would he earn, at the same rate, in 270 days; 20 guineas being equivalent to 21 sovereigns? Answ. 42. 10. If the rent of a farm of 26 a. 2 r. 23 p. be £50 - 89, what would be the rent of another, containing 17 a. 3 r. 2 p., if 6 acres of the latter be worth 7 of the former? Answ. £39 - 4 - 7. 11. If a puncheon of rum containing 85 gallons, cost £58 - 8-9, what would be the value of a hogshead containing 63 gallons, and composed of four parts of the same rum, and one part of water? Answ. £34 - 13 -0. 12. In what time would 23 men reap a field which 40 women would reap in 6 days, if 7 men can reap as much as 9 women? Answ. 8 days. 69 13. If a person, walking 13 hours each day, travel 191 miles in 7 days; in how many days of 10 hours will he complete the remainder of a journey of 500 miles, at the same rate each hour? Answ. 141378. 14. If 63 lbs. of tea cost £20 106, what cost 70 lbs. of a different quality, 9 lbs. of the former being equal in value to 10 fbs, Answ. £20 10 - 6.* of the latter? INTEREST. The sum to be paid by a person for the use of money which he owes, is called the INTEREST of that money. The money due is called the PRINCIPAL. The sum of the principal and interest is called the AMOUNT. The RATE is the money allowed for the use of one hundred pounds for any given time, but usually for a year. When interest is charged on the original principal only, it is termed SIMPLE INTEREST. When interest is charged, not only on the original principal, but also on the interest as it becomes due, it is called COMPOUND INTEREST. *Many of the questions usually proposed under the head of compound proportion are quite misplaced, as some of them are merely anticipations of what is afterwards delivered in interest, and others belong to mensuration, or other subjects, with the principles of which the pupil is supposed to be unacquainted. Hence, the number and variety of exercises here given, are purposely less than in several other works on arithmetic. It is scarcely necessary to remark that per cent., a contraction for per centum, means by the hundred; and that per annum means by the year. SIMPLE INTEREST.* RULE I. To find the interest of a given sum for a year, at a given rate per cent. per annum: Multiply the principal by the rate, and divide the product by 100.+ Or, As £100 is to the rate per cent. per annum, so is the principal to its interest for one year. Exam. 1. Required the interest of £5765-81, for 1 year, at 6 per cent. per annum.‡ * In interest five quantities are concerned, the principal, the rate, the time, the interest, and the amount; and any three of these, except the principal, the interest, and the amount, being given, the rest can be found. Hence, computations in interest admit of several problems. The most useful, however, is that in which the principal, the time, and the rate are given, to find the interest or the amount. This problem may be resolved, in all cases, by means of the first or second rule: but the third and fourth present modified, and, in many cases, shorter methods of effecting the same. £ S. d. 100)8947 13 8 £899-61 61 The following rule will be found useful: To divide money by 100, for the pounds of the quotient, take the pounds of the dividend, except the last two figures, which are to be divided by 5 for shillings: from the remainder, with half the shillings of the dividend annexed, reject a twentyfifth part, and regard what remains as farthings. If there be pence, or an odd shilling, their effect in modifying the quotient may be estimated as nearly as possible. Thus, let it be required to divide £8947 - 13 - 8 by 100. Here, by cutting off two figures, we have £89; and one fifth of 47 is 9, the shillings required, and the remainder is 2. This remainder, with 7, the half of 14 shillings, annexed, becomes 27, from which 1, nearly its twenty-fifth part, being rejected, we have 26 farthings, or 6d. We use 14 shillings in this example, because 13s. 8d. is nearly 14s. £2658-16 10 The reason of this process will be understood from the rule given in page 190. As another example, let it be required to divide £2658 - 16 - 10 by 100. Here, after cutting off two figures, we have £26, and the fifth of the remainder is 11 for shillings, with 3 remaining. This remainder being prefixed to 81, the half of 17s., to which 16s. 10d. is nearly equal, we have 381, the twenty-fifth part of which is obviously about 11. Then 381 being diminished by this quantity, the remainder is 37 farthings, or 1d., which completes the quotient. £26-11- 93 1 The rate of interest has varied much at different periods, and in different countries, but it has been generally observed to diminish as commerce extends. In Italy, about the beginning of the thirteenth century, it varied between 20 and 30 per cent. per annum; and, in the Netherlands, it was fixed by Charles V. in 1560 at 12 per cent. By an Act of the 37th year of Henry VIII., interest in England was not to exceed 10 per cent. By the 21st of James I. it was reduced to 8 per cent. Soon after the Restoration it was reduced farther, to 6 per cent. ; and in the 12th of Anne, to 5 per cent. The legal rate of interest in Ireland was 6 per cent. The Usury Laws imposing these restrictions were entirely repealed in 1854 by 17 & 18 Vic., c. 90.. Or thus, according to second note in page 225. 6 £576-5-8 100)3457 - 14- 3 £576-5-8 6 £34157-14- 3 11/54 12 6|51 2/04 The reason of the operation is quite evident, as it is nothing more than this: as the principal, £100, is to its interest, £6, so is the principal, £576-5 - 81, to its interest; and it is evident that, as often as the one principal contains its interest, so often will the other contain its interest: that is, by the nature of proportion, the interest will be proportional to the principal. Exam. 2. Required the interest and the amount of £619 - 9-6, for 1 year, at 5 per cent. per annum. In this operation the principal is multiplied by 5, for 5 per cent.; and for per cent. half the principal is added to the product. At the conclusion of the contracted division, it gives the result more nearly true to reject one, than nothing, from 21, though less than 25, and more especially as there is 3d. in the dividend. In the examples that follow, the division at full length will be omitted; it may be proper, however, for the pupil occasionally to work exercises both ways. Exam. 3. What is the interest of £1374-19, for 1 year, at 5 per cent. per annum? Exercises. Find the interests of the following sums, for 1 year, at the given rates per cent. per annum : RULE II. To find the interest of a given principal for any other time than a year: (1.) Find the interest for a year, by Rule I. (2.) As one year is to the given time, so is the interest for one year to the interest required. The work may frequently be abbreviated by finding the interest for months or other fractional parts of a year, by the method of aliquot parts. In using this method, the answer will often be found with more ease, or with a greater degree of correctness, by multiplying by the rate; then multiplying, or taking aliquot parts for the time; and, last of all, dividing by 100. Exam. 4. Required the interest of £99 - 24, for 23 years, at 4 per cent. per annum. tiplied by 2, the number of years. It might have been done by multiplying by 3, and subtracting a fourth of £3 - 19 - 31. The formal analogy would £319 31, interest for 1 year. Exam. 5. Required the interest of £179 - 12 - 11, 7 months, at 5 per cent. per annum. have been, as 1 y. 2 y.:: £3 - 19-3: £10-18-03 The interest for 1 year is found, by the method already explained, to be £8-19 - 7. The rest of the work, by aliquot parts, is as follows: the answer thus found is more nearly true, than that which, in many cases, would be obtained by the other method.* Exercises. Find the interests of the following principals, for the given times, and at the given rates per cent. per annum : Exam. 6. Required the interest of £342 - 11 - 8, for 86 days, at 4 per cent. per annum, In this exercise, the interest for a year is found (by Rule I.) to be £1314-03, nearly; and as 365 days: 86 days :: £13 14-03 : £3 - 4 - 7, nearly, the interest required. *The interest of a sum for any number of months, at 6 per cent. per annum, may be very easily found by multiplying the sum by half the number of months, and dividing the result by 100: and hence the interest at other rates may be derived by means of aliquot parts. Thus, to find the interest of £250 for 10 months, at 4 per cent. per annum, we multiply £250 by 5; and dividing the product by 100, we find the interest at 6 per cent. to be £12 -10 -0. We then take from this a third of itself, and the remainder is £8 - 6 - 8, the interest required. The reason of this is plain, since the number of pounds in the rate (£6) is half the number of months in the year. Other contractions in the computation of interest will be found in the article on mental arithmetic. |