INTEREST. 175. The charge made for a loan of money is called INTEREST. 176. The money lent is termed the PRINCIPAL; the sum of the Principal and its Interest for a given time is called the AMOUNT for that time; and the interest of £100 for a year is known as the RATE PER CENT. per annum-the words "per annum," however, being usually omitted. 177. Interest is either SIMPLE or COMPOUND. When charged upon the Principal only, and paid in yearly, half-yearly,* or quarterly instalments, (according to agreement,) Interest is called Simple; but when such instalments, on becoming due, are added to the Principal, and made to bear interest at the same rate, money is said to be at Compound Interest. SIMPLE INTEREST. EXAMPLE I.-What interest would £78 9s. 10d. produce in a year, at 3 per cent. (per annum)? At the given rate, £100 would produce £3 in a year: at the same rate, therefore, and in the same time, £78 9s. 10d. would produce a sum as many times less than £3 as £789s. 10d. is less than £100. The answer being represented by a, we thus have the proportion £100 £78 98. 10d.:: £34: a, : or, by Alternation (see p. 169), £100 £3:: £78 9s. 10d. : a =£78 98. 10d. X31÷ From this last proportion† we find a=, =£2 14s. 111d. 100= Or thus: £78 9s. 10d.=£78·492 (§ 87); £31=£3*5; a= £78 492 × 35÷100=£2·74722=£2 14s. 114d. (§ 88). Here the Principal is £78 9s. 10d.; the Interest, £2 14s. 114d.; the Amount, (£78 98. 10d.+£2 14s. 114d.=) £81 4s. 91d.; and the Rate per cent., £31, or £3 10s. * As a general rule, (Simple) Interest-like rent-is paid half-yearly. The first proportion would involve the reduction of the first two terms (100 and £78 9s. 10d.) to pence. 178. To find the Interest of any Principal for a Year, at any Rate per cent.: Multiply the Principal by the Rate per cent., and divide the product by 100. NOTE I. To find the interest for two or more years, we multiply the interest for one year by the number of years. Thus, at 3 per cent., £78 9s. 10d. would produce in 2 years, £2 14s. 11d. X2=£5 98. 103d.; in 3 years, £2 14s. 111d. X3 £8 48. 93d.; &c. NOTE 2.-- When the rate is 5 per cent., a year's interest can be found with great facility. Instead of multiplying the principal by 5, and dividing the product by 100, we simply take of the principal; and this we do by setting down Is. for every pound, 3d. for every crown, and d. for every five-pence in the principal. Thus, in £345 128. 7d. there are 345 pounds, 2 crowns, and (2s. 7d.=31d.=) 6 five-pences. As the interest of this sum for a year, therefore, at 5 per cent., we write 345 shillings+2 three-pences+6 farthings=17 5s.+6d.+1}d.= £17 58. 7 d. From 5 per cent. we can easily pass to either 4 or 6 per cent., by subtracting or adding the interest at I per cent.-obtained from the division of the interest at 5 per cent. by 5. Thus, interest of £345 128. 7d. (for a year) at 5 per cent. 17 5s. 71⁄2d.; at 1 per cent. 17 5s. 73d.÷5=£3 9s. 1d.; at 4 per cent.= £17 58. 74d.-£3 98. 14d.=£13 16s. 6d.; and at 6 per cent.= £17 5s. 74d.+£3 9s. 11⁄2d.=£20 148. 9d. NOTE 3.--The time, when not an exact number of years, is always expressed in DAYS, or in years and DAYS. Notwithstanding what several treatises on Arithmetic imply to the con trary, there is no such thing in the commercial world as a month'sor a number of months'-interest.* 'Take, for instance, three "months"-(a) from the 1st of February to the 1st of March; (b) from the 1st of March to the 1st of April; and (c) from the 1st of April to the 1st of May. In the calculation of interest, the first of these periods would be expressed as 28 days, the second as 31 days, and the third as 30 days. Because money borrowed on the 1st of February, and repaid on the 1st of March, would have been at interest during the last 27 days of February and the first day of March; money borrowed on the 1st of March, and repaid on the 1st of April, would have been at in *This appears to be contradicted by the fact that bills of exchange are usually "drawn" for a number of months; but when a bill comes to be discounted, the discount-as we shall find hereafter-is invariably calculated for days. (See p. 202.) terest during the last 30 days of March and the first day of April; and money borrowed on the 1st of April, and repaid on the 1st of May, would have been at interest during the last 29 days of April and the first day of May: The first day-that on which the borrower received the principal is never included in the number of days for which interest is charged; the reason being that money lent on (say) the 1st of February would not have been at interest for a day until the 2nd of February. EXAMPLE II.-What interest ought to be paid for the use of £678 from the 17th of April till the 23rd of July, at 4 per cent. (per annum)? dys. Having found the "time" to be 97 days, we can put the question in this way: The interest of £100 for 365 days being £4, what is the interest of £678 for 97 days? This is merely an exercise in compound proportion, by means of which -as shown below-we find the answer to be 678 × 97×4 36500 or (the terms of the frac tion being both multiplied by 2) 678×97X8 April, ... 13 ... 31 ... 30 It will be seen that the numerator of this last fraction is the continued product of the principal (£678), the number of days (97), and twice the rate (8=4×2) : a= 100 36500: 678 X 97 :: 4 : a •678×97×4 678X97X4X2 ~678×97×8 36500 =£ 36500X2 73000 179. To find the Interest of a given Principal for a given number of Days, at any Rate per cent.: Multiply the Principal by the number of Days, and the product by twice the Rate; then, divide the result by 73,000. NOTE. By doubling the terms of the fraction obtained from the compound proportion, we have 73,000 for divisor, instead of 36,500, which would be found less convenient; and it is quite as easy to multiply by twice the rate as by the rate itself-indeed, easier in many cases. In dividing by 73,000, a schoolboy would naturally proceed as follows-employing as factors 1,000 and 73: 678 × 97x8=526128; 526128÷1,000=526.128; 526128 Here, however, is a less troublesome way of dividing 526128 by 73,000: Taking a third of 526128, we obtain 175376; taking a tenth of 175376,* we obtain 17537; and, taking a tenth of 17537,* we obtain 1753. Then, adding the four numbers together, and dividing their sum by 100,000, we find the answer to be, as before, £7.207, or £7 4s. 1d. 175376 17537 1753 7.20794 We The principle involved in this is easily understood. merely, before performing the division, multiply both the dividend and the divisor by 1+3+30 +300: I 73000 X 30% of 73000 73000 × (1+1+30+300)= 73000=10 of of 73000= 73000X300=3}ʊ of 73000=; of 3 of 73000= Money lent on a Deposit Receipt.-Instead of being lent for a definite period, and at a fixed rate of interest, money is often given to a banker upon the understanding that it can be with * By simply removing each figure a place to the right. This number being larger by 10 than 100,000, the quotient obtained when we divide by 100,000 is slightly in excess of the truth. For practical purposes, however, the error is so small as to be unworthy of notice. The last example is an illustration of this: £7.20794 exceeds £7.207 by less than a farthing. 24333 24333 24331 10 1% of 2431 24333= 100010† drawn at any time, and that, if not withdrawn within a month, it shall bear interest at a variable rate--to be determined, from day to day, at the banker's discretion, by the state of the money-market. Money employed in this way is said to be lent on a deposit receipt"--the name of the acknowledgment given to the lender by the banker. 66 EXAMPLE III.-A sum of £234 was lent on a deposit receipt, and the rate was 2 per cent. for the first II days, 1 per cent. for the next 17 days, 2 per cent. for the next 8 days, 3 per cent. for the next 25 days, 4 per cent. for the next 9 days, and 3 per cent. for the remaining 13 days; find the interest. per cent. dys. 2 for II 2 X2=4; 4× II= 44 The 2,, 3 4 ,, 171X2=3; 3×17= 51 8 2X2=5; 5X 8= 40 ,, 25 3 X2=6; 6X25=150 94 X2=8; 8X 9= 72 ,, 1332 X2=7; 7×13= 91 448 Here we have no fewer than six different rates each involving, apparently, as much work as was necessary in the last example. answer can be found, however, in the way shown in the margin. Multiplying each number of days by twice the corresponding rate, and adding the products together, we obtain 448, which, when the last figure (8) is "cut off, " becomes 44. We then set 32 5 Interest of £234 for 44 days, at per cent.= £234X44X 10 73000 1 8s. 2 d., the answer. down, as the required interest, what the given principal would produce in 44 days, at 5 per cent.* Let us now consider the reason of this: In how many days, at 5 per cent., would a principal produce as much interest as would be produced by the same principal in 11 days, at 2 per cent.? In how many days, at 5 per cent., would a principal produce as much interest as would be produced by the same principal in 17 days, at 1 per cent.? In how many days, at 5 per cent., would a principal produce as much interest as would be produced by the same principal in 8 days, at 2 per cent.? * A bank clerk, instead of calculating this, would take it from an Interest table. |