CHAPTER VIII. INTEREST, DISCOUNT, PER CENTAGE, &c. INTEREST. 203. When one person lends another a sum of money, this sum is called the Principal. The money which the borrower pays to the lender for the use of his money is called the Interest. Interest is generally reckoned at so much for every £100 lent, for every year during which the loan continues, this is called the Ratio per cent. per annum. Thus, if A lends B a sum of money at 3 per cent. per annum, B will pay A£3 10s. for every £100 he has borrowed, every year as long as he keeps it. When the Interest of the Principal alone is taken, it is called Simple Interest; but if the interest, as soon as it becomes due, be added to the principal, and interest be charged upon the whole, it is called Compound Interest. 204. The solution of questions in Simple Interest depends upon an easy practical rule, deduced from Simple Proportion, e.g., if it be required to find the simple interest on £885 for 1 year at 4 per cent. we should say, "If £100 in one year gain £4 as interest, what will £885 gain in the same time?" Multiply the Principal by the rate per cent., and divide by 100. It is therefore unnecessary to state each sum as a proportion, because the division by 100 being effected in whole numbers by cutting off with a decimal point the last two figures, and in decimals by shifting the decimal point two places to the left, it is the easiest plan, generally, instead of reducing the quantities by cancelling, to multiply the principal at once by the rate per cent., and then effect the division by 100 (§ 58 (4), page 53). Y Here, in dividing the 1687 10s. by 100, we do not put down the 100, but cut off the last two figures of the pounds, giving a quotient 16, and remainder 87. We then reduce to shillings, &c., as in ordinary Long Division (see page 53). To find the simple interest for a term of years we may either multiply the interest found, as in the last example, by the number of years, or multiply the principal before we divide by the 100. Ex. 2. Find the simple interest on 2275 for 3 years, at 5 per cent. 62275 5 11375 interest for 1 year. 34125 56875 205. Discount is the abatement made upon a debt in consequence of money being paid before it was due, the value of this accommodation being reckoned in the same manner as simple interest. EXAMPLE. Ex. 1. What is the discount of £612 178. 6d., due 9 months hence, at 5 per cent. per annum ? PER CENTAGE. 205. In the preceding examples in the chapter, wherever the term "Per Cent." occurs, it referred to £100 money; there are, however, cases in which the term Per Cent. occurs where the reference is not to £100 money, but to the number 100 where the unit is an abstract number of a different kind from the above-mentioned. All such examples depend on the principles of proportion; some examples will now be worked by way of illustration, and others subjoined for practice. In other words, the question is, "Find what number bears the same ratio to 100 that 7 bears to 16." Ex. 2. In standard gold 11 parts out of 12 are pure gold, how much per cent. is dross? Ex. 3. If 8 per cent. of the persons employed in a factory in which 625 persons were engaged were women, what was the total number of the women employed? As 8 100: the women in the factory: the total number of persons engaged therein, i.e., As 8 : 100 the Ans. : 625 persons, :: Ex. 4. Two years ago an engineer's wages were 16 per month, and were then increased 12 per cent., but now being slack times, they are to be decreased by 15 per cent., what will be the present wages? Ex. 5. An engineer is to be allowed 3 per cent. on all coal left at the end of the voyage; what will be the money value to him if he has 95 tons left, coals selling at £1 28. 6d. ? 95 tons of coal at £1 2s. 6d. per ton 1 2s. 6d. X 95 = £100 I £106 178. 6d. Ex. 6. The consumption of coal is 42 tons per day, after a few days it increases to 44 tons, what is the increase per cent. ? Here the increase is 2 tons on 42 tons. 42 : 100 :: 2 : the Answer. 100 X 2 42 21 =40476 per cent. Ex. 7. The distance sailed as found from the pitch and revolutions of the propeller is 46 miles, but the actual distance sailed as found by the chart is 44 miles: required the slip of the screw per cent. ? I. 2. On £475 for 3 years at 5 per cent.: on £936 78. 3d. for 2 years at 4 per cent. On £556 138. 4d. for 6 years at 5 per cent. 3. On £945 10s. for 2 years at 4 per cent. Find the discount of 4. £31 139. 4d. due 4 months at 4 per cent: £132 38. due 2 years at 4 per cent: £1649 due 6 months at 4 per cent. 5. How much per cent. is 15 of 96; 19 of 81; 23 of 256; 185 of 7321°75. 6. An engineer's wages was £17 per month which was afterwards increased by 18 per cent., but times having become slack they are decreased by 16 per cent.: what is present rate of wages? ΤΟ EXTRACT THE SQUARE ROOT. 207. If a number be multiplied by itself, the product is called the second power, or the square, of that number. If this also be multiplied by the same number, the product is called the third power, or the cube, of that number; and so on for the fourth power, fifth power, &c. This raising of powers, which is also called Involution, is therefore nothing more than multiplying together equal factors, and is easy enough. Powers are denoted by a small figure placed above the given number at the right hand. This figure is called the index or exponent. It shows how many times the given number is employed as a factor to produce the required power. Thus the index of the first power is 1; but this is commonly omitted, that is The index of the second power is 2, the index of the third 21 = 2. the index of the fifth is 5, &c. 313, the first power of 3. That is, taking 3 as the root:— 32 3 X 3, the square, or second power of 3. 33 = 3X 3X 3, the cube, or third power of 3. 3*3 X 3 X 3 X 3, the bi-quadrate, or fourth power of 3. 3° = 3X 3X 3X 3X 3, the fifth power of 3, &c. power But the reverse operation—that is, to find the factor which, being involved in this manner, shall produce a given number—is a problem not so readily disposed of. The factor referred to is called the root of the power; so that the reverse problem spoken of, i.e., the process of resolving numbers into equal factors, is called Evolution, or the Extraction of Roots.* Powers and Roots are therefore correlative terms. If one number is a power of another the latter is a root of the latter. Thus 27 is the cube of 3 and 3 is the cube root of 27. The learner will be careful to observe that in subtraction a number is resolved into two parts; in division, a number is resolved into two factors; in evolution, a number is resolved into equal factors. Roots, as well as powers, are divided into different orders:-Thus, when a number is resolved into two equal factors, each of these factors is called the second or square root; when resolved into three equal factors, each of these factors is called the third or cube root, and so The name of the root expresses the number of equal factors into which the given number is to be resolved. on. Roots are expressed in two ways; one by the radical sign (V) placed before a number, the other by a fractional index placed above the number on the right hand. Thus V4, or 4, denotes the square or 2nd root of 4; 27, or "The extraction of the square root answers the two following questions:1. Let there be three quantities, such that the first second second third, where the first and third are given, what is the second? Evidently the result of the proportion is that the product of the second into itself is equal to the product of the first and third, or, in other words, the second is the square root of the latter product. This is called finding a mean proportional. Thus, to find a mean proportional between 7 and 63, we take the square root of 7 X 63, or of 441, viz., 21. This evidently satisfies the proportion 7: 21: 21: 63. 2. Geometrically it solves the following question. Given the length and breadth of a rectangular figure, to find the side of a square which will have the same surface. This is merely the geometrical statement of the first question." |