£ £ 1000 : 370 :: 1 400 12 lation, we find that B will receive £96, 4s., and C £74. | certain rate per hundred pounds per annum, or for the Another method of comwhole year. The hundred pounds, for shortness, is called cent., a contraction of centum (Latin), a hundred. If the money is lent for a less term than a year, then it becomes a question in arithmetic what is the proportion of interest chargeable. According to the existing laws (1849), £5 per cent. is the highest legal interest that is chargeable on money lent in any other form than by a bill or promissory-note. The interest, commonly called discount, that may be taken for advances on bills or promissory-notes is left unlimited. In general it varies from about 4 to 6 per cent., according to the state of the money market or trustworthiness of the borrower. £5 being equal to 100 shillings, the interest on £1 for a year at 5 per cent. is consequently 1s.; and if the interest be £2, 10s., or 2 per cent., the charge will of course be 6d. per £1. This is putation would consist in 1000)4800 (4 800 4 1000)3200(3 3000 200 } viding 7400 by 1000, the same result will be found if so very simple a matter of calculation, that interests Practice and Mental Arithmetic. Under these names are comprehended short practical methods of working arithmetical questions, partly or wholly by the mind, or by the jotting of a few figures. Ordinary business questions are seldom solved by the Rule of Three, but principally by Practice or Mental Arithmetic. The kind of questions most commonly occurring are computations of the aggregate value of a certain number of articles at a certain price, and the adding of the whole together to find the sum-total. The plan usually adopted is to calculate the value of any number of articles by the nearest round sum, and then to apply the difference. For instance, a lady is buying a gown from a haberdasher; she has received 15 yards at 54d. per yard; the haberdasher tells her in an instant she has to pay 6s. 104d. He knows it is so by saying internally to himself 15 yards at 6d. would be 7s. 6d.; then if I take 15 halfpence, that is, 74d., from the 7s. 6d., I find that 6s. 104d. will remain. of 5 or 2 per cent. per annum may in most instances be calculated mentally, or at least with a few figures. When the interest is 3, 3, 4, or 44, per cent., and for a less period than a year, the calculation is more complex, and will require to be wrought as a question in Rule of Three, or the amount may be determined by an appeal to Interest Tables. Interest on broken periods is not calculated by calendar months, but for days-the exact number of days from the day of lending to the day of paying; and therefore the calculation of the number of days is an important preliminary in the transaction. Bankers and merchants, to save the trouble of calculation, appeal to a table which shows the number of days from one day to another in the different months of the year. The following is an example of the mode of working a question of interest for a whole year. What is the amount of interest payable on £649, at 4 per £100 or per cent. per annum ? £ Another principle followed in this practical arithmetic is to work by aliquot parts. By remembering that a penny is the 12th of a shilling, or the 240th of a pound; that 6s. 8d. is the third of a pound; that 3s. 4d. is the sixth of a pound, and so on, we are able to save much of the ordinary figuring. As an illustration, let it be required to find the value of 3567 articles at 3s. 4d. 73000) 39564 each. By the usual rules of arithmetic, this question would be performed by multiplying the 3567 by 40 (there being 40 pence in 3s. 4d.), and then dividing by 12 to bring it into shillings, and by 20 to bring it into pounds. The practical method is much shorter; 3s. 4d. being the sixth of a pound, if we divide 3567 by 6, we at once obtain the amount thus Interest. Interest is an allowance for the use of money, paid by the borrower to the lender. The amount of allowance on £100 is called the rate of interest. The amount of money lent is styled the principal. In the United Kingdom, it is customary to reckon the interest at a = 73000 134 1928 134 1,20 £29 4 1} When the period consists of less than a year (365 days), multiply the principal by the number of days, and by twice the rate, and divide by 73000. (We get this 73000 by multiplying 365 by 2 and by 100). For example -What is the interest of £235, 10s. for 28 days, at 3 per cent. per annum? Here, for convenience, we begin by multiplying by 7 and by 4, instead of 28. The 6 is twice the interest, 3. When partial payments are made, it is necessary to deduct them from the principal, care being taken not to confuse principal with interest. The following is a rule to follow in such a case:-Calculate the interest on the principal up to the time at which the first par tial payment is made, and add it to the principal; from this sum subtract the money paid, and the remainder is a new principal; compute the interest on this principal from the time of the first payment up to the time of the second payment, add it to the latter principal; from the sum subtract the second sum paid, and the remainder is again a new principal; and continue this process till the last payment. Mensuration. Mensuration refers to the measurement of objects, and is of three kinds-lineal, or measuring by mere length; superficial, which respects breadth as well as length; and solid, which includes length, breadth, and thickness. In the United Kingdom, the foot of twelve inches is the common standard of measurement. A draper measures cloth with a rod of three feet or one yard, and workmen usually measure the dimensions of walls, or the superficies of apartments, by a ribbon marked in feet and inches. In common usage, the inch is divided into eighths and sixteenths. As lineal measurement requires no explanation, we pass to a consideration of superficial measurement, or that of both length and breadth. A superficial foot, which is the basis of this kind of measurement, is either a square of a foot in length and a foot in breadth -in other words, a foot each way-or it is any dimension in which the length multiplied by the breadth will form a foot. For example, the surface of a piece of wood, 2 feet in length and 6 inches in breadth, is a superficial | foot. A superficial foot is generally called a square foot, and is a superficies consisting of 12 times 12, or 144 square inches. Sometimes the term square feet is confounded with that of feet square, which is quite a different thing. A piece of cloth said to measure six square feet, consists of six squares of a foot each; but a piece said to measure six feet square would be six feet along each side, and comprise thirty-six squares of a foot each. Inattention to these distinctions has often led to awkward errors and disputes. The method of finding the superficial contents of any oblong surface, is to multiply the length by the breadth; but other points require attention in the calculation. To arrive at exactness, the inch is reckoned to contain 12 seconds or parts, each second contains 12 thirds, and each third contains 12 fourths. Feet multiplied by feet, give feet; feet multiplied by inches, give inches; feet multiplied by seconds, give seconds; inches multiplied by inches, give seconds; inches multiplied by seconds, give thirds; and seconds multiplied by seconds, give fourths. Rule for working questions-1. Write the multiplier under the multiplicand, feet under feet, inches under inches, seconds under seconds, &c. 2. Multiply each denomination of the length by the feet of the breadth, beginning at the lowest, and place each product under that denomination of the multiplicand from which it arises, always carrying 1 for every 12. 3. Multiply by the inches, and set each product one place farther to the right hand. 4. Multiply by the seconds or parts, and set each product another place towards the right hand. 5. Proceed in this manner with all the rest of the denominations, and their sum will be the answer. Example.-Multiply 6 feet 3 inches by 3 feet 2 inches. In working, we begin by multiplying the 3 inches by 3, and then the 6 feet also by the same 3 below it; this gives 18 feet 9 inches. This makes 18, 9. We now multiply by the 2 inches, placing the 6 one remove to a side. By then multiplying the 6 by 2, we have 12 inches or 1 foot, and 19 setting down the one below the 18, we add up. The answer is 19 feet 9 inches and 6 seconds. Questions of this kind may also be wrought by decimals. Ft. In. 6 3 3 2 18 9 0 6 9 6 To measure the solid contents of an object, a different process is pursued. Suppose we take a piece of wood measuring a square foot, and cover it with dice, each die an inch square and an inch high, the wood will be covered with exactly 144 dice. Let us now put a second layer of dice on the first, and the number will be doubled, or 288 dice; and if we thus go on adding layer above layer till we have 12 layers, the number of dice will be finally 12 times 144, or 1728 dice; in other words, we shall have formed a cube consisting of 1728 solid inches. Such is solid measure. Solid measure is computed arithmetically, by multiplying the length by the breadth, and the product by the thickness. Civil engineers, who require to calculate the solid contents of masses of earth, with a view to excavation, resort to this simple rule; it is likewise 29 20 580 14 2320 580 followed by builders in reference to walls, plumbers to cisterns, and other artificers. The following is a question not unlikely to occur. Required the contents of a cistern 9 yards 2 feet in length, 6 yards 2 feet in breadth, and 4 yards 2 feet in depth. Analyse these dimensions-9 yards 2 feet are 29 feet; 6 yards 2 feet are 20 feet. Multiply the 29 by 20, and then multiply by 14 (14 feet being equal to 4 yards 2 feet); next divide by 27, which is the number of solid feet in a solid yard. The question is thus wrought in the margin. The answer is seen to be 300 yards 20 feet. 27)8120 300 y. 20 f. Evolution. The extraction or discovery of the square and cube roots of numbers, forms a department of arithmetic called Evolution, and is useful in some kinds of measurement. In the following table, the squares, cubes, and fourth and fifth powers of the nine units are given. The square of any number, it will be observed, is gained by multiplying the number by itself; the cube, by multiplying the square by the number; the biquadratic by multiplying the cube by the number; and the sursolid by multiplying the biquadratic by the number: Root, or 1st power, . 1 2 27 81 The powers of numbers are usually expressed by adding a small figure to them; thus 32 signifies the second power or square of 3-that is, 3 x 3 = 9; 33 signifies the third power or cube of 3, or 3 x 3 x 3 = 27. The small figure thus added is called an index or exponent, because it indicates or exposes to view the powers of the quantity to which it is affixed. The square root of any small quantity may easily be ascertained by means of the multiplication table; for instance, 6 is at once seen to be the square root of 36, because 6 x 6 = 36; but when larger numbers occur, it is requisite to have recourse to another process. Supposing it were required to find the length of the side of a square which contains 59,049 square inches on its surface, it is evident that it cannot at once occur to the mind what number, by being multiplied into itself, produces this quantity. It is therefore requisite to find the square of one part of the number, and then the square of the other part. To effect this, the whole number is divided into parts, by a dot being placed over each alternate figure, beginning at the unit. The reason for thus separating it into periods of two figures each, is because the square of a single figure never consists of more than two places, the square of a number of two figures of not more than four places, &c. After the whole number has been thus divided, the root whose square comes nearest to the first period must be placed in the quotient, and its square subtracted from the first period. Thus 5 being the first period, 59049(243 2, as being the root of 4, is placed in the quotient, and the square of 2 (2 x 2 = 4) is subtracted from 5, because 5 here forms the first period. The next period (90) is then annexed to the remainder for a new dividend. The root is doubled for a divisor, and 4 44)190 176 483)1449 1449 after ascertaining how often it is contained in the dividend, omitting its last figure, the result (4) is placed both in the quotient and also in the unit's place of the divisor: the whole number (44) thus formed, must be multiplied by 4, and the product must be subtracted from the dividend. The same process must be repeated until there is no remainder, and the quotient will contain the root sought, which in this case is 243. The rule for extracting the cube root is equally tedious, and indeed almost too complex for practical purposes. The methods of extracting the roots of higher powers are so complicated, that they are usually omitted in arithmetical works. The 4th, or biquadratic root, may be ascertained by extracting the square root twice, because it is the square root of the square root. The 6th root is the square root of the cube root, or the cube root of the square root. The 8th root, being the square root of the biquadratic root, may be found by extracting the square root three times. In the same way, the 9th root may be called the cube root of the cube root, and may be found by extracting the cube root twice. But the common arithmetical operations furnish no method of obtaining the 5th, 7th, 10th, 13th, 14th, 15th, 17th, 19th, 20th, and most other roots; and this part of arithmetic continued, therefore, involved in practical difficulties, until the invention of logarithms. 4 22 8 23 16. 24 64 25 26 It is evident that this might have been expressed still more concisely by omitting the 32 number signifying the ratio (2) throughout, retaining only the indices or exponents, they alone being sufficient to indicate the degree of power to which the ratio is raised in each term. Exponents thus placed in order, opposite to a series of numbers in geometrical progression, are, as we have said, called Logarithms, or literally, number of the ratio. The most tedious sums in multiplication can, by means of logarithms, be solved simply by addition. For instance, if it were required to multiply 256 by 32, it would merely be requisite to add the logarithms standing opposite to those two numbers-their sum (13) stands opposite the product required, namely, 8192: Again, to multiply 128 by 16, we take the 7th + 4th, or 11th, power of 2, because 128 is the 7th and 16 the 4th power of 2; opposite to the sum of 7 and 4 (11) is 2048, equal to the result of the multiplication of the two numbers. This also affords a quick and easy method of dividing one number by another: thus if it be required to divide 4096 by 16, it is only requisite to ascertain the difference between the logarithm of these two numbers, which in this case being 8, the figure opposite to 8 is the required quotient, indicating how often 16 is contained in 4096-namely, 256 times. The indices or exponents, 1, 2, 3, 4, &c. might, however, denote the powers of any other number or ratio. Every different ratio or geometrical progression gives a different system of logarithms. Soon after the invention of logarithms by Lord Napier, it occurred to Briggs, then professor of geometry at Oxford, that a system whose base or ratio is 10 is preferable to all others, on account of its being analogous to the general method of notation. In A. D. 1624, Briggs published the tables of logarithms which are now in common use. In this system, 10 being the ratio or multiplier, the terms may be thus expressed 1 0 1 2 3 10,000, &c. 104 &c. The exponents, 1, 2, 3, 4, are, as was shown in the previous table, the logarithms of the opposite numbers, and might therefore have been written thus10 100 1000 10,000, &c. 4 &c. The logarithms of all intermediate numbers, such as those between 1 and 10, 10 and 100, &c. are found by ascertaining the geometrical mean proportions between any two numbers, and likewise the corresponding arithmetical means between the indices of those numbers. In every system, 0 is the logarithm of 1. Hence the logarithm of any number between 1 and 10 must be less than a whole number, and therefore a decimal fraction; and the logarithm of any number between 10 and 100 must be one with a fraction. In the same way, the logarithm of any number between 100 and 1000 must be 2 and a fraction, and so on, through the rest of the series. The integers, 0, 1, 2, 3, &c. to the left of the decimals in logarithms, are called the characteristics of those logarithms. Thus 0 is the characteristic of all numbers between 1 and 10; 1 is the characteristic of all between 10 and 100; 2 that of all between 100 and 1000, &c. In tables of logarithms the characteristics are generally omitted. [For a complete and accurate collection of MATHEMATICAL TABLES, the reader is referred to CHAMBERS'S EDUCATIONAL COURSE.] The method of using a table of logarithms is the following:-To ascertain, in the most common tables, the logarithm of a number less than 10,000, it is merely necessary to find the first three figures of the number in the left-hand column, and the fourth figure in the horizontal line at the top; then opposite to the former, and under the latter, is the required logarithm. If the given number consist of five figures, find, as before, the logarithm belonging to the first four figures; the difference between it and the next logarithm may be ascertained from the margin; this difference is to be multiplied by the fifth figure of the given number, and one figure cut off from the right of the product; and the other figures of this product being added to the preceding logarithm, and the index 3 prefixed, the sum will be the required logarithm. The process of finding the natural number of any given logarithm is precisely the converse of that just described. The given logarithm must be sought in the table; and against it, in the left-hand column, will be found the natural number. It has already been shown that tables of logarithms afford an easy method of finding the product in multiplication, and the quotients in division, by means of simple addition and subtraction. Logarithms are equally applicable to the solution of questions in Proportion or the Rule of Three, it being merely requisite to add the logarithms of the second and third terms together, and from the sum to subtract the logarithm of the first: the remainder is the logarithm of the fourth term. A number may, by means of logarithms, be raised to any required power, by multiplying the logarithm of the number by the index of the power. Thus to square any number, multiply its logarithm by 2; to cube a number, multiply its logarithm by 3; and so on. To extract the root of any given number, the logarithm of the number must be divided by the proposed fraction; thus denotes that a is divided by b. index, and the quotient is the logarithm of the required | be divided, with a small line between, in the form of a root. Thus to find the square root of 169, it is only necessary to divide its logarithm, which is 2-2278867 by 2; opposite to the quotient will be found the number 13, which is the square root of 169, because = 169. [To those who desire a thorough knowledge of ARITHMETIC, we refer to the complete and accessible treatise, by Mr Marr, published in CHAMBERS'S EDUCATIONAL COURSE.] 13 x 13: mmmm ALGEBRA. It has been shown in Arithmetic that the powers of quantities are denoted by a small figure, called the exponent or index of the power. Thus a xa, or the square of a, is expressed by a2; b xbx b, or the cube of b, is expressed by 63, &c. The cube of a + b, or of (a + b) (a + b) is expressed thus: (a + b)3. The roots of quantities are represented by the sign Again, a is the cube root of a2, or of the square of a. We have seen that logarithmic tables may be used as a substitute for many lengthened operations in arithmetic. It is evident that the value of all methods of computation lies in their brevity. Algebra must be considered as one of the most important departments of mathematical science, on account of the extreme rapidity and certainty with which it enables us to determine the most involved and intricate questions. The term algebra is of Arabic origin, and has a reference to the resolution and composition of quantities. In the manner in which it is applied, it embodies a method of performing calculations by means of various signs and When two or more letters or quantities are conabbreviations, which are used instead of words and nected together by signs, the combination is called an phrases, so that it may be called the system of sym-algebraic expression, and each letter or quantity is bols. Although it is a science of calculation, yet its operations must not be confounded with those of arithmetic. All calculations in arithmetic refer to some particular individual question, whereas those of algebra A quantity of two terms, as b+c, is called a binomial. refer to a whole class of questions. One great advan- When a binomial expresses the difference between tage in algebra is, that all the steps of any particular two quantities, it is called a residual, as a — - b. course of reasoning are, by means of symbols, placed A quantity consisting of 3, 4, or many terms, are at once before the eye; so that the mind, being unim-called respectively trinomials, quadrinomials, multinopeded in its operations, proceeds uninterruptedly from one step of reasoning to another, until the solution of the question is attained. Symbols are used to represent not only the known, but also the unknown quantities. The present custom is to represent all known quantities by the first letters of the alphabet, as a, b, c, &c. and the unknown quantities by the last letters, x, y, z. The symbols used in arithmetic to denote addition, subtraction, &c. belong properly to algebra. Thus the sign+plus, denotes that one quantity is to be added to another, and is called the positive or additive sign; all numbers to which it is prefixed are called positive. The sign minus, denotes that one quantity is to be subtracted from another; it is called the negative or subtractive sign, and all quantities to which it is prefixed are called negative. If neither + be prefixed to a quantity, then the sign + plus is understood. nor called a term. Quantities of one term are called simple quantities; as a, 2a, 3b, &c. mials. = The sign placed between two quantities shows, as in arithmetic, the equality of those quantities. When quantities are connected by this sign, the expression is called an equation: thus, 2 + 4 = 6, is an equation; as also, a + b = c-f. The symbol or is called that of inequality, it being placed between two quantities, of which one is greater than another; the open part of the symbol is always turned towards the greater quantity: thus, ab denotes a to be greater than b; and cd denotes d to be greater than c. The sign of difference is~, only used when it is uncertain which of two quantities is the greater; thus, e~ f denotes the difference between e and ƒ when it is uncertain which is the greater. The word therefore, or consequently, often occurring in algebraical reasoning, the symbol.. has been chosen to represent it: thus the sentence, 'Therefore a + b is equal to cd,' is thus expressed in algebra, Like quantities are such as consist of the same letter or letters, or powers of letters: thus, 6a and 2a are like quantities, and also 4 abc and 9 abc. Unlike quantities are such as consist of different letters; as, 4a, 5b, 6 ax, 4 cd, which are all unlike quantities. The general sign to denote that one number is to be multiplied by another, is x; but it often occurs that. a + b = c + d. one letter has to be multiplied by another, and this is represented by placing those letters one after the other, generally according to the order in which they stand in the alphabet; thus a multiplied by b, is expressed by ab. The multiplication of quantities consisting of more than one term, as for instance a + b by c+d, may be represented by any one of the following methods: a + bx c + d, or a + b. c + d, or (a + b) (c + d). The bar drawn over a + b and c +d, which in the two first examples marks them as distinct quantities, is called a vinculum, but brackets or parentheses, as in the last example, are now in more frequent use. When a letter is multiplied by any given number, it is usual to prefix that number to the letter. Thus twice a, three times b, four times c, six times x, &c. are expressed thus: 2a, 3b, 4c, 6r; and the numbers 2, 3, 4, 6, thus prefixed, are called the coefficients of the letters before which they stand. The sign between two numbers shows, as in arithmetic, that the former of those numbers is to be divided by the latter; thus ab, means that a is to be divided by b. It is, however, more usual to place the number to be divided above that by which it is to Addition. The operation of addition in arithmetic consists, as has been shown, simply in joining or adding several quantities together: thus, 4 + 8 + 7 + 6 = 25. This same process is always used in algebra, whenever like quantities with like signs are required to be added: thus, 2a+3a+6a= 11a; and 7 b — 4 b — 6 b — — 17 6. But as it often happens that like quantities which are to be added together have unlike signs, addition has in algebra a far more extended signification than in arithmetic. Thus to add 7 a + 4 a to 8a-3 a, it is evident that, after 7 a + 4 a + 8a have been added according to the usual method, 3 a must be subtracted. Hence the general rule for the addition of like quantities with unlike signs is to add first the coefficients of the positive terms, and then to add those of the negative terms; the less sum must be subtracted from the greater, and to this difference the sign of the greater must be annexed, with the common letter or letters. Thus let it be required to add 7a-3 a+4a+5a6 a 2a and 9a; 25 a will be found the sum of the positive terms, and 11 a that of the negative; 11 a being the less number, must therefore be subtracted from 25 a, the greater, leaving a remainder of 14 a, which is the required amount. The reason of this mode of procedure may be shown by a simple illustration :-As a letter may represent any quantity, let a represent £5; and suppose that a gentleman has in one bank 7 a, or seven five pounds; in another bank 4 a, and in another 9 a, or nine five pounds: let us suppose, too, that his tailor's bill is 6 a, or six five pounds; his baker's 3 a, or three five pounds; and his butcher's 2a, or two five pounds; it is evident that, to ascertain how much money is really his own, he must first compute the whole value of his property dispersed in the different banks, then the amount of the bills of his creditors, and then find the difference between the two sums by subtracting the less from the greater. When the aggregate of the positive terms is equal to that of the negative ones, the sum of the two quantities will be equal to 0: thus, if a man possesses £2000, and owes £2000, it is evident that when his debts have been deducted from his property, nothing will remain. Unlike quantities can only be added by collecting them in one line, and prefixing the proper sign of each; thus the sum of 3 a +26+4c-2d can only be rendered 3a+2b+4c2d; this will be evident by reflecting that different letters in the same algebraical expression always represent different quantities, which cannot of course be added into one sum unless their precise value be known. Thus the addition of a and b can not be represented by 2 a or 2 b, because that would imply that a is equal to b, which it is not necessarily neither could it be represented by ab, because ab denotes the multiplication of the two quantities; the only method, then, of expressing these sums is thus, a + b. When like and unlike quantities are mixed together, as in the following example, the like quantities must first be collected together according to the method above described, and all unlike quantities must be annexed in order : When two like quantities, having like signs, are to be subtracted the one from the other, the process is precisely the same as that already described in Arithmetic: thus, 3 a subtracted from 7 a, leaves as a remainder 4 a. From 8a+ 5 a take 6 a + 2 a, and the remainder will be 2 a + 3 a, or 5 a. But supposing it were required to subtract 6 a4 a from 9 a, it is evident that some other process must be adopted; because if 6 a be subtracted from 9 a, the proposed operation will not be performed; for it is not 6 a, but 6 a- -4 a, that is, 2 a, which is required to be subtracted from 9 a; 6 a subtracted from 9 a leaves 3 a, which is 4 a less than would result from subtracting 2 a from 9 a; but if to 3 a we add the other termnamely, 4 a-the sum will be the remainder sought, because 3 a + 4 a = 7 a; and if 2 a be subtracted from 9 a, which is just the same question in another form, for 6 a-4 a is = 2 a, the remainder is just 7 a, as before. So if ab is to be subtracted from c, the son. When like quantities are to be subtracted from each other, it is usual to place them in two rows, the one above the other; the signs of the quantities to be subtracted must, for the reason above adduced, be conceived to be changed; and the several quantities must be added, as shown in the following examples:From 5 ax + 7 xy-2y Take 3 y + 3 ax- 6 xy Remainder, 2 ax + 13 xy 5 y Multiplication. The multiplication of two quantities is performed by multiplying, as in arithmetic, the coefficients of the quantities, and then prefixing the proper sign and annexing letters: thus the product of 3 a, multiplied by 5 b, is 15 ab, and 7 a × 4 ab = 28 ab. When the signs of both quantities are alike, the sign + is to be prefixed; but when unlike, the sign must be prefixed, which may be thus shown at one view :— 1. +multiplied by + produces + multiplied by produces + produces multiplied by + produces 2. 3. +multiplied by 4. abx 6 a5 ÷ 2 a3, or remainder would be ca + b, and for the same rea-ceeds that of the same letter in the dividend, the latter When the exponent of any letter in the divisor exIt may therefore be given as a general rule, exponent must be subtracted from the former, and the that all the signs of a quantity which is required to quotient will be in the form of a fraction; thus, be subtracted from another, must be changed: thus when 4 x - 3 y is subtracted from 7 a + 5 b, the remainder is written thus, 7 a + 5 b − 4 x + 3y. 12 a3x38 ax5 12 a3x2 3 a2 |