question: What is the interest of $120 for 10 months, at 6 per cent.? The statement, by the general rule given, would be as below: In the above question it is readily seen that 120×10 on the right is equal to 100×12 on the left; therefore, these four terms are cancelled, and the number remaining, 6, is the answer. But suppose the question to be, What is the interest of $-for- months, at 6 per cent. per annum ? Here the statement would be, and by cancelling the 12 and 6, the number 6 will be exterminated, and the 12 reduced to 2. Hence the only numbers remaining will be 100 and 2, and as these are both divisors, the interest of any sum for any given number of months at 6 per cent. may be found,-First, by dividing the product of the given sum and given months by 200. Secondly, by dividing half the product of the sum and months given by 100. Thirdly, by multiplying half the given sum by the months and dividing the product by 100. Fourthly, by multiplying the given sum by half the months and dividing the product by 100. To divide by 100 is simply to cut off two right hand figures from the dividend. So in like manner we may proceed with any other rate per cent. Again, to find the interest (or discount) of any given sum as computed at the banks, at 6 per cent., we have by the general rule the statement thus: Here, by cancelling, the dividing terms become 100 and 60; hence, when the number of days given are 20, we find the interest by dividing the given sum by 3, and pointing off two of the right hand figures in the quotient. When B the days are 30, we divide by 2, and point off as above.. When the days are 60, we simply point off two of the right hand figures from the given sum, &c. for any other number of days. We might give other instances of this method of abridging the general rules which we have given in this work, under the heads of Reduction, Proportion, Interest, Exchange, &c. suited to particular cases, but we trust the above will suffice. The subject may be further pursued by teachers and those for whom this work is particularly intended. Many of the questions which are usually solved by the rule of Position may be answered by the method of Equations in a very brief manner. Thus in the following examples. What number is that which being multiplied by 7, and the product divided by 6, the quotient will be 14? We know that multiplication and division are operations directly the reverse of each other. Hence, if a number is produced by multiplication, to perform a retrograde or backward operation, we use division; but if the number arises from division, we use multiplication; consequently, in the above question, the terms that are given as multipliers, we make divisors, and those given as divisors, we make multipliers, thus Again, What number is that which being divided by 15 will make? In some questions it will only be necessary to find some equation arising from the given conditions; thus-What sum, at 6 per cent. per annum, will amount to 860 dollars in 12 years? Here 100=106 in one year, and in 12 years 100=172. Hence the statement would be With these remarks the work is respectfully submitted to teachers and others interested in the instruction of youth, and the promotion of useful knowledge. Trenton, N. J. Nov. 1834. C. P. Notation and Numeration,. Contractions in Division,.. Prime and Composite Numbers,.. Notation of Decimal Fractions,.. Reduction of Moneys, Weights, and Compound Reduction, Page 17 Simple Proportion, or Single Rule of 19 Three, 23 Practice, ib. Insurance,Commission,andBrokerage 114 Page 90 22 21 Compound Proportion, or Double Rule of Three, 97 102 25 Interest,.. 105 ib. Simple Interest.. ib. 28 Compound Interest,. 115 30 Discount,.. 116 31 Equation of Payments.. 117 32 Barter, 118 33 Loss and Gain, 121 122 125 129 132 134 142 ib. Fellowship, 34 Tare and Trett, 35 Exchange,.. 39 Domestic Exchange, ib. Involution, or Raising of Roots,. 42 Evolution, or Extracting of Roots... 143 The Square Root,. 43 Of the Cube Root, 44 Extraction of the Cube Root, ib. Extraction of the Cube Root by a short way,.. 45 ib. Extraction of Roots of all Powers,. ib. Double Position,. 144 149 ib. 151 152 153 155 47 Permutation, 159 48 Combination,.. 157 51 Duodecimals, ib. NOTE.-The other signs used will be found explained throughout the work. 16 ARITHMETIC is the art of computing by numbers. The principal rules for its operation are Notation, Numeration, Addition, Subtraction, Multiplication, and Division. NOTATION. Notation is the method of expressing, by means of certain characters or figures, any proposed number or quantity. The figures used in Arithmetic are as follow: 1, one; 2, two; 3, three; 4, four; 5, five; 6, six; 7, seven; 8, eight; 9, nine; 0, cipher, naught, or nothing. These characters or figures were formerly all called ciphers, and hence the art of arithmetic was called ciphering. NUMERATION Is the reverse of Notation, and teaches the manner of reading or expressing, in words, the value of any number represented by the characters or figures above, any how combined or repeated. Besides the value which those figures have respectively, they have also another value, which depends on the place they stand in when joined together, as exhibited in the following 4 54 6 5 4 6 5 4 6 54 1 One 2 1 Twenty-one 3 2 1 Three hundred and twenty-one 321 4 Thousand 321 3 2 1 54 Thousand 321 321 7 Million 654 Thousand 321 |