APPENDIX. A. To prove the rule for finding the G.C.M. of two expressions. (See Arith. § 35; Alg. § 32.) Let X, Y, denote the two expressions. Divide Y by X, and let p denote the quotient, and Z the remainder, that is, Y-pX. Divide X by Z, and let q denote the quotient, and W X qZ, the remainder. Repeat this process until there is no remainder. Suppose W(= X XqZ), the last divisor. = X)Y (p pX Z)X (1 W)Z(r rW Then W divides Z without remainder; therefore it must divide W+qZ, or X, without remainder; and therefore also it must divide Z + pX, or Y, without remainder. Hence W is a common measure of X and Y. And as W is the greatest measure of itself, and as every measure of X and Y must be a measure of Y-pX, or Z, and therefore of X - qZ, or W, it follows that W is the greatest common measure of X and Y. B. To illustrate the rule for converting a vulgar fraction into a decimal. (Arith. § 16.) a Let represent the vulgar fraction; æ, the significant figures of the decimal; y, the number of O's assumed in the division. Hence, if the numerator be multiplied by any power of 10, and this product be divided by the denominator, the quotient will give the significant figures of the decimal, and the number of decimal places will be the same as the index of the power of 10. C. To prove the rule for finding a vulgar fraction equivalent to a given recurring decimal. (Arith. §57.) Let a represent the non-recurring decimal figures, and let m their number. Let k represent the recurring figures, and let n = their number. Let Frepresent the whole decimal, and let its value. (10)m, where n is greater than m, will m 9's, followed by m O's. Now (10)" always consist of n Hence the rule: Take for the numerator of the vulgar fraction, the figures to the end of the first period, diminished by the non-circulating part; and for the denominator, as many 9's as there are circulating figures, followed by as many O's as there are circulating figures. non D. Annuities.--(Arith., § 96.)—By aid of algebraical symbols we can find a formula which enables us to find the present value of any annuity, whether the interest be reckoned as simple or compound. In both cases, however, we have to assume results beyond the stage to which we are limited in this volume. We will merely give the formula when the interest is reckoned as simple. Here we must assume that the sum of the first n natural numbers (1, 2, 3 ... n) = n(n + 1). 2 Now let P denote the value of the annuity, which consists of 4 pounds, payable yearly for n years; and let r denote the ratio of increase of the simple interest. = (Thus, if interest be reckoned at 5 per cent., r = τόσ Then the simple interest on P pounds for n years, at the rate r, = Pnr. And as the interest must be reckoned on each payment of the annuity except the last, the sum of these interests 1) n(n will be (1 + 2 + 3 + ... + 22- - 1) rA = rA, Suppose the annuity to be £100, for 100 years, at 5 SOLUTIONS OF THE QUESTIONS SET AT THE GOVERNMENT EXAMINATIONS OF SCIENCE SCHOOLS, FROM 1867 TO 1872. 1867. [In this and the following year, students were divided into five classes, and a paper of twelve questions was set to the lowest three. Of these questions the following three only were of a nature corresponding to the present First Stage.] (i.) L.C.M. 6. Multiply by 6 throughout. 3. Find the least common multiple of 16, 24, and 30, and explain the method. |