LOWER DIVISION CLERKS. ARITHMETIC. 1. To of 5-8 add 2 of 904761, and from their sum subtract 43 of 5692307. 2. A person estimates that four-sevenths of his income will be required for housekeeping, foursevenths of the remainder for the education of his children, and four-sevenths of what still remains for rent and taxes; and he finds that he will then have £135 left for general expenses. What is his income? 3. Find by duodecimals the area of a rectangle which measures 27 ft. 9 in. 4 pts. by 6 ft. 2 in. 3 pts. If these figures represent the length and height of a nine-inch wall, and the dimensions of a brick are 9 inches by 4 in. by 21⁄2 in., how many bricks will be required to build the wall? 4. Find the square root of 845355625. If a piece of silk costs £15 15s. 24d, and the number of yards in its length is the same as the number of pence in the cost of one yard, what is the length of the silk and its price per yard? 5. Given the cubic metre is equal to 35-316581 cubic feet, find the length in feet of a linear metre correct to four places of decimals. 6. A question being proposed in an examination to find the simple interest on a certain sum of money for 2 years at 31 per cent., a candidate by mistake reckoned it for 24 years at 3 per cent., and so obtained a result too little by £26 4s. 8d; what ought the answer to have been? 7. Find three numbers which are to each other as 1 2 3, and such that the sum of their cubes is 4500. 8. State the rule for transforming a number from one scale of notation to another. Transform 45678 from the scale of 9 to the scale of 11. How many times is the greatest number of three figures in the scale of 4 contained in the greatest number of four figures in the scale of 8? 9. If I invest in Three per Cent. Consols at 951, and sell out when the price has risen to 97, paying one-eighth per cent. brokerage on each transaction; by how much do I thereby increase my capital? If the interval between buying and selling be exactly a year, so that one year's dividends are received, putting the dividends and increase of capital together, what rate per cent. have I made. on the sum invested? 10. The profits on a capital of £10,000 used in trade for four years are equivalent to compound interest at 25 per cent. per annum for that time, how much do the profits amount to? A sum of money is laid out at compound interest at a rate which causes it to be doubled in four years, how many fold will it be increased in 16 years? 11. The nominal weight of a truck of coals is 8 tons 10 cwts., the actual weight is 8 tons 18 cwts.; if 10 per cent. profit be made by selling the coal by the truck at 22s. per ton nominal weight, what rate of profit will be made by retailing it at 28s. 2d. per ton actual weight? 12. There are two rectangular wells, one of them 5 ft. by 4 ft. in section, the other 6 ft. by 3 ft. a hose which fills the first in 4 minutes, would take 7 minutes to fill the other, and the water which would fill the smaller well, if poured into the larger one would stand 15 ft. below the What is the depth of each well? top. 13. The amount of a certain sum put out at compound interest (payable yearly) exceeds the amount that would have been obtained from the same sum at the same rate per cent. simple interest by £2 12s. 1d. at the end of the second year, and by £5 6s. 3d. at the end of the third year; find the original sum. 14. A and B engage to walk a match of 10 miles round a course of 5 furlongs; A, during his first 8 rounds, gains 110 yards each round upon B; and then their rates being reversed, B gains similarly on A at the rate of 110 yards on each round for the rest of the race; which comes in first, and by how many yards does he win? 15. An ornament is made of gold and silver; the weight of the gold used is that of the silver, and the cost is £14 13s. Assuming the volume of gold to be that of an equal weight of silver, and the value of gold to 17.5 that of an equal volume of silver; find the value of gold and silver respectively used in the Ir is also evident from the figure, that as the angle increases from 90° to 180°, the numerator of the fraction expressing the sine (which within these limits is still positive) decreases, while the denominator remains unchanged, the sine, therefore, decreases, and when the measure of the angle reaches 180° the numerator vanishes; wherefore, sine of 180° = 0. The cosine of an angle between 90° and 180° is, as we have seen, negative, and as the angle increases from 90° to 180° the numerator of the fraction expressing its cosine increases in absolute magnitude, while the denominator remains the same, the cosine, therefore, increases numerically, and when the measure of the angle reaches 180° the numerator becomes numerically equal to the denominator, but is of opposite sign; the value then is -1; therefore, The tangent of an angle between 90° and 180° is, as has been shown, negative, and as the angle increases from 90° to 180°, the numerator of the fraction expressing its tangent diminishes, while the denominator increases, the tangent, therefore, decreases numerically, and when the measure of the angle reaches 180°, the numerator becomes 0; therefore, tangent of 180° = 0. The cotangent has the same algebraic sign and varies inversely as the tangent which is its reciprocal; the cotangent, therefore, of an angle between 90° and 180° is negative, and increases in absolute magnitude from 0 at 90° to co (infinity) at 180°; wherefore, In like manner the secant has the same algebraic sign, and varies inversely as its reciprocal the cosine; the secant, therefore, of an angle between 90° and 180° is negative, and decreases, in absolute magnitude, from infinity (∞), its value for an angle of 90°, to -1 for an angle of 180°; wherefore, Similarly the cosecant has the same algebraic sign and varies inversely as its reciprocal the sine; the cosecant, therefore, of an angle between 90° and 180° is positive and increases from 1, its value for an angle of 90°, to infinity (c) for an angle of 180°; wherefore, cosecant of 180° = ∞. The reader will readily perceive from the figure that while the measure of the angle increases from 180° to 270° the sine, which is then negative, increases in absolute magnitude from 0 at 180° to -1 at 270°; hence, sine of 270° = −1, B3 Similarly the figure shows that while the measure of the angle increases from 180° to 270° the cosine, which is still negative, decreases in absolute magnitude from -1, (its value for an angle of 180°), to 0, for an angle of 270°; wherefore, It is also plain from an examination of the figure, that as the angle increases from 180° to 270° the tangent, which within these limits is positive, increases from 0 at 180° to ∞ (infinity) at 270°; therefore, tangent of 270° = ∞. It may be similarly shown, and also follows from the above, that as the angle increases from 180° to 270° the cotangent diminishes from infinity (∞) at 180° to 0 at 270°; therefore, cotangent of 270° = 0. The secant, the reciprocal of the cosine, is negative for all angles between 180° and 270°, and increases in absolute magnitude as the angle increases from -1 (its value for an angle of 180°) to ∞ (infinity) for an angle of 270°; therefore, secant of 270° = oo. The cosecant, the reciprocal of the sine, is, as we have seen, negative for angles between 180° and 270°, and as the angle increases from 180° to 270° its cosecant decreases in absolute magnitude from infinity (∞), (its value for an angle of 180°), to -1 for an angle of 270°; therefore, cosecant of 270° = -1. While the angle increases from 270° to 360° the numerator of the fraction expressing its sine is negative, and diminishes in absolute magnitude from the measure of the denominator to 0; therefore as the angle increases from 270° to 360° its sine decreases numerically from -1 at 270° to 0 at 360°; wherefore, sine of 360° = 0 = sine of 0°. . As the angle increases from 270° to 360° the numerator of the fraction expressing its cosine (which within these limits is positive) increases from 0, at 270° to the equal of the denominator at 360°; therefore, cosine of 360° = 1 = cosine of 0°. It is also evident from the figure that as the angle increases from 270° to 360° the numerator of the fraction expressing its tangent (which within these limits is negative) decreases numerically, while the denominator increases the tangent therefore diminishes in absolute magnitude from ∞ (infinity) its value for an angle of 270° to 0 for an angle of 360°; wherefore, The cotangent of an angle has the same algebraic sign and varies inversely as its reciprocal the tangent, therefore as the angle increases from 270° to 360° its cotangent increases numerically from O at 270° to co (infinity) at 360°; wherefore, cotangent of 360° = cocotangent of 0°. Since, as the angle increases from 270° to 360° its cosine is positive and increases from 0 to 1, its secant the reciprocal of the cosine is positive and diminishes from ∞ (its value for an angle of 270°) to 1 for an angle of 360°; therefore, secant of 360° = 1, secant of 0°. Since the cosecant is the reciprocal of the sine, and that as the angle increases from 270° to 360° its sine which is negative decreases numerically from 1 to 0, the cosecant is negative and increases numerically from 1 to ∞ (infinity); therefore, Since the versed sine of an angle is equal to 1 minus the cosine of the angle, the versed sine is always positive, and varies from 0, its value when the cosine is 1, to 2, its value when the cosine is -1; when the cosine of an angle is 1 its versed sine is 0; therefore, vers 0° 0, vers 90° 1, vers 180° 2. = A similar remark will apply to the coversed sine. NOTE.-The student should trace all the changes in sign and magnitude of the functions of an angle as it varies from 0° to 360°, and verify every one of the above results by a careful examination of the figure. Sine The principles above established may be briefly stated as follows: The sine increases with the increase of the angle from 0, its value when the measure of the angle is 0 to 1 when the angle increases to 90°; from which while the angle increases to 180° the sine decreases to 0. When the angle increases from 180° to 360° its sine is negative and increases numerically to -1 its value for an angle of 270°; it then decreases numerically as the angle increases from 270° to 360° for which its value again becomes 0. Cosine As the measure of the angle increases from 0 to 90° its cosine decreases from 1 to 0; while the angle increases from 90° to 180° its cosine, which is then negative, increases numerically from 0 to -1; between 180° and 270° the cosine, which is still negative, decreases numerically from -1 to 0; its value for an angle of 270°, between which and 360° it is positive, and increases to 1, its value for an angle of 360°; or for the corresponding reduced angle whose measure is 0. When the angle lies between 45° and 135° (for which two angles the sine and cosine are numerically equal) its sine is greater than its cosine; between 135° and 225° the sine is numerically less, and between 225° and 315° numerically greater than the cosine; while between 315° and 405° (= 360°+45°) the sine is numerically less than the cosine. Hence sin Acos A gives a positive result when A lies between 45° and 225°, but a negative result when A lies between 225° and 405°. The tangent increases with the increase of the angle from 0, its value when the measure of the angle is 0, to ∞ (infinity), when the angle increases to 90°; while the angle increases from 90° to 180° the tangent is négative, and decreases numerically from ∞ to 0, its value for an angle of 180°; as the angle increases from 180° to 270° its tangent is positive, and increases from 0 to ∞, then between 270° and 360° the tangent is negative and decreases numerically from co to 0, its value for an angle of 360°, which is identical with that for the corresponding reduced angle 0. Cotangent The cotangent has the same algebraic sign, the same limits of magnitude, and varies inversely as the tangent which is its reciprocal; when, therefore, the tangent is 0 the cotangent is co, and decreases to 0 while the tangent increases to ∞. With these alterations the above statement respecting the changes in the tangent will apply also to those of the cotangent as the angle increases from 0 to 360°. The changes and limits of the tangent and cotangent, as the angle varies, may also be traced from sin A cos A those of the sine and cosine by substituting for tan A, and for cot A. cos A sin A When the angle lies between 45° and 135°, or between 225° and 315° (for which the numerical value of the tangent and cotangent is unity), its tangent is greater than its cotangent, but for angles between 135° and 225°, or between 315° and 405° (=360°+45°), the tangent is less than the cotangent. |