and 100 from P, and get it by itself on one side of a new equation. 298. 1. Remove the divisor 100 by multiplying both sides by 100. This gives I × 100 = PRY. To multiply a fraction by the denominator, we simply cancel the denominator. 299. 2. We next remove from P the multipliers RY, by dividing both sides by RY. This gives PRY÷RY gives P; for when we divide a product by the product of any of its factors, the result or answer is-the other factors. 300. At each step an equation was still formed, for the same thing was done to both sides. The unknown term is now by itself on one side; on the other side are the known terms, with signs denoting what to do with them to get the quantity sought. 301. Thus, from the formula for finding interest, a rule to find the principal has easily been extracted— namely, To find the principal which will yield a given interest at a given rate, and given time expressed in years,multiply the interest by 100, and divide that product by the rate and by the years. 302. In like manner, from this last, or from the original equation, we may form an equation expressing the formula to find R, or that for Y. 303. Similarly, also, from other equations we may extract equations showing the values of, or formulæ for finding any quantities they contain. 304. If the unknown quantity is one of the terms of a proportion, an equation is easily formed, from which its value can be found. When four quantities are proportional, the product of the extremes (first and last terms) is always equal to the product of the means (the two middle terms). Thus, 47: 12 : 21 ; and accordingly we find that 4 x 21 7 × 12, = the product in each case being 84. abcd, then 305. From this equation, by division, the value of any of the four quantities may be found. If the fourth term in the proportion, d, is the unknown quantity, then, dividing both sides of the equation by a, to clear off a as a multiplier of d, we have the formula, This example, it is manifest, is the common Rule of Three, where, from three terms given, we find a fourth : -Divide the product of the second and third terms by the first term. 306. In the same manner, equations may be formed to express the rule for finding any other of the four terms in a proportion-always understanding that they are arranged in proper order, so that the first is to the second as the third is to the fourth. 307. It is not necessary to form the equation: its result may be found by simple inspection of the proportion. If one extreme is the quantity sought, the other extreme is the divisor, and the product of the means is the dividend; if one of the means is the unknown quantity, the other mean is the divisor, the product of the extremes the dividend. EQUATIONS. Examples and Exercises. 87 308. When two forces act on a lever, there is equilibrium (they balance each other), if one force (F) is to the other force (f) as the perpendicular from the fulcrum on the direction of the second force (p), is to the perpendicular from the fulcrum on the direction of the first force (P); that is, when Ffp P. Extract the formulæ, or equational rules for finding each of these four quantities. 309. When several terms on one side of an equation consist of the same kind of quantity, they are to be collected into one term. Thus 6x+x are the same as 7x; 8x-3x (8 times x less 3 times x) are to be collected into one term, 5x. we wish to find the value of x; multiplying by 3 to get rid of the fraction, 84; 7x 84; X 12. = x = abc + d, we wish to find the formula for c, we must get c by itself on one side. 310. When the pupil feels embarrassed by the sign prefixed to x towards the close of the working of an equation, he may change the signs of all the terms on both sides; the two sides will continue equal. INTRODUCTION TO MENSURATION. 311. Mensuration is the art of measuring the lengths of lines, areas of surfaces, and volumes (or capacities) of solids or spaces. The inch is divided into eighths or into twelfths, or decimally into tenths. In land-surveying a chain is used. Its length is 4 poles -that is, 22 yards, 66 feet, or 792 inches. It consists of 100 links, each of which is 7.92, or 73 inches long. 313. Metric System.-The length of the French standard, the metre, is 3.28089 feet, or 39.37079 inches. The kilometre (1000 metres) is 1093-633 yards very nearly 1093 yards. The decimetre (tenth of a metre) is 3.937 inches. The English foot is 30479 decimetres; the yard, 0.91438 metre; the mile, 1609.3149 metres. 314. The mean length of a degree of latitude is 69.0444 miles. A nautical or geographical mile is the 60th part of this, or 6075.6 feet. |