PERCENTAGE Percentage means by or on the hundred. Thus, 1% = 100 = .01, 3% = 180 = .03. To Find the Percentage, Having the Rate and the Base. Multiply the base by the rate expressed in hundredths; thus, 6% of 1,930 is 1,930X.06=115.80. To Find the Amount, Having the Base and the Rate.-Multiply the base by 1 plus the rate; thus, the amount of $1,930 for 1 yr. at 6% is $1,930X1.06 $2,045.80. = To Find the Base, Having the Rate and the Percentage. Divide the percentage by the rate; thus, if the rate is 6% and the percentage is 115.80, the base is 115.80.061,930. To Find the Rate, Having the Percentage and the Base. Divide the percentage by the base; thus, if the percentage is 115.80 and the base 1,930, the rate is 115.80÷1,930.06, or 6%. FORMULAS The term formula, as used in mathematics and in technical books, may be defined as a rule in which symbols are used instead of words; in fact, a formula may be regarded as a shorthand method of expressing a rule. The signs used are the ordinary signs indicative of operations and the signs of aggregation; all of which are used in arithmetic. The use of formulas can best be shown by means of an example; therefore, the well-known rule for finding the horsepower of a steam engine will be taken. This rule may be stated as follows: Rule.-Divide the continued product of the mean effective pressure, in pounds per square inch, the length of the stroke, in feet, the area of the piston, in square inches, and the number of strokes per minute by 33,000: the result will be the horsepower. An examination of the rule will show that four quantities (viz., the mean effective pressure, the length of the stroke, the area of the piston, and the number of strokes) are multiplied together, and the result is divided by 33,000. Hence, the rule might be expressed as follows: This expression can be greatly shortened by representing each quantity by a single letter, thus representing horsepower by the letter H, the mean effective pressure, in pounds per square inch, by P, the length of the stroke in feet, by L, the area of the piston, in square inches, by A, the number of strokes per minute by N, and substituting these letters for the quantities that they represent, the following formula is obtained, PXLXAXN The formula just given shows that a formula is really a shorthand method of expressing a rule. It is customary, however, to omit the sign of multiplication between two or more quantities when they are to be multiplied together, or between a number and a letter representing a quantity, it being always understood that when two letters are adjacent with no sign between them, the quantities represented by these letters are to be multiplied. Bearing this fact in mind, the formula just given can be further simplified to The sign of multiplication, evidently, cannot be omitted between two or more numbers, as it would then be impossible to distinguish the numbers. Use of Formulas.-The area of any segment of a circle that is less than (or equal to) a semicircle is expressed by the formula πr2E с A = (r− h), r = radius; E= angle obtained by drawing lines from center to extremities of arc of segment; c = chord of segment; h = height of segment. EXAMPLE.-What is the area of a segment whose chord is 10 in. long, angle subtended by chord is 83.46°, radius, is 7.5 in., and height of segment is 1.91 in.? SOLUTION.-Applying the formula just given, =40.968-27.95 13.018 sq. in., nearly The area of any triangle may be found by means of the following formula, EXAMPLE.-What is the area of a triangle whose sides are 21 ft., 46 ft., and 50 ft. long? SOLUTION.-In order to apply the formula, let a represent the side that is 21 ft. long; b, the side that is 50 ft. long; and c, the side that is 46 ft. long. Then, substituting, =25 √441-8.252=25. √441-68.0625=25 √372.9375 =25X19.312=482.8 sq. ft., nearly These operations have been extended much further than was necessary; this was done in order to show the reader every step of the process. Rankine-Gordon Formula.-The Rankine-Gordon formula for determining the least load in pounds that will cause a long column to break is in which P=load (pressure), in pounds; S ultimate strength of material composing column, in pounds per square inch; A area of cross-section of column, in square inches; q=a factor (multiplier) whose value depends on shape of ends of column and on material com posing column; 1= length of column, in inches; G= least radius of gyration of cross-section of column. EXAMPLE. What is the least load that will break a hollow steel column whose outside diameter is 14 in., inside diameter 11 in., length 20 ft., and whose ends are flat? SOLUTION. For steel, S=150,000, and q= 1 25,000 for flat ended steel columns; A = .7854(d12 — d22), di and d2 being the outside and inside diameters, respectively; l=20×12=240 in.; Logarithms are designed to diminish the labor of multiplication and division, by substituting in their stead addition and subtraction. A logarithm is the exponent of the power to which a fixed number, called the base, must be raised to produce a given number. The base of the common system is 10, and, as a logarithm is the exponent of the power to which the base must be raised in order to be equal to a given number, all numbers are to be regarded as powers of 10; hence, 100= 1, therefore logarithm of 103 = 1=0 10=1 100=2 1,000 = 3 10,000 = 4 The logarithms of numbers between 1 and 10 are less than unity, and are expressed as decimals. The logarithm of any number between 10 and 100 is more than 1 and less than 2, hence it is equal to 1 plus a decimal. Between 100 and 1,000 it is equal to 2 plus a decimal, etc. The integral part of a logarithm is its characteristic, the decimal part is its mantissa. For example, the log of 67.7 is 1.83059; the characteristic of this logarithm is 1 and the mantissa is .83059. The characteristic of a logarithm is always 1 less than the number of whole figures expressing that number, and may be either negative or positive. The characteristic of the logarithm of 7 is 0; of 17 is 1; of 717 is 2; etc. mantissa is always considered positive. The To Find Logarithm of Any Number Between 1 and 100. Look on the first page of the table, along the column marked No., for the given number; opposite it will be found the logarithm with its characteristic. To Find Logarithm of Any Number of Three Figures.-Find the decimal in the first column to the right of the number; prefix to this the characteristic 2. Thus, the logarithm of 327 is 2.51455. As the first two figures of the decimal are the same for several successive figures, they are only given where they change. Thus, the decimal part of the logarithm of 302 is .48001. The first two figures remain the same up to 310, and are therefore to be supplied. To Find Logarithm of Any Number of Four Figures.-Look in the column headed No. for the first three figures, and then along the top of the page for the fourth figure. Down the column headed by the fourth figure, and opposite the first three, will be found the decimal part. To this prefix the characteristic 3. To Find Logarithm of Any Number of More Than Four Figures. Place a decimal point after the fourth figure from the left, thus changing the number into an integer and a decimal. If the decimal part contains more than two figures, and its second figure is 5 or greater, add 1 to the first figure in the decimal. Find the mantissa of the first four figures, and subtract it from the next greater mantissa in the table. Under the heading P. P., find a column headed by the difference first |