COMPLEX FRACTIONS. Wherr a fraction is complex, it may be changed into a vulgar, or simple fraction, by the following RULE. First. If the complex fraction has a whole number in it, the integer should be reduced to the fractional form, or if it contains an improper fraction, it should be reduced to a mixed number; then, Second. Place the different fractions composing the complex fraction, on a line from left to right, INVERTING all except the top fraction, and proceed (by cancelling,) as in multiplication, the PRODUCT will be the answer. EXAMPLES. Find the value of s. in the following equations, in which s. represents the simple quantity, to which the complex fraction in the left hand member is equivalent. When quantities bear the relation to each other expressed by the preceding examples, it will be more simple to write them in the usual order from left to right, e. g., if we wish to divide 9 by the quotient of divided by 12, we should write thus, 9(1), which is called a SYMBOLICAL, or COMPLEX QUANTITY, because the parts are connected by arithmetical signs, or SYMBOLS, in a COMPLICATED form. The mode of WRITING complex quantities, must always be suggested by the conditions of the problem, and therefore requires no rule; but, since a correct solution, of a quantity of this kind, depends almost entirely, upon reading it properly we may adopt the following RULE for Reading Complex Fractions, and Reducing them to their Simplest Form. First. Read each particular TERM with reference to the FIRST, i. e. with reference to the LEFT HAND term; and, Second. Read the parts, in such a manner, as to DESIGNATE as nearly as possible, the increments, decrements, multipliers and divisors of the FIRST term, and, at the same time, to point out what PROCESS these increments &c. must pass through, before they are added to, subtracted from, multiplied or divided, into the FIRST or leading term. This RULE will be sufficiently illustrated by the following EXAMPLES. 1. Read (÷÷÷(7÷1)+}×8=s, and find the simple quantity represented by s. Thus : First. We say the QUOTIENT of (1), one half, divided by (3), two thirds, is the FIRST term; second the DIVISOR, Of the first term is the quotient of (2), three fourths divided by four fifths; third, the INCREMENT, of the first term, is (3). one-fifth; and, fourth, the MULTIPLIER is (8) eight. Now first the QUOTIENT of÷3 = 1.3 3 2.2 4 the FIRST 1x 8-8 s., which is the simple quantity required. 2. Read (÷)÷(3÷4)÷(9÷8)×12+5—43=s., and find the value of s. Here the first term is ÷=3. The second term is÷4. The third is 38. The fourth term is 14. The fifth term is 5; and the sixth 36 term is 4=13. Now, if these simple terms (3, §, 31, 4, 7, and 13, are joined by the same signs which united them in the given expression, we shall have 3÷÷3×+¥—13= s.; but ÷=; and ÷3=3; and 7×7=1=2; and 2+ Y==3; and hence, we have 39-13-39-26-13=21 =S., which is the ANSWER required. The pupil may read, and find the value of s., in the following equations. 3. (8+6+7+4+9)+(12+25)=s. 4. (3x4) (2x5) x 8x 6x 3-9-s. 14. 2 53 (42)÷(8343)÷1=s. 15. 5÷17÷(7}÷÷11)÷(2÷14÷3)=s. 16. (343) (24÷14)÷(8÷5÷21÷7)=s. 17. 12÷[× (÷ 1) ÷ (} × } ÷ })] × ( of §) +2} ARTICLE XIII. RATIO. 1. Ratio means the COMPARATIVE size, or magnitude, which quantities of the same kind bear to each other. Now the comparative SIZE of quantities, is found by measurement, and since division is the art of measuring quantities of the same kind, and since the quotient, is the RESULT of division, it follows, that, RATIO is the QUOTIENT, arising from the DIVISION of two quantities of the SAME KIND; as the ratio of 5 to 8 is; vice versa, the ratio of 8 to 5 is =13 2. The Antecedent of a ratio, is that quantity, which comes first, in the statement, and answers to the dividend, in division. Thus: If the ratio of 3 to 4 is required, 3 is the antecedent, and 4 the consequent. 3. The consequent of a ratio, is the quantity which follows the antecedent, and answers to the divisor in division, as, in the above example, where 4 is the consequent, because it follows the antecedent (3), and becomes its divisor, as, 3. 4. DIRECT ratio is that in which the ANTECEDENT is compared with the CONSEQUENT, where more requires more, and less requires less; as, the DIRECT ratio of 5 to 8 is §, and, vice versa, 8 to 5 is =13, i. e, the greater the ANTECEDENT (8), the greater will be the RATIO,(13), and vice versa. 5 89 5. INVERSE ratio, is that, in which the consequent is compared with the antecedent, where less requires more, and more requires less, as the inverse ratio of 8 to 5 is §, vice versa, 5 to 8 is -13, i. e, the greater the antecedent (5), the less the ratio (13), for if 5 were increased to 16, the inverse ratio would become =; but if 5 were decreased to 4, the inverse ratio would become &=2. 8 16 6. Compound ratios, are such as have their antecedents and consequents consisting of two or more FACTORS, as the compound ratio of 2 x 3 x 4, to 3× 4× 5, is 2.3.4 2 = which 3.4.5 5 is compounded of the ratio of 2 to 3, 3 to 4, and 4 to 5. 7. MIXED ratios, are such as have their antecedents and consequents composed of factors, some of which have an INVERSE and others a DIRECT ratio to each other. 8. A proportion is an orderly arrangement of the terms of two equal ratios; Thus, if 3, we have the proportion, 3 to 5 as 9 to 15, which may be written thus, 3:5::9:15. When either or both the ratios are compound, their orderly arrangement will give a compound proportion, or in general the proportion will take the same name as the ratios which form it. 9. A Rational Equation is an equation used for finding the RATIO of numbers, or the RATIONAL EQUIVALENT of a required quantity. 10. THE RATIONAL EQUIVALENT of a required quantity, is a quantity which bears the same RATIO to the REQUIRED quantity, as the EQUIVALENT of the GIVEN quantity bears To the GIVEN quantity. 11. A GIVEN quantity, is ONE which is equal to some KNOWN quantity and therefore has an equivalent (or known) value, as $1=100 cents, i. e., one dollar equals one hundred cents, where one dollar is the GIVEN quantity, and one hundred cents its equivalent. 12. A required quantity is one, whose RATIONAL EQUIVALENT is not known; but has the SAME demanded or required to be found The MODE of conducting ARITHMETICAL calculations, by the use of EQUATIONS may be explained as follows: 1. Required the DIRECT ratio of 3 to 5. Here 5 is a GIVEN quantity, because the ratio of EVERY number to itself is KNOWN to be ONE, (1), i. e., EVERY quantity must be the WHOLE of itself, and therefore, when COMPARED with itself, will always givé a UNIT, or ONE, (1). Now, since the RATIO of 5 (compared with 5,) is known to be ONE, (1), it follows that the ratio of 1, compared with 5, is of one, (3), and therefore, the ratio of 3 to 5, is of one, i. e.,, which is the result SOUGHT, and may be found by EQUATIONS; thus, = first, or GIVEN equation. If then, hence, second equation. third, or REQUIRED equation. First. Place the GIVEN quantity (55), in the RIGHT hand member of the EQUATION, and its EQUIVALENT (1={), |