Rule. — Change the given numbers to their equivalent common fractions. Multiply them together, and reduce the product to its equivalent decimal. Examples. 2. Multiply 87.32586 by 4.37. Ans. 381.6140338. 3. Multiply 58-2.347 by .03. 4. Multiply 8.145 by 4.297. Ans. 13.5i69533. 5. What is the value of .285714 of a guinea? Ans. 8s. 6. What is the value of .461607142857 of a ton? Ans. 9cwt. Oqr. 23+1b. 7. What is the value of .284931506 of a year? Ans. 104<L DIVISION OF CIRCULATING DECIMALS. ,° Ex. 1. Divide .54 by .15. Ans. 3.506493. Operation. "We change 54 6 • If 14 7 the given num "= 99 = IT 10 " 90 " 45* bers to their equivalent com TT A = Vf- = 3?f = 3.506493 Ans. nx'n fractions, and, dividing, obtain *fj'-, which, reduced to its equivalent decimal, gives 3.506493, the answer required. Rule. — Cliange the given numbers to their equivalent common fractions. Divide, and reduce the quotient to its equivalent decimal. Examples. 2. Divide 345.8 by .6. Ans. 518.83. 3. Divide 234.6 by .7. 4. Divide 13.5169533 by 3.145. Ans. 4.297. 5. Divide 381.6140338 by 4.37. Ans. 87.32586. 6. Divide .42857 i by .625. 7. Find the value of 2.370 4.923076. Ans. .48i. 8. Find the value of .09 -f- .230769. Ans. .39. 9. Find the value of 316.31015 H- .3. 10. Find the value of 100006 -f- .6. 11. Divide .36 by .25. Ans. 1.422924901185770750988L CONTINUED FRACTIONS. 307. A Continued Fraction is a fraction having for its numerator 1, and for its denominator a whole number plus a fraction whose numerator is 1, and whose denominator is a whole number plus a fraction, and so on. Thus, 1 3+1 4 + is a continued fraction. The partial fractions composing the parts of a continued fraction are called its terms. Thus, in the fraction given above, ±, &c. are its terms. 308. Continued fractions are used in obtaining, in smaller numbers, the approximate values of fractions whose terms, when reduced to their simplest forms, are expressed in numbers inconveniently large. 309. ° To transform a common fraction into a continued fraction, and to find, in smaller numbers, its approximate values. Ex. 1. Transform into a continued fraction, and find its several approximate values. (6 X 3) + 1 19 . . — —, the original value, (19 x 3)+ 3 60' Dividing both terms of ^| by the numerator, which operation will not change the value expressed (Art. 217), the fraction becomes _—; the denominator of which being between 3 and 4, the value of the given fraction must be between J and \; and neglecting the fraction fo, for the present, in the denominator, we have J for the first approximate value. This approximation, however, is greater than the true value, since the denominator, 3, is less than the true denominator 3^. We therefore divide both terms of the T8T by its numerator, and it becomes i-, which is between J and \. By neglecting the fraction £ in the denominator, and taking the J instead of the fy, we have - j ^ = ^ = for the second approximate value of the given fraction; which approximation is too small, since in the denominator, instead of fy, we used ^, which is greater than the TST. If we now include in the calculation the remaining partial fraction J, we have = !_ = the original fraction. By the processes of the operation it will be seen that the first approximate value sought was obtained by disregarding all the partial fractions after the first, the second approximate value by disregarding all the partial fractions after the second, &c. Rule. — Divide the greater term of the given fraction by the less, and the divisor by the remainder, and so on, as in finding the greatest common divisor. The quotients thus found will be the denominators of the several terms of the continued fraction, and the numerator of each will be 1. For the First approximation, tale the first terms of the continued fraction. For the Second approximation, multiply the terms of the first approximate fraction by the denominator of the Second term of the continued fraction, adding 1 to the product of the denominators. For each Succeeding approximation, multiply the terms of the approximation last found by the denominator of the Next term of the continued fraction, and add the corresponding terms of the preceding approximation. Note 1. — When the fraction given is improper, the true approximations will be the reciprocals of the fractions found by the rule. Note 2.— In a scries of approximations the first is larger, the second smaller, and so on, every odd fraction being larger, and every even one smaller, than the given fraction.. Each successive approximate fraction, however, approaches more nearly than the one preceding it to the value of the given fraction. When the continued fraction indicates many approximations, it is generally sufficient for ordinary purposes to find only from three to six of them. Note 3. — A continued fraction may, for convenience, be expressed by writing its terms one directly after another, with the sign plus between the denominators; thus, the continued fraction equivalent to J| may be written h + l + h Examples. 2. Transform J$ into a continued fraction. Ans. 1 3 + f 3. Transform ?8^- into a continued fraction. 4. Find the approximate values of ff. Ans. y, §, §, f ?, 5. Find the first five approximate values of |f %\. 6. Find the first three approximate values of |fy. Ans. f, f, J, or 2, 2*, 2*. 7. Find the first six approximate values of . Ans. ,j'j, y'j, j2r, ^5, -jVj, iVr* 8. What are the first four approximate values of 1.27? Ans. |, |, {, ft, £f. or 1, H. lT3r, 1 A RATIO. 31©i Ratio is the relation, in respect to magnitude or value, which one quantity or number bears to another of the same kind. 311. The comparison by ratio is made by considering how often one number contains, or is contained in, another. Thus, the ratio of 10 to 5 is expressed by 2, the quotient arising from the division of the first number by the second, or it may be expressed by -j^ = the quotient arising from the division of the second number by the first, as the second or the first number shall be regarded as the unit or standard of comparison. In general, of the two methods, the first is regarded as the more simple and philosophical, and therefore has the preference in this work. Note. — Which of the two methods is to be preferred, is not a question of so much importance as has been by some supposed, since the connection in which ratio is used is usually such as to readily determine its interpretation. 312. The two numbers necessary to form a ratio are called the terms of the ratio. The first term is called the antecedent, and the last, the consequent. The two terms taken together are called a couplet; and the quotient of the two terms, the index or exponent of the ratio. 313. The ratio of one number to another may be expressed either by two (lots ( :) between the terms; or in the form of a fraction, by making the antecedent the numerator and the consequent the denominator. Thus, the ratio 6 miles to 2 miles may be expressed as 6 : 2, or as f. 314. The terms of a ratio must be of the same kind, or such as may be reduced to the same denomination. Thus, cents have a ratio to cents, and cents to dollars, &c.; but cents have not a ratio to yards, nor yards to gallons. 315. A simple ratio is that of two whole numbers; as, 3 : 4, 8 : 16, 9 : 36, &c. 316. A complex ratio is that of two numbers, of which one or both are fractional; as, 6 : 4£, f: 4£ : 2^, &c. o 317. A compound ratio is the product of two or more ratios. Thus, the ratio compounded of 4 : 2 and 6 : 3 is j- X § 25t = 4, or 4 X 6 : 2 X 3 = 4. A compound ratio is generally expressed by writing the ratios composing it, in a column, with the antecedents in one 4 • 2 ) vertical line, and the consequents in another; .thus, 6 ! g r expresses a compound ratio. Note. — If a ratio be compounded of two equal ratios, it is called a duplicate ratio; of three ratios, a triplicate ratio, &c. 318. A ratio is either direct or inverse. A direct ratio is the quotient of the antecedent by the consequent; an inverse ratio, or reciprocal ratio, as it is sometimes called, is the quotient of the consequent by the antecedent, or the reciprocal of the direct ratio. Thus the direct ratio of 6 to 2 is f or 3; and the inverse or reciprocal ratio of 6 to 2 is § or ^, which is the same as the reciprocal of 3, the direct ratio of 6 to 2. Note 1. — One quantity is said to vary directly as another, when both increase or decrease together in the same ratio; one quantity is said to vary t» |