IDENTITY OF DIVISION, FRACTIONS AND RATIOS. 409. The dividend in an example in division, the numerator of a fraction and the antecedent of a ratio are identical in office; so, also, are the divisor, denominator and consequent. Hence, whatever operations are performed on dividend and divisor, numerator and denominator, or antecedent and consequent, will affect the quotient, value of the fraction or ratio precisely alike; thus, multiplying the dividend (59, a), numerator (125) or antecedent (244, a), multiplies the quotient, fraction or ratio. (a) Again multiplying dividend and divisor (60, Cor.), numerator and denominator (133, a, Note 1) or antecedent and consequent (244, e) by the same number, does not alter the quotient, value of the fraction or ratio; .. two examples in division may, without altering the quotients, be so changed as to have a common divisor or a common dividend; two fractions may be reduced to a common denominator or a common numerator; and two ratios, to a common consequent or a common antecedent. CORALLARY TO (a).—Since we may multiply dividend and divisor, numerator and denominator, or antecedent and consequent by any number, integral or fractional, it follows that we may add to or subtract from these corresponding terms any numbers that have the same ratios, and the quotient, value of fraction or ratio will remain unchanged; thus, REMARK. It is on this principle that the rule for reducing pence and farthings to the decimal of a pound by inspection, is founded; thus, 1 qr.=£, but if 960. be increased by 24 of itself the sum will be 1000; .., increasing the numerator and de nominator, each, by 4 of itself, we shall have 1 qr. 124 = £, and, for a like reason, 2 qr.: = 1 104 = 960 1000' i. e., 1000 410. To reduce shillings, pence and farthings to the decimal of a pound, by inspection, RULE. Write half the greatest even number of shillings as so many tenths of a pound; write the odd shilling, if there be one, as .05 of a pound; and write the number of farthings in the given pence and farthings, increased by 1 if the number is 12 or more, and by 2 if it is 36 or more, as so many thousandths of a pound. The sum of these will not vary more than 1⁄2 of .001£ (or less than qr.) from the true value of the given shillings, pence and farthings. Ex. 1. Reduce 15s. 3d. 2qr. to the decimal of a pound by inspection? 3d. 2qr..015£, nearly, .. 15s. 3d. 2qr. = .765£, nearly, Ans. as before. Ex. 2. Reduce 19s. 8d. 1qr. to the decimal of a pound. 3. Reduce 12s. 11d. 3fr. to the decimal of a pound. 411. Reversing this operation we may reduce the first 3 figures of the decimal of a pound, back to shillings, pence and farthings; thus, Ex. 1. Reduce .875£ to shillings and pence. To reduce the first 3 figures of a decimal of a pound to shillings, pence and farthings, by inspection, RULE.-Double the tenths for shillings; if the hundredths be 5 or more, add another shilling; then, after the .05 is deducted the remaining figures of the 2d and 3d places, abating 1 when the remainder is 12 or more, and 2 when 36 or more, will represent the farthings, which may be reduced to pence and farthings. 2. Reduce .784£ to shillings, pence and farthings. Ans. 15s. 8d. 1qr. 3. Reduce .247£ to shillings, pence and farthings. BARTER. 412. Barter is an exchange of commodities in trade. Questions in barter are solved by analysis. Ex. 1. How much coffee, at 25c. per pound, must be given in exchange for 300 pounds of sugar, at 15c. per pound? Ans. 180lb. 2. How many bushels of oats, at 50c. per bushel, are equal in value to 1000 bushels of wheat at $2.37 per bushel? Ans. 4750. 3. A has flour worth $10 per barrel; but, in exchanging it with B, for broadcloth, he asks $12. Now, B's broadcloth being worth $4 per yard, what shall he charge for it that he may not suffer loss? Ans. $4.80 per yard. 4. C has 1937 pounds of tea, worth 624c. per pound, which he will put at 561c. provided he can get coffee, worth 25c. per pound, for 23c. Does he gain or lose, and what per cent.? Ans. Loses 2 per cent. PRACTICE. 413. Practice is a mode of finding the value of any number of articles at any price, by assuming the value of the whole or a part, at the given or some other price, and then modifying the assumption according to circumstances. Ex. 1. What is the value of 9a. 3r. 20rd. of land, at $40 per acre? 2. What is the value of 356 barrels of flour, at $9 per bar 414. A series consists of 3 or more terms, following each other in accordance with some law (109*). 415. An infinite series is one which is without end. 416. The sum of an ascending infinite series is infinite; i. e. great beyond limits. 417. The sum of a descending infinite series in geometrical progression, may be found by the rule in Art. 363, except that in the infinite series the least term may be considered 0, and diregarded. A quantity that is small beyond any determinate limits is an infinitesimal; as, e. g. the smaller terms of a descending infinite series. Ex. 1. What is the sum of the infinite series, 6, 2, 3, 3, etc.? 623; and 3+69, Ans. 2. What is the sum of the infinite series, 1, 1, 1, etc.? Ans. 13. 3. What is the sum of the infinite series, fo, Tổσ, Tooʊ, etc.? Ans. . 418. There are various methods of finding the sums of different series, but they are Algebraic and cannot be investigated in this treatise. Rules for summing two species of series, only, will be given here. (a) To find the sum of the squares of any number of terms in the natural series, 1, 2, 3, 4, etc., RULE.-Multiply the number of terms in the series by that number plus one; then multiply the product by twice the number plus one, and of the product will be the sum sought. Ex. 1. What is the sum of 12 terms of the series, 12, 22, 32, etc.? 2. What is the sum of 12 terms of the series, 62, 72, 82, etc.? Ans. 1730. The rule is not directly applicable to this example, but we must get the sum of 17 terms of the series, 12, 22, 32, etc., and also of 5 terms, and the difference of these sums will be the |