324, Ex. 26. The denominator of the first fraction should be 332, Ex. 39. The answer = √ ¦ (√2 − 1) should be a =\ (V2-1). 350, Prob. 5. For all after the second paragraph substitute the following: These hours are those indicated by the clock: and since the clock gains 0.1 in 24 hours, 24×60×60′ +0·1' of clock time = 24×60×60' of real time. 24×60×60+0·1 .. 1 hour of real time hours of clock time; 24×60×60 = 0.1 Let a be the gaining rate of the watch per hour: then since is the 24 359, last line, and p. 360, lines 1, 5, 6, for whole number next less than, 496, line 3 from bottom, for £211. 13s. 4d. read £417. 10s., and supply the following Answer: Cost of type at Cambridge = £50. .................... INTRODUCTION. VULGAR FRACTIONS. ART. 1. A FRACTION is a quantity which represents a part or parts of an integer or whole, 2. A simple fraction consists of two members (or terms), the numerator and the denominator; the denominator shews into how many equal parts the whole, or unity, is divided; and the numerator the number of those parts taken. The numerator is usually placed over the denominator with a line 2 between them. Thus (two thirds,) signifies that unity is 3' divided into three equal parts, and that two of those parts are taken. It must be observed, that we suppose every integer to be divisible into any number of equal parts at pleasure, 3. A proper fraction is one whose numerator is less than 4. An improper fraction is one whose numerator is equal to, or greater than, its denominator, as 6 7 5. A compound fraction is a fraction of a fraction, as 2 6 9 taken; also of of is a compound fraction; &c. 11 6. A quantity consisting of a whole number and a fraction is called a mixed number, as 7%, which signifies 7 integers 7. Every integer may be considered as a fraction whose denominator is 1; thus 5, or 5 units, is 5 5 1 And may be read 5 integers, or 5 wholes. 8. A continued fraction is one whose denominator is continued by being itself a mixed number, and the denominator of the fractional part again continued as before, and so on: thus 1. 2+ 3+ 9. To multiply a fraction by any whole number. RULE. Multiply the numerator by that number and retain the same denominator. 2 14 the fractions and is divided into 15 equal parts, and 7 - 15 - 15 times as many of those parts are taken in the latter case as in the former. 10. To divide a fraction by any whole number. RULE. Multiply the denominator by that number and retain the same numerator. the parts in the latter case is four times as great as in the former, and the same number of parts is taken in both cases; therefore the former fraction is one fourth of the latter. * To avoid repetition the reader is referred to the first section of the Algebra for the explanation of the signs +, −, ×, =, ÷, &c. 11. A simple fraction may be considered as representing the quotient arising from the division of the numerator by the denominator. 3 Thus the fraction-represents the quotient of 3 divided 4 by 4; for 3 is (Art. 7), and this divided by 4 is the fraction 3 3 (Art. 10). If the integer be supposed a pound, or twenty 3 shillings, of £1, which is 15 shillings, is equal to 4 which is also 15 shillings. 12. If the numerator and denominator of a fraction be both multiplied by the same number, the value of the fraction is not altered. For, if the numerator be multiplied by any number, the fraction is multiplied by that number (Art. 9); and if the denominator be multiplied by the same number, the fraction is divided by it (Art. 10); and if a quantity be both multiplied and divided by the same number, its value is not altered. COR. Hence, if the numerator and denominator of a fraction be both divided by the same number, its value is not The operation by which a quantity is changed from one denomination to another, or by which a fraction has its terms diminished, without altering its value, is called Reduction. 13. To reduce a whole number to a fraction with a given denominator. RULE. Multiply the proposed number by the given denominator, and the product will be the numerator of the fraction required. |