: the upper fraction, we have multiplying by 5, to get quit of the denominator of the lower fraction, we have 40; dividing both terms of this fraction by 8, there results for the simple fraction required. Case 5. To reduce fractions of different denominators to equivalent fractions having a common denominator. Multiply each numerator into all the denominators except its own, for new numerators; and all the denominators together for a common denominator. Ex.-Reduce,, and, to equivalent fractions having a common denominator. 2 X 7 X 9 = 126 6 × 3 × 9= 162 the numerators. 5 x 3 x 7 = 105 3 X 7 X 9= 189, the common denominator. Ex.-Reduce of a penny, and of a shilling, each to the fraction of a pound; and then reduce the two to fractions having a common denominator. of a penny = 2 300 10 of a pound. Hence of a shilling are 10 times as much as of a penny. Note. Other methods of reduction will occur to the student after tolerable practice, and still more after the principles of algebra are acquired. Addition and Subtraction of Fractions. RULE. If the fractions have a common denominator, add or subtract the numerators, and place the sum or difference as a new numerator over the common denominator. If the fractions have not a common denominator, they must be reduced to that state before the operation is performed. In addition of mixed numbers, it is usually best to take the sum of the integers, and that of the fractions, separately; and then their sum, for the result required. Examples. 1. Find the sum of 3, 4, , and 3. = 18 2. Take of a shilling from of a pound sterling. = 11 240 = 11 pence. = 40 480 Hence 480 3. Find the difference between 12 and 83. 12-837-43-385-258 480 22 = 18 480 480= Multiplication and Division of Fractions. RULE 1. To multiply a fraction by a whole number, multiply the numerator by that number, and retain the denomi nator. 2. To divide a fraction by a whole number, multiply the denominator by that number, and retain the numerator. 3. To multiply two or more fractions is the same as to take a fraction of a fraction; and is, therefore, effected by taking the product of the numerators for a new numerator, and of . the denominators for a new denominator. (The product is evidently smaller than either factor when each is less than unity.) 4. To divide one fraction by another, invert the divisor, and proceed as in multiplication. (The quotient is always greater than the dividend when the divisor is less than unity.) 3. Multiply £2 13s. 4d. by 31, and divide £4 15s. by 31. = , and × 31 = 1 × 10 = 57 = The embarrassment and loss of time occasioned by the computation of quantities expressed in vulgar or ordinary fractions, have inspired the idea of fixing the denominator so as to know what it is without actually expressing it. Hence originate two dispositions of numbers, decimal fractions and complex numbers. Of the latter, such, for example, as when we express lineal measures in yards, in feet (or thirds of a yard), and inches (or twelfths of a foot), we shall treat after a few pages. We shall now treat of the former. Decimal fractions, or substantively, decimals, are fractions expressed as whole numbers, but whose values decrease from the place of units progressively to the right hand in the same decuple or tenfold proportion as the common scale of whole numbers increase to the left. They are usually separated from the integers by a dot placed between the upper part of the figures. Thus, 22 expressed according to the decimal notation is 22.7. The value of a decimal fraction is not altered by ciphers on the right hand for 500, or is in value the same as, or •5, that is 1. 500 10009 When decimals terminate after a certain number of figures, they are called finite, as 125 = 125 1000 = , 958 958 = 1000 When one or more figures in the decimal become repeated, it is called a repeating or circulating decimal; as 333333, &c. 1,66666, &c. = ,428571428571, &c., and many = others. 39 Rules for the management of this latter kind of decimals are given by several authors; but, in general, it is more simple and commodious to perform the requisite operations by means of the equivalent vulgar fractions, from which circulating decimals are educed. Reduction of Decimals Reduction of Decimals is a rule by which the known parts of given integers are converted into equivalent decimals, and vice versa. Case 1. To reduce a given vulgar fraction to an equivalent decimal. Annex ciphers to the numerator, divide by the denominator, and the quotient will be the decimal required These two are evidently circulating decimals, in the former of which the figures 148 become indefinitely repeated, in the latter the figures 174603. 3. Reduce 14s. 6d. to the decimal of a pound. 14 First 148. 6d. 18 + 1 of 2% = 28 +26= 28. = 4. Reduce to its equivalent decimal. 399 410 399 110 57 530 513 170 114 56 = 11 Note. The above fraction is 4 1, of which the two denominators are both prime numbers (that is, divisible by no other number than unity), the entire equivalent decimal is a circulator of 18 places, i. e. one less than the last prime . . . 771929824561403508, 7719, &c. over again ad infinitum.* |