to a whole or mixed quantity. Dividing 3 y3 by x2 + xy + y2, we have xresult. The answer is, therefore, xy. y as a Dividing we obtain 2 as a result, and a remainder of 2. 4. Hence our answer is x 2+ 2x-4 NOTE. When the first term of the remainder will not contain the first term of the divisor, the former may be written over the latter in the form of a fraction. If we have a fractional form like the following: EXAMPLES FOR PRACTICE. Reduce the following to mixed quantities: 6a3 3a2 + Sa 7 a + b a+b+c За 5m+7 4m 1 7 1. a + 3 Ans. 3m 3x 5 REDUCTION OF MIXED QUANTITIES TO 48. As this operation is the converse of the preceding, we have the following rule: Multiply the integral part by the denominator of the fraction. Add or subtract, as the sign may indicate, the numerator of the fraction to this product. Write the expression over the denomina tor of the fraction. Reduce x + x Multiplying 4 to fractional form. by 7 we have 7x+7x. Adding the numerator of the fractional part we have 7x2+7x-4. Writ 7x2+7x-4. ing this over the denominator we obtain the answer 7 NOTE. It should be remembered that the line in a fraction acts as a vinculum. It is well, therefore, to enclose the numerator in a parenthesis when the sign before a fraction is —. EXAMPLES FOR PRACTICE. Reduce the following to fractional form : 49. Reduce and 5xy3 to equivalent fractions 17c'd' c'd' 51c3d2 having the lowest common denominator. The lowest common denominator is the lowest common multiple of the denominators 17c'd, c'd3, and 51c3d. It is found to be 51c'd3. As we have seen, both terms of a fraction may be multiplied by the same quantity without changing its value. Multiplying both terms of 6a2b 17c'd 18a2bd2 by 3d we have 51c'd 51c2m'n' by 51c we have 51c'd Inspection of the above shows that the terms of each fraction are multiplied by a quantity obtained by dividing the lowest common denominator by its own denominator. We may, therefore, formulate the following rule: Find the lowest common denominator. Divide this by each denominator separately. Multiply the corresponding numerators by the quotients, and write the results over the common denominator. NOTE. Each fraction should be in its lowest terms to start with. EXAMPLES FOR PRACTICE. Reduce the following to equivalent fractions having the lowest common denominator: = 1 and + = 2 + = 3 3 1 + x For α a a a 1 1 2 1. Similarly in algebra += a if a quantity is divided into c parts and a parts are taken, rep a b с α с resents the amount taken. Similarly if the quantity is divided into c parts and b parts are taken we have to represent the amount. Again if a quantity is divided into c parts and a parts are taken and b parts are also taken, the whole amount taken is a + b a b a + b By this we see that tor. In addition or subtraction of fractions, reduce them if necessary to equivalent fractions having the lowest common denominaAdd or subtract the numerators as the case may be and write the sum or difference thus obtained over the common denominator. The final result should be reduced to its simplest form. x2 + 4 y2 — 7 Find the sum of 2xy* and 3x2y The lowest common denominator is 6x2y2. Multiplying both terms of the first fraction by 3x and boch terms of the second by 2y, we have NOTE. Remember that the line of the fraction acts as a vinculum, and it is therefore well to enclose the numerator of a fraction preceded by a - sign in a parenthesis. 2a + b The lowest common denominator is 663. 5 2a + b Sab 56 2a + b + + 4a 36 66 662 = + + 66 8ab5b+ 2a + b 662 8ab6b+2a 662 4ab + 3b+ a 362 |
