~Reference:~ The chapter on H. C. F. and L. C. M. in any algebra. FRACTIONS Define: fraction, terms of a fraction, reciprocal of a number. Look up _the law of signs_ as it applies to fractions. Except for this, fractions in algebra are treated exactly the same as they are in arithmetic. 1. Reduce to lowest terms: (_a_) 32/24; (_b_) (a^6 - x^6)/(a^4 - x^4); (_c_) [(a + b)^2 - (c + d)^2]/[(a + c)^2 - (b + d)^2]. (_M. I. T._) 2. Reduce to a mixed expression: (_a_) 756/11; (_b_) (a^3 + b^3)/(a - b). 3. Reduce to an improper fraction: (_a_) 45-1/8; (_b_) 9-11/12 qt.; (_c_) a^2 - ab + b^2 - (b^3)/(a + b). Add: 4. 5/18 + 7/9 + 11/16 + 5/8. 5. 5/(1 + 2x) - (3x)/(1 - 2x) + (4 - 13x)/(4x^2 - 1). 6. 1/[x(x - a)(x - b)] + 1/[a(a - x)(a - b)] + 1/[b(b - x)(b - a)]. Multiply: 7. 72/121 x 55/56 x 77/90. 8. (b - y)/(a^3 + y^3) x (ca + cy)/(b^2 + by) x (b^6 + y^6)/(b^2 + y^2) x b/c. Divide: 9. (12/25) / (6/50). 10. [1 - (ab)/(a^2 - ab + b^2)] [1 - (ab)/(a^2 + 2ab + b^2)] / (a^3 - b^3)/(a^3 + b^3). (_Yale._) 11. [(x^4 - y^4)/(x^2 - y^2) / (x + y)/(x^2 - xy)] / [(x^2 + y^2)/(x - y) / (x + y)/(xy - y^2)]. (_Sheffield._) Simplify: 12. [(4y)/x - (15y^2)/(x^2) + 4] / [4 - (16y)/x + (15y^2)/(x^2)] x [3 - (4x + 20y)/(2x + 5y)]. ~Reference:~ The chapter on Fractions in any algebra. COMPLEX FRACTIONS AND FRACTIONAL EQUATIONS Define a complex fraction. Simplify: 1. (3/7 + 4/5)/(2 - 3/7 . 4/5). 2. (2 - 3/2 + 2/3)/(5 - 2/3 + 3/2). 3. 2 - 2/(1 - 1/[1 - 1/(1 + 1/2)]). 4. a/(b^2) - a/[b^2 + (cb)/(a - c/b)]. (_Harvard._) 5. If m = 1/(a + 1), n = 2/(a + 2), p = 3/(a + 3), what is the value of m/(1 - m) + n/(1 - n) + p/(1 - p)? (_Univ. of Penn._) 6. Simplify the expression {x + y - 1/[x + y - xy/(x + y)]}(x^3 - y^3)/(x^2 - y^2). (_Cornell._) 7. Simplify [1 - (2xy)/((x + y)^2)]/[1 + (2xy)/((x - y)^2)] / {(1 - y/x)/(1 + y/x)}^2. 8. Solve (7y + 9)/4 - [y - (2y - 1)/9] = 7. 9. Solve 2-1/3 - (2/5)(x^2 + 3) = (10x)/3 + 1 - (2x^2)/5. 10. How much water must be added to 80 pounds of a 5 per cent salt solution to obtain a 4 per cent solution? (_Yale._) ~Reference:~ See Complex Fractions, and the first part of the chapter on Fractional Equations in any algebra. FRACTIONAL EQUATIONS 1. Solve for each letter in turn 1/b = 1/p + 1/q. 2. Solve and check: (5x + 2)/3 - (3 - (3x - 1)/2) = (3x + 19)/2 - ((x + 1)/6 + 3). 3. Solve and check: (1/2)(x - a/3) - (1/3)(x - a/4) + (1/4)(x - a/5) = 0. 4. Solve (after looking up the special _short_ method): (3x - 1)/30 + (4x - 7)/15 = x/4 - (2x - 3)/(12x - 11) + (7x - 15)/60. 5. Solve by the special _short_ method: 1/(x - 2) - 1/(x - 3) = 1/(x - 4) - 1/(x - 5).