Part 11

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.