Algebraic Fractions (B) worksheet with problems for simplifying and combining algebraic expressions.

Here are the solutions for the algebraic fractions worksheet. I have worked through each problem step-by-step to ensure accuracy.

Section A: Work out the following and simplify where possible

1) $\frac{ac}{b} \times \frac{b}{c}$
* Multiply numerators and denominators: $\frac{abc}{bc}$
* Cancel $b$ and $c$ from top and bottom.
* Answer: $a$

2) $\frac{a+b}{3ab} \times \frac{6a^2}{2a+2b}$
* Factorize the denominator of the second fraction: $2a+2b = 2(a+b)$.
* Expression becomes: $\frac{a+b}{3ab} \times \frac{6a^2}{2(a+b)}$
* Cancel $(a+b)$ from top and bottom.
* Simplify numbers: $\frac{6}{3 \times 2} = \frac{6}{6} = 1$.
* Simplify variables: $\frac{a^2}{a} = a$.
* Answer: $a$

3) $\frac{x^2}{x^2-2x} \times \frac{3+x}{x}$
* Factorize the first denominator: $x^2-2x = x(x-2)$.
* Expression becomes: $\frac{x^2}{x(x-2)} \times \frac{3+x}{x}$
* Combine into one fraction: $\frac{x^2(3+x)}{x^2(x-2)}$
* Cancel $x^2$ from top and bottom.
* Answer: $\frac{3+x}{x-2}$ (or $\frac{x+3}{x-2}$)

4) $\frac{(3x+2)^2}{6x} \times \frac{x^4}{6x+4}$
* Factorize the second denominator: $6x+4 = 2(3x+2)$.
* Expression becomes: $\frac{(3x+2)(3x+2)}{6x} \times \frac{x^4}{2(3x+2)}$
* Cancel one $(3x+2)$ term.
* Simplify numbers: Denominator is $6x \cdot 2 = 12x$.
* Simplify $x$ terms: $\frac{x^4}{x} = x^3$.
* Remaining numerator: $(3x+2)x^3$.
* Answer: $\frac{x^3(3x+2)}{12}$

5) $\frac{x^2y^2}{y} \div \frac{x^4y}{x}$
* Flip the second fraction to multiply: $\frac{x^2y^2}{y} \times \frac{x}{x^4y}$
* Multiply across: $\frac{x^3y^2}{x^4y^2}$
* Cancel $y^2$.
* Simplify $x$: $\frac{x^3}{x^4} = \frac{1}{x}$.
* Answer: $\frac{1}{x}$

6) $\frac{b-3}{2b^2-5b-3} \div (2b-6)$
* Write divisor as a fraction: $\frac{2b-6}{1}$. Flip it: $\frac{1}{2b-6}$.
* Factorize the quadratic denominator $2b^2-5b-3$. We need numbers that multiply to $-6$ and add to $-5$. They are $-6$ and $1$.
* $2b^2 - 6b + b - 3 = 2b(b-3) + 1(b-3) = (2b+1)(b-3)$.
* Factorize the flipped divisor: $2b-6 = 2(b-3)$.
* Expression: $\frac{b-3}{(2b+1)(b-3)} \times \frac{1}{2(b-3)}$
* Cancel one $(b-3)$ from the first fraction's top and bottom.
* Remaining: $\frac{1}{2b+1} \times \frac{1}{2(b-3)}$
* Answer: $\frac{1}{2(2b+1)(b-3)}$

7) $\frac{1}{y^2+3y+2} \div \frac{2}{y^2-4}$
* Flip the second fraction: $\frac{1}{y^2+3y+2} \times \frac{y^2-4}{2}$
* Factorize quadratics:
* $y^2+3y+2 = (y+1)(y+2)$
* $y^2-4 = (y-2)(y+2)$ (Difference of two squares)
* Expression: $\frac{(y-2)(y+2)}{2(y+1)(y+2)}$
* Cancel $(y+2)$.
* Answer: $\frac{y-2}{2(y+1)}$

8) $\frac{a^2(a-2)}{y^2} \div \frac{a(a-2)}{y^5}$
* Flip the second fraction: $\frac{a^2(a-2)}{y^2} \times \frac{y^5}{a(a-2)}$
* Cancel $(a-2)$.
* Simplify $a$: $\frac{a^2}{a} = a$.
* Simplify $y$: $\frac{y^5}{y^2} = y^3$.
* Answer: $ay^3$

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Section B: Write the following as single fractions

1) $\frac{x}{5} + 4$
* Common denominator is 5.
* $4 = \frac{20}{5}$.
* Answer: $\frac{x+20}{5}$

2) $y - \frac{6}{y}$
* Common denominator is $y$.
* $y = \frac{y^2}{y}$.
* Answer: $\frac{y^2-6}{y}$

3) $4 - \frac{5}{x-2}$
* Common denominator is $x-2$.
* $4 = \frac{4(x-2)}{x-2} = \frac{4x-8}{x-2}$.
* Subtract: $\frac{4x-8-5}{x-2}$.
* Answer: $\frac{4x-13}{x-2}$

4) $\frac{1}{3w+5} + 9$
* Common denominator is $3w+5$.
* $9 = \frac{9(3w+5)}{3w+5} = \frac{27w+45}{3w+5}$.
* Add: $\frac{1 + 27w + 45}{3w+5}$.
* Answer: $\frac{27w+46}{3w+5}$

5) $6 + \frac{3}{x+2} + x$
* Group integers/variables: $x+6 = \frac{(x+6)(x+2)}{x+2}$.
* Expand numerator: $x^2 + 2x + 6x + 12 = x^2 + 8x + 12$.
* Add the fraction part $\frac{3}{x+2}$.
* Total numerator: $x^2 + 8x + 12 + 3 = x^2 + 8x + 15$.
* Answer: $\frac{x^2+8x+15}{x+2}$

6) $\frac{3}{x} + \frac{5}{x}$
* Same denominator, just add numerators.
* Answer: $\frac{8}{x}$

7) $\frac{7a}{4} + \frac{11a}{4}$
* Same denominator, add numerators: $18a$.
* Simplify $\frac{18a}{4}$ by dividing by 2.
* Answer: $\frac{9a}{2}$

8) $\frac{5b}{3} - \frac{9b}{2}$
* Common denominator is 6.
* $\frac{10b}{6} - \frac{27b}{6}$.
* $10b - 27b = -17b$.
* Answer: $-\frac{17b}{6}$

9) $\frac{1}{2y} + \frac{7}{y}$
* Common denominator is $2y$.
* $\frac{7}{y} = \frac{14}{2y}$.
* Add: $\frac{1+14}{2y}$.
* Answer: $\frac{15}{2y}$

10) $\frac{6}{4k} - \frac{5}{6k}$
* Simplify first fraction: $\frac{3}{2k}$.
* Common denominator for $2k$ and $6k$ is $6k$.
* $\frac{3}{2k} = \frac{9}{6k}$.
* Subtract: $\frac{9-5}{6k} = \frac{4}{6k}$.
* Simplify result.
* Answer: $\frac{2}{3k}$

11) $\frac{r}{p} - \frac{4p}{q}$
* Common denominator is $pq$.
* $\frac{rq}{pq} - \frac{4p^2}{pq}$.
* Answer: $\frac{rq-4p^2}{pq}$

12) $\frac{x}{8} + \frac{1}{x+2}$
* Common denominator is $8(x+2)$.
* Numerator: $x(x+2) + 1(8) = x^2 + 2x + 8$.
* Answer: $\frac{x^2+2x+8}{8(x+2)}$

13) $\frac{2}{x+1} + \frac{1}{x+2}$
* Common denominator is $(x+1)(x+2)$.
* Numerator: $2(x+2) + 1(x+1) = 2x + 4 + x + 1 = 3x + 5$.
* Answer: $\frac{3x+5}{(x+1)(x+2)}$

14) $\frac{3}{y-1} - \frac{1}{y+2}$
* Common denominator is $(y-1)(y+2)$.
* Numerator: $3(y+2) - 1(y-1) = 3y + 6 - y + 1 = 2y + 7$.
* Answer: $\frac{2y+7}{(y-1)(y+2)}$

15) $\frac{6}{1-2b} - \frac{b}{3+b}$
* Common denominator is $(1-2b)(3+b)$.
* Numerator: $6(3+b) - b(1-2b) = 18 + 6b - b + 2b^2 = 2b^2 + 5b + 18$.
* Answer: $\frac{2b^2+5b+18}{(1-2b)(3+b)}$

16) $\frac{2}{x^2} - \frac{1}{x(x-1)}$
* Common denominator is $x^2(x-1)$.
* First term needs $(x-1)$: $2(x-1) = 2x - 2$.
* Second term needs $x$: $1(x) = x$.
* Numerator: $(2x - 2) - x = x - 2$.
* Answer: $\frac{x-2}{x^2(x-1)}$

17) $7 - \frac{x-4}{4x(x-2)}$
* Common denominator is $4x(x-2)$.
* Numerator for 7: $7 \cdot 4x(x-2) = 28x(x-2) = 28x^2 - 56x$.
* Subtract the existing numerator: $(28x^2 - 56x) - (x - 4) = 28x^2 - 57x + 4$.
* Answer: $\frac{28x^2-57x+4}{4x(x-2)}$

18) $\frac{3y+1}{(y-2)^2} + \frac{y}{y-2}$
* Common denominator is $(y-2)^2$.
* Second term needs $(y-2)$: $y(y-2) = y^2 - 2y$.
* Numerator: $(3y + 1) + (y^2 - 2y) = y^2 + y + 1$.
* Answer: $\frac{y^2+y+1}{(y-2)^2}$

19) $\frac{w}{w^2-1} + \frac{w+1}{w-1}$
* Factorize first denominator: $w^2-1 = (w-1)(w+1)$.
* Common denominator is $(w-1)(w+1)$.
* Second term needs $(w+1)$: $(w+1)(w+1) = w^2 + 2w + 1$.
* Numerator: $w + (w^2 + 2w + 1) = w^2 + 3w + 1$.
* Answer: $\frac{w^2+3w+1}{(w-1)(w+1)}$ (or $\frac{w^2+3w+1}{w^2-1}$)

20) $\frac{1}{x^2+5x+4} + \frac{1}{x^2+7x+12}$
* Factorize denominators:
* $x^2+5x+4 = (x+4)(x+1)$
* $x^2+7x+12 = (x+4)(x+3)$
* Least Common Denominator: $(x+1)(x+3)(x+4)$.
* Adjust numerators:
* First term needs $(x+3)$: $1(x+3) = x+3$.
* Second term needs $(x+1)$: $1(x+1) = x+1$.
* Add numerators: $x + 3 + x + 1 = 2x + 4$.
* Factor numerator: $2(x+2)$.
* Answer: $\frac{2(x+2)}{(x+1)(x+3)(x+4)}$

21) $\frac{2}{2x^2+5x-3} - \frac{3}{3x^2+11x+6}$
* Factorize denominators:
* $2x^2+5x-3 = (2x-1)(x+3)$
* $3x^2+11x+6 = (3x+2)(x+3)$
* Least Common Denominator: $(2x-1)(3x+2)(x+3)$.
* Adjust numerators:
* First term needs $(3x+2)$: $2(3x+2) = 6x + 4$.
* Second term needs $(2x-1)$: $3(2x-1) = 6x - 3$.
* Subtract numerators: $(6x + 4) - (6x - 3) = 6x + 4 - 6x + 3 = 7$.
* Answer: $\frac{7}{(2x-1)(3x+2)(x+3)}$

Final Answer:
See the step-by-step solutions above for all questions in Section A and Section B.