Question

1. $\frac{a + 4}{8a} + \frac{5}{24}$
2. $\frac{3x - 1}{x - 1} - \frac{x - 1}{x - 3} + \frac{x + 1}{x^2 - 4x + 3}$

topic 3: complex fractions

1. $\frac{\frac{12m^3}{m^2 + 14m + 45}}{\frac{3m^3 - 6m^2}{m^2 + 7m - 18}}$
2. $\frac{\frac{b}{2} - \frac{32}{b}}{1 + \frac{8}{b}}$

Explanation:

Step1: Find common denominator (Q12)

Common denominator is $24a$.
$$\frac{a+4}{8a} + \frac{5}{24} = \frac{3(a+4)}{24a} + \frac{5a}{24a}$$

Step2: Combine numerators (Q12)

Expand and simplify the numerator.
$$\frac{3a+12 + 5a}{24a} = \frac{8a+12}{24a}$$

Step3: Simplify fraction (Q12)

Factor and reduce terms.
$$\frac{4(2a+3)}{24a} = \frac{2a+3}{6a}$$

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Step1: Factor denominator (Q14)

Factor $x^2-4x+3=(x-1)(x-3)$.
$$\frac{3x-1}{x-1} - \frac{x-1}{x-3} + \frac{x+1}{(x-1)(x-3)}$$

Step2: Get common denominator (Q14)

Common denominator is $(x-1)(x-3)$.
$$\frac{(3x-1)(x-3)}{(x-1)(x-3)} - \frac{(x-1)(x-1)}{(x-1)(x-3)} + \frac{x+1}{(x-1)(x-3)}$$

Step3: Expand numerators (Q14)

Calculate each expanded numerator.
$$\frac{3x^2-10x+3 - (x^2-2x+1) + x+1}{(x-1)(x-3)}$$

Step4: Simplify numerator (Q14)

Combine like terms.
$$\frac{3x^2-10x+3 -x^2+2x-1+x+1}{(x-1)(x-3)} = \frac{2x^2-7x+3}{(x-1)(x-3)}$$

Step5: Factor and reduce (Q14)

Factor numerator: $2x^2-7x+3=(2x-1)(x-3)$.
$$\frac{(2x-1)(x-3)}{(x-1)(x-3)} = \frac{2x-1}{x-1}$$

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Step1: Rewrite as multiplication (Q15)

Dividing fractions = multiply by reciprocal.
$$\frac{12m^3}{m^2+14m+45} \times \frac{m^2+7m-18}{3m^3-6m^2}$$

Step2: Factor all polynomials (Q15)

Factor quadratics and monomials.
$$\frac{12m^3}{(m+5)(m+9)} \times \frac{(m+9)(m-2)}{3m^2(m-2)}$$

Step3: Cancel common terms (Q15)

Cancel $m^2$, $(m+9)$, $(m-2)$, 3/12=4.
$$\frac{4m}{m+5}$$

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Step1: Multiply by common denominator (Q17)

Multiply numerator and denominator by $2b$.
$$\frac{\frac{b}{2}-\frac{32}{b}}{1+\frac{8}{b}} = \frac{b^2-64}{2b+16}$$

Step2: Factor terms (Q17)

Factor numerator as difference of squares, denominator as $2(b+8)$.
$$\frac{(b-8)(b+8)}{2(b+8)}$$

Step3: Cancel common terms (Q17)

Cancel $(b+8)$.
$$\frac{b-8}{2}$$

Answer:

1. $\boldsymbol{\frac{2a+3}{6a}}$
2. $\boldsymbol{\frac{2x-1}{x-1}}$
3. $\boldsymbol{\frac{4m}{m+5}}$
4. $\boldsymbol{\frac{b-8}{2}}$