Algebraic Fractions

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Notes de leçon

Simplifying Algebraic Fractions

Factorise numerator and denominator fully.
Cancel any common factors (single term or bracket).
Cannot cancel a factor unless it is common to all terms in numerator/denominator.
E.g. \frac{6x}{x+1} cannot be simplified.
Always check if numerator factorises after simplification.

Adding & Subtracting Algebraic Fractions

Find the lowest common denominator (LCD).
LCD may be product of denominators, e.g. (x+2)(x+5).
If denominators share a factor, LCD is the LCM, e.g. \frac{1}{x} and \frac{1}{2x} have LCD 2x.
Rewrite each fraction over the LCD, multiplying numerator accordingly.
Combine into a single fraction and simplify numerator.
Check if numerator factorises for further cancellation.

Multiplying Algebraic Fractions

Factorise all numerators and denominators first.
Cancel any common factors between any numerator and any denominator.
Multiply numerators together and denominators together.
Check for further simplification.

Dividing Algebraic Fractions

Flip the second fraction (find its reciprocal) and change ÷ to ×.
Then follow the rules for multiplying algebraic fractions.
E.g. \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}.

Solving Equations with Algebraic Fractions

Method 1: Combine fractions into a single fraction, then cross-multiply and solve.
Method 2: Multiply every term by each denominator to clear fractions.
Use brackets when multiplying by algebraic expressions.
Solve the resulting polynomial equation (often quadratic).

Common Mistakes & Tips

Always factorise before cancelling or adding fractions.
When subtracting, be careful with negative signs across brackets.
Check for difference of two squares (e.g. x²-25).
Leave answer in factorised form to see if cancellation is possible.

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1.Simplify \frac{5}{3x} \times \frac{9x}{20}.
Easy
- A\frac{3}{4}
- B\frac{45x²}{60x}
- C\frac{3x}{4}
- D\frac{45}{60}
2.Simplify \frac{x²+5x}{x²-25}.
Easy
- A\frac{x}{x-5}
- B\frac{x+5}{x-5}
- C\frac{x}{x+5}
- D\frac{x+5}{x+5}
3.Simplify \frac{x²-25}{x²-2x-35}.
Easy
- A\frac{x-5}{x-7}
- B\frac{x+5}{x+7}
- C\frac{x-5}{x+7}
- D\frac{x+5}{x-7}
4.Simplify \frac{5p²-20}{p} \div \frac{2p²-32}{p}.
Medium
- A\frac{5}{2}
- B\frac{5(p²-4)}{2(p²-16)}
- C\frac{5(p-2)}{2(p-4)}
- D\frac{5(p+2)}{2(p+4)}
5.Write \frac{x-5}{3} + \frac{6}{x+2} as a single fraction in its simplest form.
Medium
- A\frac{x²-3x-10+18}{3(x+2)}
- B\frac{x²-3x-10}{3(x+2)}
- C\frac{x²-3x+8}{3(x+2)}
- D\frac{x²-3x-10+6}{3(x+2)}
6.Simplify \frac{p}{2q} \times \frac{4pq}{t}.
Medium
- A\frac{2p²}{t}
- B\frac{4p²q}{2qt}
- C\frac{2p²q}{t}
- D\frac{4p²}{t}
7.Simplify \frac{ab-b²}{a²-b²}.
Medium
- A\frac{b}{a+b}
- B\frac{b}{a-b}
- C\frac{a-b}{a+b}
- D\frac{a}{a+b}
8.Simplify \frac{x²-5x}{2x²-50}.
Hard
- A\frac{x}{2(x+5)}
- B\frac{x}{2(x-5)}
- C\frac{x-5}{2(x+5)}
- D\frac{x}{2x+10}

Plus de questions

9.Write as a single fraction: 2 - \frac{2x-1}{x+1}.
Hard
- A\frac{3}{x+1}
- B\frac{2x+2-2x+1}{x+1}
- C\frac{2x+2-2x-1}{x+1}
10.Simplify \frac{2x²-x-1}{2x²+x}.
Hard
- A\frac{2x+1}{2x}
- B\frac{2x-1}{2x}
- C\frac{x-1}{x}
- D\frac{2x-1}{x}
11.A car travels 200 km at an average speed of x km/h. The return journey of 200 km is at an average speed of (x+10) km/h. The difference between the times for the two journeys is given by which expression?
Hard
- A2000/(x(x+10))
- B2000/(x(x+10)) hours
- C200/(x(x+10))
- D200/(x+10) - 200/x
12.Simplify: \frac{x² - 25}{x² - 17x + 60}
Medium
- A\frac{x+5}{x-12}
- B\frac{x-5}{x-12}
- C\frac{x+5}{x+12}
- D\frac{x-5}{x+12}
13.Given f(x)=7x-4 and g(x)=\frac{2x}{x-3}, x \neq 3, find \frac{f(x)}{2} + g(x) as a single fraction in simplest form.
Hard
- A\frac{7x-4}{2} + \frac{2x}{x-3}
- B\frac{7x² - 33x + 12}{2(x-3)}
- C\frac{7x² - 33x + 12}{x-3}
- D\frac{7x² - 29x + 12}{2(x-3)}
14.Write \frac{x}{2} - \frac{2x+4}{x+1} as a single fraction in its simplest form.
Hard
- A\frac{x² - 3x - 8}{2(x+1)}
- B\frac{-x² - 3x - 8}{2(x+1)}
- C\frac{x² - 3x + 8}{2(x+1)}
- D\frac{x² - 5x - 8}{2(x+1)}
15.Solve \frac{1}{x} - \frac{2}{x+1} = 3. Give your answers correct to 2 decimal places.
Hard
- Ax = 0.43 or x = -0.77
- Bx = 0.43 or x = 0.77
- Cx = -0.43 or x = 0.77
- Dx = -0.43 or x = -0.77
16.Given f(x) = \frac{3}{x+2}, x \neq -2, and g(x) = 8x-5, write g(x) - f(x) as a single fraction in simplest form.
Hard
- A\frac{8x²+11x-7}{x+2}
- B\frac{8x²+11x-13}{x+2}
- C\frac{8x²-5x-3}{x+2}
- D\frac{8x²+11x-7}{x-2}
17.Simplify \frac{2x² - x - 1}{2x² + x}.
Hard
- A\frac{2x+1}{x}
- B\frac{x-1}{x}
- C\frac{2x-1}{x}
- D\frac{2x+1}{2x}
18.Simplify \frac{3y² + 18y}{4y² - 144}.
Medium
- A\frac{3y}{4(y-6)}
- B\frac{3y}{4(y+6)}
- C\frac{y+6}{4y}
19.Simplify \frac{2x² + x - 15}{ax + 3a - 2bx - 6b}.
Hard
- A\frac{2x-5}{a-2b}
- B\frac{2x+5}{a-2b}
- C\frac{2x-5}{a+2b}
- D\frac{2x+5}{a+2b}
20.Given f(x)=8-3x and g(x)=\frac{10}{x+1}, x \neq -1, write f(x) + g(x) as a single fraction in simplest form.
Hard
- A\frac{-3x²+5x+18}{x+1}
- B\frac{-3x²+5x-2}{x+1}
- C\frac{8-3x+10}{x+1}
- D\frac{-3x²+11x+18}{x+1}
21.Show that \frac{1}{6x²+7x-5} \div \frac{1}{4x²-1} simplifies to \frac{ax+b}{cx+d} where a, b, c, d are integers. Find a, b, c, d.
Hard
- Aa=2, b=1, c=3, d=5
- Ba=2, b=-1, c=3, d=5
- Ca=2, b=1, c=3, d=-5
- Da=2, b=-1, c=3, d=-5
22.Write as a single fraction in its simplest form: \frac{1}{x+2} - \frac{2}{3x-1}.
Medium
- A\frac{x-5}{(x+2)(3x-1)}
- B\frac{-x-5}{(x+2)(3x-1)}
- C\frac{x+5}{(x+2)(3x-1)}
23.Simplify \frac{ab - b²}{a² - b²}.
Medium
- A\frac{b}{a+b}
- B\frac{b}{a-b}
- C\frac{a}{a+b}
- D\frac{a-b}{a+b}
24.Write as a single fraction in its simplest form: \frac{1}{x} - \frac{1}{x+1}.
Medium
- A\frac{1}{x(x+1)}
- B\frac{2x+1}{x(x+1)}
- C\frac{-1}{x(x+1)}
25.Simplify \frac{3+x}{9-x²}.
Medium
- A\frac{1}{3-x}
- B\frac{1}{3+x}
- C\frac{3+x}{9-x²}
- D\frac{1}{x-3}
26.Write as a single fraction in its simplest form: \frac{1}{y-1} - \frac{1}{y}.
Medium
- A\frac{1}{y(y-1)}
- B\frac{-1}{y(y-1)}
- C\frac{2y-1}{y(y-1)}
27.What is the first step in simplifying an algebraic fraction?
Easy
- AFactorise the numerator and denominator fully
- BCancel common factors
- CMultiply numerator and denominator by the same expression
- DAdd the numerator and denominator
28.Simplify \frac{ab - b²}{a² - b²}.
Easy
- A\frac{b}{a+b}
- B\frac{b}{a-b}
- C\frac{a-b}{a+b}
- D\frac{a}{a+b}
29.Write as a single fraction: \frac{1}{x} - \frac{1}{x+1}.
Easy
- A\frac{1}{x(x+1)}
- B\frac{2x+1}{x(x+1)}
- C\frac{1}{x²+x}
30.Simplify \frac{3+x}{9-x²}.
Easy
- A\frac{1}{3-x}
- B\frac{1}{3+x}
- C\frac{3+x}{(3-x)(3+x)}
- D-\frac{1}{x-3}
31.Write as a single fraction: \frac{1}{y-1} - \frac{1}{y}.
Easy
- A\frac{1}{y(y-1)}
- B\frac{2y-1}{y(y-1)}
- C\frac{1}{y²-y}
32.Write as a single fraction: \frac{x}{2} - \frac{2x+4}{x+1}.
Medium
- A\frac{x²-3x-8}{2(x+1)}
- B\frac{x²-3x-8}{2x+2}
- C\frac{x²-3x+8}{2(x+1)}
33.Simplify \frac{3y²+18y}{4y²-144}.
Medium
- A\frac{3y}{4(y-6)}
- B\frac{3y}{4(y+6)}
- C\frac{3y}{4y-24}
34.Simplify \frac{x²-25}{x²-17x+60}.
Hard
- A\frac{x+5}{x-12}
- B\frac{x-5}{x-12}
- C\frac{x+5}{x+12}
35.Given f(x)=7x-4 and g(x)=\frac{2x}{x-3}, find \frac{f(x)}{2}+g(x) as a single fraction.
Hard
- A\frac{7x²-17x+12}{2(x-3)}
- B\frac{7x²-17x-12}{2(x-3)}
- C\frac{7x²-17x+12}{x-3}
36.Solve \frac{1}{x} - \frac{2}{x+1} = 3. Give your answers correct to 2 decimal places.
Hard
- Ax = 0.39 or x = -0.89
- Bx = 0.39 or x = 0.89
- Cx = -0.39 or x = 0.89
37.Given f(x)=8-3x and g(x)=\frac{10}{x+1}, write f(x)+g(x) as a single fraction.
Hard
- A\frac{-3x²+5x+18}{x+1}
- B\frac{-3x²+5x-18}{x+1}
- C\frac{3x²+5x+18}{x+1}
38.Simplify \frac{x² - 5x}{2x² - 50}.
Medium
- A\frac{x}{2(x+5)}
- B\frac{x}{2(x-5)}
- C\frac{x-5}{2(x+5)}
- D\frac{x}{x+5}

Notes de révision

## Algebraic Fractions - Night Before Exam Summary ### Simplifying - **Factorise** numerator and denominator fully. - **Cancel** common factors (single term or bracket). - Example: \(\frac{4x+6}{2x^2-7x-15} = \frac{2(2x+3)}{(2x+3)(x-5)} = \frac{2}{x-5}\) ### Adding & Subtracting 1. Find **lowest common denominator (LCD)**. 2. Write each fraction over LCD. 3. Combine numerators, simplify. 4. Check if final fraction can be simplified. - Example: \(\frac{x}{x+4} - \frac{3}{x-1} = \frac{x(x-1)-3(x+4)}{(x+4)(x-1)} = \frac{x^2-4x-12}{(x+4)(x-1)} = \frac{(x+2)(x-6)}{(x+4)(x-1)}\) ### Multiplying 1. Factorise all numerators and denominators. 2. Cancel common factors across fractions. 3. Multiply tops together, bottoms together. - Example: \(\frac{x}{3x+6} \times \frac{2x+4}{x+7} = \frac{x}{3(x+2)} \times \frac{2(x+2)}{x+7} = \frac{2x}{3(x+7)}\) ### Dividing - Flip the second fraction (reciprocal) and multiply. - Then follow multiplication steps. - Example: \(\frac{x+3}{x-4} \div \frac{2x+6}{x^2-16} = \frac{x+3}{x-4} \times \frac{(x-4)(x+4)}{2(x+3)} = \frac{x+4}{2}\) ### Solving Equations **Method 1:** Combine fractions into a single fraction, then cross-multiply and solve. **Method 2:** Multiply every term by all denominators to clear fractions, then solve. - Example: \(\frac{4}{x-3} + \frac{5}{x+1} = 5\) → multiply by \((x-3)(x+1)\) → \(4(x+1)+5(x-3)=5(x-3)(x+1)\) → solve quadratic. ### Key Tips - Always factorise first. - Watch for difference of two squares: \(a^2-b^2 = (a-b)(a+b)\). - Be careful with negative signs when subtracting. - Check if final answer can be simplified further.

Algebraic fraction: A fraction with an algebraic expression in the numerator and/or denominator.
Lowest common denominator (LCD): The smallest expression that is a multiple of all denominators in a set of fractions.
Reciprocal: The fraction obtained by swapping the numerator and denominator; used for division.
Factorising: Writing an expression as a product of its factors (e.g., common factor, quadratic factorisation, difference of two squares).
Cross-multiplying: Multiplying both sides of an equation by the denominators to eliminate fractions.

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