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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 11 · 11.11 Algebraic Fractions

Problem-solving Pack

Name: _________________________________
Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12
Addition. Simplify

3x+2x+1\frac{3}{x} + \frac{2}{x + 1}


(a) Find a common denominator.
(b) Combine into a single fraction.
(c) State any restrictions on xx.

Working space

2Problem 2 of 12
Subtraction. Simplify

4x−1−3x+2\frac{4}{x - 1} - \frac{3}{x + 2}


(a) Combine as a single fraction.
(b) State restrictions on xx.

Working space

3Problem 3 of 12
Multiplication & division. Simplify

(a) x+1x⋅2xx+1\dfrac{x + 1}{x} \cdot \dfrac{2x}{x + 1}
(b) x2x+1÷xx+1\dfrac{x^2}{x + 1} \div \dfrac{x}{x + 1}
(c) x2−4x−2⋅1x+2\dfrac{x^2 - 4}{x - 2} \cdot \dfrac{1}{x + 2}

Working space

4Problem 4 of 12
Simplify with factoring. Simplify fully

x2−9x2−x−6\frac{x^2 - 9}{x^2 - x - 6}


State any restrictions on xx.

Working space

5Problem 5 of 12
Rational equation. Solve

2x−1+3x+2=1\frac{2}{x - 1} + \frac{3}{x + 2} = 1


(a) Multiply through by (x−1)(x+2)(x - 1)(x + 2).
(b) Solve the resulting quadratic.
(c) Check each candidate against the original equation.

Working space

6Problem 6 of 12
Complex fraction. Simplify fully

1x−1x+11x(x+1)\frac{\dfrac{1}{x} - \dfrac{1}{x + 1}}{\dfrac{1}{x(x + 1)}}

Working space

7Problem 7 of 12
Partial fractions [EXT]. Find constants AA and BB such that

5x+1(x−1)(x+3)=Ax−1+Bx+3\frac{5x + 1}{(x - 1)(x + 3)} = \frac{A}{x - 1} + \frac{B}{x + 3}

Working space

8Problem 8 of 12
Speed–time. A car drives the first 60 km of a trip at vv km/h, then the next 40 km at v−10v - 10 km/h. The total time is 2 hours.

(a) Write an equation in vv.
(b) Multiply through to clear fractions, and use the quadratic formula.
(c) Give vv to 3 s.f.

Working space

9Problem 9 of 12
Pipe rate. Pipe A fills a swimming pool in 8 hours; pipe B in 12 hours.

(a) What fraction does each pipe fill in 1 hour?
(b) Write an equation for TT, the time for both pipes together.
(c) Solve.

Working space

10Problem 10 of 12
Identity. Find constants AA and kk such that

3x−7x−2=A+kx−2\frac{3x - 7}{x - 2} = A + \frac{k}{x - 2}


for all x≠2x \neq 2.

Working space

11Problem 11 of 12
Restrictions. Simplify fully and state restrictions on xx.

(a) x2−1x2+2x+1\dfrac{x^2 - 1}{x^2 + 2x + 1}
(b) x2−4x+4x2−4\dfrac{x^2 - 4x + 4}{x^2 - 4}
(c) x3−xx2−1\dfrac{x^3 - x}{x^2 - 1}

Working space

12Problem 12 of 12
Rational equation modelling. A photo print costs aa francs each, and developing a roll of film costs bb francs. The average cost per print on a roll with nn prints is na+bn\frac{na + b}{n}.

(a) Write the average cost as a+bna + \frac{b}{n}.
(b) If a=0.40a = 0.40 and b=6b = 6, find the number of prints needed for the average cost to be at most CHF 0.70.
(c) Comment on what happens to the average cost as n→∞n \to \infty.

Working space