IMC 2013 Q19
19. Which of the expressions below is equivalent to \((x \div (y \div z)) \div ((x \div y) \div z)\) ?
- A 1
- B \(\dfrac{1}{xyz}\)
- C \(x^2\)
- D \(y^2\)
- E \(z^2\)
Intermediate Mathematical Challenge 2013, Question 19. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E \(z^2\)
\((x \div (y \div z)) \div ((x \div y) \div z) = \left(x \div \tfrac{y}{z}\right) \div \left(\left(\tfrac{x}{y}\right) \div z\right) = \left(x \times \tfrac{z}{y}\right) \div \left(\tfrac{x}{y} \times \tfrac{1}{z}\right)\)
\(= \tfrac{xz}{y} \div \tfrac{x}{yz} = \tfrac{xz}{y} \times \tfrac{yz}{x} = z^2.\)
\(= \tfrac{xz}{y} \div \tfrac{x}{yz} = \tfrac{xz}{y} \times \tfrac{yz}{x} = z^2.\)
Solution © UK Mathematics Trust, from the IMC 2013 solutions. Longer versions with investigations: Solutions and investigations (PDF).