IMC 2019 Q18
18. Two numbers \(x\) and \(y\) are such that \(x + y = \dfrac{2}{3}\) and \(\dfrac{x}{y} = \dfrac{2}{3}\).
What is the value of \(x - y\)?
- A \(-\dfrac{2}{3}\)
- B \(-\dfrac{2}{15}\)
- C \(\dfrac{2}{25}\)
- D \(\dfrac{2}{5}\)
- E \(\dfrac{2}{3}\)
Intermediate Mathematical Challenge 2019, Question 18. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B \(-\dfrac{2}{15}\)
From the equation \(\dfrac{x}{y} = \dfrac{2}{3}\) we deduce that \(3x = 2y\). We are given that \(x + y = \dfrac{2}{3}\), so \(3x + 3y = 2\).
Substituting for \(3x\) gives \(2y + 3y = 2\), so \(y = \dfrac{2}{5}\).
Therefore \(x - y = x + y - 2y = \dfrac{2}{3} - \dfrac{4}{5} = \dfrac{10}{15} - \dfrac{12}{15} = -\dfrac{2}{15}\).
Solution © UK Mathematics Trust, from the IMC 2019 solutions. Longer versions with investigations: Solutions and investigations (PDF).