IMC 2019 Q16
16. Two numbers \(x\) and \(y\) have a product which is equal to their sum.
Which of these expressions gives \(x\) in terms of \(y\)?
- A \(\dfrac{y}{y - 1}\)
- B \(\dfrac{y}{y + 1}\)
- C \(\dfrac{y + 1}{y}\)
- D \(\dfrac{y - 1}{y}\)
- E \(\dfrac{y^2}{y + 1}\)
Intermediate Mathematical Challenge 2019, Question 16. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A \(\dfrac{y}{y - 1}\)
We are given that \(xy = x + y\). So \(xy - x = y\). Therefore \(x(y - 1) = y\) and hence \(x = \dfrac{y}{y - 1}\).
(Note that division by \(y - 1\) is allowed because if \(y = 1\), then we have \(x = x + 1\) which is impossible.)
Solution © UK Mathematics Trust, from the IMC 2019 solutions. Longer versions with investigations: Solutions and investigations (PDF).