IMC 2019 Q17
17. Which of these is equal to \(0.\dot{8} + 0.0\dot{7}\)?
- A \(0.8\dot{7}\)
- B \(0.88\)
- C \(0.9\dot{5}\)
- D \(0.9\dot{6}\)
- E \(0.9\dot{8}\)
Intermediate Mathematical Challenge 2019, Question 17. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D \(0.9\dot{6}\)
It may be shown that \(0.\dot{1} = \dfrac{1}{9}\). So \(0.\dot{8} = \dfrac{8}{9}\) and \(0.0\dot{7} = \dfrac{7}{90}\).
Therefore \(0.\dot{8} + 0.0\dot{7} = \dfrac{8}{9} + \dfrac{7}{90} = \dfrac{80}{90} + \dfrac{7}{90} = \dfrac{87}{90} = \dfrac{29}{30}\). Now \(\dfrac{29}{3} = 9 + \dfrac{2}{3} = 9 + 0.\dot{6} = 9.\dot{6}\).
So \(\dfrac{29}{30} = 0.9\dot{6}\). Hence \(0.\dot{8} + 0.0\dot{7} = 0.9\dot{6}\).
(It is left as an exercise to the reader to prove the assertion in the first sentence.)
Solution © UK Mathematics Trust, from the IMC 2019 solutions. Longer versions with investigations: Solutions and investigations (PDF).