IMC 2013 Q14
14. Which of the following has the greatest value?
- A \(\left(\dfrac{11}{7}\right)^3\)
- B \(\left(\dfrac{5}{3}\right)^3\)
- C \(\left(\dfrac{7}{4}\right)^3\)
- D \(\left(\dfrac{9}{5}\right)^3\)
- E \(\left(\dfrac{3}{2}\right)^3\)
Intermediate Mathematical Challenge 2013, Question 14. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D \(\left(\dfrac{9}{5}\right)^3\)
As all the fractions are raised to the power 3, the expression which has the largest value is that with the largest fraction in the brackets.
Each of these fractions is a little larger than \(1\tfrac{1}{2}\). Subtracting \(1\tfrac{1}{2}\) from each in turn, we get the fractions \(\tfrac{1}{14}\), \(\tfrac{1}{6}\), \(\tfrac{1}{4}\), \(\tfrac{3}{10}\), 0, the largest of which is \(\tfrac{3}{10}\) (because \(0 < \tfrac{1}{14} < \tfrac{1}{6} < \tfrac{1}{4} = \tfrac{2\frac{1}{2}}{10} < \tfrac{3}{10}\)). Hence \(\left(\tfrac{9}{5}\right)^3\) is the largest.
Solution © UK Mathematics Trust, from the IMC 2013 solutions. Longer versions with investigations: Solutions and investigations (PDF).